Stock Price Dynamics - Stochastic Interest Rate Model 115

B. Stochastic Interest Rate Model 115

B.4. Stock Price Dynamics

For the dynamics of stock price, we introduce a two-dimensional setup, because in most asset pricing models we have that there are underlying processes which affect each other, such as price processes for a different asset, i.e. a bond’s price dynamics.

Adding the time-setting adds a second standard Brownian motion. In our system, we consider two assets, a bond and a stock whereB_tand S_t is the notation for each, respectively. Let:

dB_t

B_t =µ_B_tdt+σ_B,1_tdz_1,t+σ_B,2_tdz_2,t dS_t

S_t =µStdt+σS,1tdz1,t+σS,2tdz2,t

where z₁ = (z_1t) and z₂ = (z_2t) are independent Brownian motions. And µ_S_t = r+ψσ_S, whereψ is the Sharpe Ratio. Firstly, we want to investigate the correlation between the two stochastic processes, so we derive the covariance function of these two as follows:

Cov(X, Y) = E(XY)−E(X)E(Y)

Substitute in our processes:

Cov_t(dB_t, dS_t) = E[dB_t·dS_t]−E[dB_t]·E[dS_t]

=Eh

(µ_B_tdt+σ_B_1tdz_z1+σ_B_2tdz_2t)·(µ_S_tdt+σ_S_1tdz_z1+σ_S_2tdz_2t)i

−E[µ_B_tdt+σ_B_1tdz_z1 +σ_B_2tdz_2t]·E[µ_S_tdt+σ_S_1tdz_1z +σ_S_2tdz_2t] After multiplying

Cov_t[dB_t, dS_t] =Eh

µ_Bµ_Sdt²+µ_Bdtσ_S_1tdz_1t+µ_Bdtσ_B_2tdz_2t +σB1tdz1tµS1tdt+σB1tσS1tdz_1t² +σB1tdz1tσS1tdz2t

+σ_B_2tdz_2tµ_S_tdt+σ_B_2tdz_2tσ_S_1tdz_1t+σ_B_2tσ_S_2tdz_2t²i

−µ_B_tµ_Sdt²

Imposing the rules of stochastic calculus, where we remember the rules of stochastic calculus:

dt² = 0 dz·dt = 0 and (dzt)² =dt Using the rules below

Cov_t[dB_t, dS_t] =Eh

µ_Bµ_Sdt²

| {z }

+µ_Bdtσ_S_1tdz_1t

| {z }

+µ_Bdtσ_B_2tdz_2t

| {z }

+σ_B_1tdz_1tµ_S_1tdt

| {z }

+σ_B_1tσ_S_1tdz_1t² +σ_B_1tdz_1tσ_S_1tdz_2t

| {z }

+σ_B_2tdz_2tµ_S_tdt

| {z }

+σ_B_2tdz_2tσ_S_1tdz_1t

| {z }

+σ_B_2tσ_S_2tdz_2t²i

−µ_B_tµ_Sdt²

| {z }

Now we can further reduce the expression to:

Cov_t[dB_t, dS_t] =E[σ_B_1tσ_S_1tdz_1t² +σ_B_2tσ_S_2tdz²_2t] Using the rule(dz)² =dt again, and taking expectations, we get:

Cov_t[dB_t, dS_t] =σ_B_1tσ_S_1tdt+σ_B_2tσ_S_2tdt.

So the covariance function for the two processes is:

Cov[dB , dS] = (σ σ +σ σ )dt

In order to find the correlation function, we must first find the variances of the two processes,dB_t and dS_t. Applying the variance formula to each process:

V ar[X] =E[(X−E[X])²] =E[X²]−(E[X])² FordB_t:

V ar_t[dB_t] =E[dB_t²]−(E[dB_t])²

=E[µ²_B_tdt²+σ_B²

1tdz_1,t² +σ_B²

2tdz_2,t² ]−[E[µ_B_tdt+σ_B,1_tdz_1,t+σ_B,2_tdz_2,t]]² Again taking expectations and using the rules of stochastic calculus:

dt² = 0 dz·dt = 0 and (dzt)² =dt Then we get:

V ar_t[dB_t] =E[µ²_B_tdt²

| {z }

dt²=0

+σ_B,1² _tdt+σ²_B,2_tdt]−[E[µ_B_tdt+σ_B,1_tdz_1,t

| {z }

+σ_B,2_tdz_2,t

| {z }

]]²

=σ_B,1² _tdt+σ²_B,2_tdt−µ²_B_tdt²

| {z }

= (σ²_B_2t +σ_B²_2t)dt

This is the variance for for the bond price process. Now we want to carry out the same procedure for the variance of the stock price:

V ar_t[dS_t] =E[µ²_S_tdt²+σ_S²_1tdz_1t² +σ_S²_2tdz²_2t]−[E[µ_S_tdt+σ_S_1tdz_1t+σ_S_2tdz_2t]]²

=E[µ²_S_tdt²

| {z }

+σ_S²_1tdt+σ_S²_2tdt]−[E[µ_S_tdt+σ_S_1tdz_1t

| {z }

+σ_S_2tdz_2t

| {z }

]]²

=σ²_S_1tdt+σ²_S_2tdt−µ²_S_tdt²

| {z }

= (σ²_S_1t+σ_S²_2t)dt

Now that we have the variances for each processes, and the covariance between them, we are now able to find the correlation function of the two processes. The

correlation function is given as:

Corr_t[dB_t, dS_t] = Cov_t[dB_t, dS_t] pV ar_t[dB_t]·V ar_t[dS_t]

= (σ_B_1tσ_S_1t +σ_B_2tσ_S_2t)dt q

[(σ_B²_2t+σ_B²_2t)dt]·[(σ²_S_1t +σ_S²_2t)dt]

= σB1tσS1t +σB2tσS2t

q (σ²_B

2t+σ_B²

2t)·(σ_S²

1t+σ_S²

2t)

However, from this derivation of the correlation of the two stochastic processes, we can see that when specifying the two-dimensional process [dBt, dSt], and character-izing the first-order local and second-order local moments, which are:

• The first-order local moments are µ_B_t and µ_S_t.

• The second-order local moments are the volatility coefficients (shock) coeffi-cients σ_B_1t,σ_B_2t, σ_S_1t, and σ_S_2t, which defines the two instantaneous variances and the instantaneous correlation coefficient.

From the above derivation, it was evident that the four volatility (shocks) coefficient would give rise to the same variances and the same correlation. This implies, that we have one degree of freedom extra by fixing the coefficients, which mean that we choose to fix one of the volatility (shock) coefficients for the process of the bond price, specific σ_B_2t = 0, since we want to see how the stock price dynamics are affected the bond price dynamics. So we can simplify our two processes as such:

dB_t

B_t =µ_B_tdt+σ_B_tdz_1t dS_t

S_t µ_S_tdt+σ_S_t(ρ_tdz_1t+p

1−ρ²dz_2t) whereρ_t is the correlation between the market returns of stock and bond. Now we can again define the new instantaneous variances and the instantaneous covariance of the newly defined stochastic processes. Now using prior definition of µ_S_t = r+ ψσ_S, we can obtain the same stock price dynamics as in Munk (2013), which are represented by the following process:

dS_t

S_t = (r+ψσS)dt+σSt(ρtdz1t+p

1−ρ²dz2t)

This shows the dynamics of the stock price, where the drift term is defined as a mean return with addition of the Sharpe ratio with risk adjustment term, and the

Further, in this appendix we want show the relationship between the processes.

Firstly, we consider the instantaneous variance ofdB_t is:

V art[dBt] =E[(µBtdt+σBtdz1t)²]−[E[µBtdt+σBtdz1t]]²

=E[µ²_B_tdt²

| {z }

+σ²_B_tdz_1t²]−[E[µ_B_tdt+σ_B_tdz_1t

| {z }

]]²

=σ_B²_tdt−µ²_B_tdt²

| {z }

=σ_B²_tdt

And the same is done for the instantaneous variance of dS_t: V art[dSt] =E[[µStdt+σSt(ρtdz1t+p

1−ρ²dz2t)]²]

−[E[µ_S_tdt+σ_S_t(ρ_tdz_1t+p

1−ρ²dz_2t)]]²

=E[µ²_S

tdt²

| {z }

+σ_S²

t(ρ_tdz_1t+p

1−ρ²_tdz_2t)²

−[E[µ_S_tdt+σ_S_t(ρ_tdz_1t

| {z }

1−ρ²dz_2t)

| {z }

]]²

=σ_S²_t(ρ_tdz_1t+p

1−ρ²_tdz_2t)²−µ²_S_tdt²

| {z }

=V ar_th

σ_S²_t(ρ_tdz_1t+p

1−ρ²_tdz_2t)²i

=σ_S²_tV ar_th

(ρ_tdz_1t+p

1−ρ²_tdz_2t)²i

=σ_S²_th

(ρ²_tdz_1t² + (1−ρ²_t)dz²_2ti

=σ_S²_t h

(ρ²_tdt+ (1−ρ²_t)dt i

=σ_S²_tdt

Next, we can also find the instantaneous covariance under this new specification:

Cov_t[dB_t, dS_t] =E[dB_t·dS_t]−E[dB_t]·E[ES_t]

=E[(mu_B_tdt+σ_B_tdz_1t)(µ_S_tdt+σ_S_t(ρ_tdz_1t+p

1−ρ²dz_2t))]

−E[mu_B_tdt+σ_B_tdz_1t

| {z }

]·E[µ_S_tdt+σ_S_t(ρ_tdz_1t

| {z }

1−ρ²dz_2t

|{z}=0

)]

=Eh

µ_B_tµ_S_tdt²

| {z }

+µ_B_tσ_S_t(ρ_tdz_1tdt

| {z }

1−ρ²dz_2tdt

| {z }

) +σ_B_tµ_S_tdz_1tdt

| {z }

+σ_B_tσ_S_t(ρ_tdz_1t² +p

1−ρ²dz_1tdz_2t)i

−µ_B_tµ_S_tdt²

| {z }

=σ_B_tσ_S_t(ρ_tdz²_1t+p

1−ρ²dz_1tdz_2t)

=Covt

σBtσSt(ρtdz_1t² +p

1−ρ²dz1tdz2t) i

=σ_B_tσ_S_tCov_th

z_1t, ρ_tdz_1t² +p

1−ρ²dz_1tdz_2t)i

This is the instantaneous covariance between the two stochastic processes, dB_t and dS_t. However, we can transform the embedded covariance, which are inside the instantaneous covariance to an embedded correlation in the following way:

Cov_t[dB_t, dS_t] =

σBtσStCovt

z1t, ρtdz_1t² +p

1−ρ²dz1tdz2t) i q

σ_B²_tdt·σ_S²_tdt

=σ_B_tσ_S_tρ_tdt

From this expression of the instantaneous covariance, we can see if σ_B_t and σ_S_t are both positive or both negative, this implies that the instantaneous correlation is ρ_t between the two processesdB_t and dS_t, while if they have opposite signs, then the correlation will be−ρ_t.

Appendix C

Justification for Numerical Integration

In Section 7.4.2 it is chosen to use numerical integration to estimate the loss from choosing a suboptimal allocation. The numerical integration of A⁰₀ is done by the Trapez-formula

Z t⁰ t

A⁰₀(s)ds= (t⁰−t)A⁰₀(t) +A⁰₀(t⁰)

2 ,

where t⁰ > t. To have a high precision of the estimation of the integral, the size of the subintervals is set to t⁰ −t = 0.01. To verify the precision, the two functions A₁(τ) andA₂(τ)have been estimated by the Trapez-formula and then compared to the correct, closed-form solution. Estimation of the value at time t = 3 has been done for several sizes of the subinterval. As seen from Table C.1 the marginal effect from a smaller size of subintervals i decreasing. At t = 3 the formula estimates 98.0511% of the actual value of A₂(τ). For t= 30 the estimation finds 98.0506% of the actual value ofA₂(τ). Even for a long time horizon, the precision is high, and it does not even seem to decrease very much as time increases. ForA₁(τ)for precision at 30 years is 99.5066%.

Subunit 1 0.75 0.5 0.25 0.15 0.1 0.05 0.01

Estimated 31.6% 38.1% 47.9% 65.3% 76.2% 83.0% 90.8% 98.1%

Table C.1: The table illustrates for 8 different subintervals,t⁰−t, how much of the correct value is estimated by use of numerical integration fort= 3.

Appendix D R-code

All R-programs and Excel files used for this thesis can be provided upon request.

Below is the full R-code provided. We apologise in advance for the length.

Listing D.1: R-code

########################

# PROGRAM 1 TABLE 2.1#

########################

#Load packages r e q u i r e(s t a r g a z e r)

#Set working d i r e c t o r y getwd( )

setwd("C:/Users/d i t l e/Desktop/")

#Load Excel f i l e AUM_DK_WW AUM.df < data.frame(AUM_DK_WW)

#Use s t a r g a z e r

s t a r g a z e r(AUM.df, t y p e = " l a t e x ", summary = F A L S E)

##################################################

# PROGRAM 2 MEAN VARIANCE ANALYSIS : FIGURE 3 . 1 #

##################################################

#CONTENT:

#Plot : E f f i c i e n t F r o n t i e r o f only r i s k y a s s e t

#Plot : E f f i c i e n t F r o n t i e r o f a l l a s s e t s

#Load Excel Mean_Variance

mean.v a r i a n c e.df < data.frame(mean_v a r i a n c e) h e a d(mean.v a r i a n c e.df)

t a i l(mean.v a r i a n c e.df)

g e o m_l i n e(aes(x = SD_2 , y = Exp_ret_2 , c o l = " red ") , s i z e = 1) + g e o m_p o i n t(aes(x = SD_tan, y = Exp_ret_tan) , s i z e = 3) +

g e o m_p o i n t(aes(x = SD_GMV, y = Exp_ret_GMV) , s i z e = 3) + g e o m_p o i n t(aes(x = A1_SD, y = A1_ret) , s i z e = 3) + g e o m_p o i n t(aes(x = A3_SD, y = A3_ret) , s i z e = 3) + s c a l e_x_c o n t i n u o u s(" Standard Deviation (%)") + s c a l e_y_c o n t i n u o u s(" Expected Return (%)") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c( " E f f i c i e n t F r o n t i e r o f r i s k y ← -a s s e t s ", " E f f i c i e n t F r o n t i e r o f a l l a s s e t s ") ,

v a l u e s = c(" blue ", " red ") ) + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

legend.t i t l e = e l e m e n t_b l a n k( ) , a x i s.t e x t.x=e l e m e n t_t e x t(s i z e=9) , a x i s.t i t l e.x=e l e m e n t_t e x t(s i z e=9) , a x i s.t e x t.y=e l e m e n t_t e x t(s i z e=9) , a x i s.t i t l e.y=e l e m e n t_t e x t(s i z e=9) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=8, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 0 . 4 8 , 0 . 8 5 ) , legend.key.s i z e = u n i t( 0 . 3 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 11) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=a l p h a(' white ', 0 . 4 ) ) )

mv.p l o t + g e o m_t e x t(aes(x = SD_tan, y = Exp_ret_tan, l a b e l = "Tangency P o r t f o l i o "← -) , h j u s t=1.1 , v j u s t=0, s i z e = 3) +

g e o m_t e x t(aes(x = SD_GMV, y = Exp_ret_GMV, l a b e l = "GMV P o r t f o l i o ") , h j u s t←

-= 0 . 1 , v j u s t=0, s i z e = 3)

##########################################

# PROGRAM 3 INTEREST RATES: FIGURE 5 . 1 #

##########################################

#'===========================================================================

#' Varies i n t e r e s t r a t e s f o r the y e a r s 1986 to 2016

#'===========================================================================

#i n s t a l l . packages (" t s e r i e s ") ; i n s t a l l . packages (" quantmod ") ; i n s t a l l . packages ("← -gg p lot 2 ")

#' Neded packages

# '

l i b r a r y(t s e r i e s) ; l i b r a r y(q u a n t m o d) ;l i b r a r y(g g p l o t 2) r e q u i r e(t s e r i e s) ; r e q u i r e(q u a n t m o d) ;r e q u i r e(g g p l o t 2)

# '

# ' 1 0 Year Treasury Constant Maturity Rate

# '

g e t S y m b o l s( c("DGS10") , src="FRED") a d j y e a r 1 0 < D G S 1 0[ 6 2 9 7 :nrow(D G S 1 0) , ]

# '

#'3 month Treasury i n t e r e s t r a t e

# '

g e t S y m b o l s( c("DTB3") , src="FRED") a d j m o n t h 3 < D T B 3[ 8 3 8 3 :nrow(D T B 3) , ]

# '

#'30 y e a r s Treasury i n t e r e s t r a t e

# '

g e t S y m b o l s( c("DGS30") , src="FRED") a d j y e a r 3 0 < D G S 3 0[ 2 3 5 2 :nrow(D G S 3 0) , ]

#' Due to data i s s u e s we w i l l have a time p e r i o d with i n t e r e s t r a t e

#' with the value o f z e r o . This i s because the 3 0 year t r e a s u r y bond

#' was d i s c o n t i n e d on February 19 , 2002 and r e i n t r o d u c e d February 9 , 2006.

u t i l s: :V i e w(a d j y e a r 3 0)

a d j y e a r 3 0[i s.na(a d j y e a r 3 0) ] < 5 #Just using a random number to avoid ommiting ← -l a t e r

#Combine a l l t h r e e i n t e r e s t r a t e s

c o m b i n e d < cbind(a d j y e a r 1 0, a d j m o n t h 3, a d j y e a r 3 0)

#Rename the columns

colnames(c o m b i n e d) < c(" Tenyear ", "Threemonth", " Thirtyyear ")

#I n s p e c t data

u t i l s: :V i e w(c o m b i n e d)

#We s t i l l have a high number o f missing data : sum(i s.na(c o m b i n e d$T e n y e a r) )/nrow(c o m b i n e d)∗100

#This i s due to Federal h o l d i d a y s .

#The problem i s simply s o l v e d by removing days with missing v a l u e s c o m b i n e d < na.omit(c o m b i n e d)

d a t a f r a m e c o m b i n e d < data.frame(c o m b i n e d)

p < g g p l o t( ) +

g e o m_l i n e(data = d a t a f r a m e c o m b i n e d, aes(x = index(c o m b i n e d) , y = Tenyear, c o l o r←

-= "Ten year ") ) +

g e o m_l i n e(data = d a t a f r a m e c o m b i n e d, aes(x = index(c o m b i n e d) , y = T h r e e m o n t h, ← -c o l o r = " Three month") ) +

g e o m_l i n e(data = d a t a f r a m e c o m b i n e d[ 1 : 4 0 0 2 , ] , aes(x = index(c o m b i n e d[ 1 : 4 0 0 2 , ] ) , ← -y = T h i r t y y e a r, c o l o r = " Thirty year ") ) +

g e o m_l i n e(data = d a t a f r a m e c o m b i n e d[ 4 9 9 7 :nrow(d a t a f r a m e c o m b i n e d) , ] , aes(x = ← -index(c o m b i n e d[ 4 9 9 7 :nrow(c o m b i n e d) , ] ) , y = T h i r t y y e a r, c o l o r = " Thirty year←

-") ) +

#l a b s ( t i t l e =" I n t e r e s t r a t e s from 1986 to 2016" , x="Year " , y=" I n t e r e s t r a t e i n ←

-%", c o l o r ="Treasury type ") t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

legend.t i t l e = _ ( ) ,

a x i s.t i t l e.y=e l e m e n t_t e x t(s i z e=7) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 1 , 0 . 7 5 ) , legend.key.s i z e = u n i t( 0 . 5 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") ) p

p+l a b s(x="Year", y=" I n t e r e s t r a t e in %", c o l o r=" Treasury type ")

# theme ( a x i s . l i n e = element_l i n e ( c o l o u r = " black ") ,

# #panel . g r i d . major = element_l i n e ( c o l o u r = " black ") ,

# #panel . g r i d . minor = element_l i n e ( c o l o u r = " black ") ,

# panel . border = element_blank ( ) ,

# panel . background = element_blank ( ) )

#'========================

#' I n t e r e s t r a t e v o l a t i l i t y

#'========================

r e t y e a r 1 0 < l o g(lag(a d j y e a r 1 0) ) l o g(a d j y e a r 1 0) r e t y e a r 1 0 < r e t y e a r 1 0[ 1 ]

p l o t(r e t y e a r 1 0)

r e t y e a r 1 0 < na.omit(r e t y e a r 1 0) p l o t(r e t y e a r 1 0)

r e t m o n t h 3 < l o g(lag(a d j m o n t h 3) ) l o g(a d j m o n t h 3) r e t m o n t h 3 < r e t m o n t h 3[ 1 ]

p l o t(r e t m o n t h 3)

r e t 3 < na.omit(r e t m o n t h 3) p l o t(r e t m o n t h 3)

r e t y e a r 3 0 < l o g(lag(a d j y e a r 3 0) ) l o g(a d j y e a r 3 0) r e t y e a r 3 0 < r e t y e a r 3 0[ 1 ]

p l o t(r e t y e a r 3 0)

r e t y e a r 3 0 < na.omit(r e t y e a r 3 0) p l o t(r e t y e a r 3 0)

##################################################

## PROGRAM 4 Simulate Sample Paths : FIGURE 5 . 2 #

##################################################

#Load packages r e q u i r e(g g p l o t 2)

#FUNCTION:

## Define model parameters r0 < 0 . 0 3

t h e t a < 0 . 1 0 k < 0 . 3 beta < 0 . 0 3

## s i m u l a t e s h o r t r a t e paths

n < 10 # MC s i m u l a t i o n t r i a l s T < 30 # t o t a l time

m < 200 # s u b i n t e r v a l s

dt < T/m # d i f f e r e n c e i n time each s u b i n t e r v a l

r < matrix( 0 ,m+1,n) # matrix to hold s h o r t r a t e paths r[ 1 , ] < r0

f o r(j in 1 :n) { f o r(i in 2 : (m+1) ) {

dr < k∗(theta r[i 1 ,j] )∗ dt + beta ∗ s q r t(dt)∗rnorm( 1 , 0 , 1 ) r[i,j] < r[i 1 ,j] + dr

} }

## Standard matplot : t < seq( 0 , T, dt)

rT.e x p e c t e d < t h e t a + (r0 t h e t a)∗exp( k∗ t) rT.s t d e v < s q r t( beta^2/(2∗k)∗( 1 exp( 2∗k∗ t) ) ) matplot(t, r[ , 1 : 1 0 ] , t y p e=" l ", lty=1, y l a b=" r t ") a b l i n e(h=theta, c o l=" red ", lty=2)

l i n e s(t, rT.e x p e c t e d, lty=2)

l i n e s(t, rT.e x p e c t e d + 2∗rT.stdev, lty=2) l i n e s(t, rT.e x p e c t e d 2∗rT.stdev, lty=2) p o i n t s( 0 ,r0)

#Set up f o r gg p l o t 2 p l o t r.df < data.frame(r) e x p e c t e d.r < rT.e x p e c t e d

up.l i m i t < rT.e x p e c t e d + 2∗rT.s t d e v d o w n.l i m i t < rT.e x p e c t e d 2∗rT.s t d e v

#ggpl ot 2 p l o t

g g p l o t(r.df, aes(x = t, y= X1) ) + g e o m_l i n e(aes(c o l=" darkpurple ") ) + g e o m_l i n e(aes(y=X2, c o l = " red ") ) + g e o m_l i n e(aes(y=X3, c o l = " green ") ) + g e o m_l i n e(aes(y=X4, c o l = " blue ") ) + g e o m_l i n e(aes(y=X5, c o l = " darkred ") ) + g e o m_l i n e(aes(y=X6, c o l = " darkblue ") ) + g e o m_l i n e(aes(y=X7, c o l = " orange ") ) + g e o m_l i n e(aes(y=X8, c o l = " darkgreen ") ) + g e o m_l i n e(aes(y=X9, c o l = " o l i v e ") ) + g e o m_l i n e(aes(y=X10, c o l = " purple ") ) + g e o m_l i n e(aes(y=up.l i m i t) , l i n e t y p e = 2) + g e o m_l i n e(aes(y=d o w n.l i m i t) , l i n e t y p e = 2) + g e o m_l i n e(aes(y=e x p e c t e d.r) , l i n e t y p e = 2) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s("%") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") ,

panel.g r i d. = _ ( ) ,

a x i s.t i t l e.y=e l e m e n t_t e x t(s i z e=9) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.p o s i t i o n= "none")

#################################################################

## PROGRAM 5 CONSTANT AND STOCHASTIC INVESTMENT OPPORTUNITIES #

#################################################################

#CONTENT

#( i ) g g p l o t o f E f f i c i e n t Frontier , Figure 5 . 3

#( i i ) g g p l o t o f a l l o c a t i o n r e s u l t s , RA=2 and RA= 10 , Figure 5 . 6

#( i i i ) g g p l o t o f bond a l l o c a t i o n , constant and s t o c h a s t i c model , Figure 5 . 4

#( i v ) g g p l o t o f cash a l l o c a t i o n , constant and s t o c h a s t i c model , Figure 5 . 5

#Packages

r e q u i r e(g g p l o t 2) r e q u i r e(g r i d E x t r a)

#################################################

#Create dataframe from CSV. f i l e

f r o n t i e r.df < data.frame(F r o n t i e r_R c s v)∗100

#View data

h e a d(f r o n t i e r.df)

#E f f i c i e n t f r o n t i e r

f r o n t i e r.p l o t < g g p l o t(f r o n t i e r.df, aes(x = Con_Stddev, y = Con_ret) ) + g e o m_l i n e(aes(c o l = " black ") , s i z e = 1) +

g e o m_l i n e(aes(x = Sto_d e v 2. 5 , y = Sto_r e t 2. 5 , c o l= " blue ") , s i z e = 1) + g e o m_l i n e(aes(x = Sto_dev5, y = Sto_ret5, c o l= " green ") , s i z e = 1) + g e o m_l i n e(aes(x = Sto_dev30, y = Sto_ret30, c o l = " reed ") , s i z e = 1) + c o o r d_c a r t e s i a n(x l i m = c( 0 , 2 0 ) , y l i m = c( 0 , 1 0 ) ) +

s c a l e_x_c o n t i n u o u s(" Standard Deviation ") + s c a l e_y_c o n t i n u o u s(" Expected Return ") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c(" Constant ", "T=2.5", "T=5", "T←

-=30") ,

v a l u e s = c(" black ", " blue "," green ", " red ") ) + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=8, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= " r i g h t ", #c ( 1 . 0 2 4 9 , 0 . 8 4 5 ) legend.key.s i z e = u n i t( 0 . 5 ,"cm") ,

legend.t e x t = e l e m e n t_t e x t(s i z e = 9) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=a l p h a(' white ', 0 . 4 ) ) , legend.margin = u n i t( 0 . 5 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l = " black ", s i z e=0.5 , l i n e t y p e="← -s o l i d ",

c o l o u r =" black ") )

##### Bond , Cash and Hedge ###

#Load Excel f i l e

weights.df < data.frame(weights_csv)

#View date h e a d(weights.df)

#Plot o f a l l o c a t i o n r e s u l t , RA = 10

p l o t.r i s k 1 0 < g g p l o t(weights.df, aes(x = Time, y = B o n d_10) ) + g e o m_l i n e(aes(c o l = " blue ") , s i z e = 1) +

g e o m_l i n e(aes(y = C a s h_10 , c o l=" green ") , s i z e = 1) + g e o m_l i n e(aes(y = H e d g e_10 , c o l=" red ") , s i z e = 1) + c o o r d_c a r t e s i a n(x l i m = c( 0 , 2 0 ) , y l i m = c( 0 . 7 5 , 1 . 1 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction o f Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("Bonds", " Risk Free ", "Hedge" ) ,← -v a l u e s = c(" blue "," green ", " red ") ) +

l a b s(t i t l e=" ( b ) Risk Aversion = 10") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 0 . 3 4 5 , 0 . 7 5 ) , legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") )

#Plot o f a l l o c a t i o n r e s u l t , RA = 2

p l o t.r i s k 2 < g g p l o t(weights.df, aes(x = Time, y = B o n d_2) ) + g e o m_l i n e(aes(c o l = " blue ") , s i z e = 1) +

g e o m_l i n e(aes(y = C a s h_2 , c o l=" green ") , s i z e = 1) + g e o m_l i n e(aes(y = H e d g e_2 , c o l=" red ") , s i z e = 1) + c o o r d_c a r t e s i a n(x l i m = c( 0 , 2 0 ) , y l i m = c( 0 . 7 5 , 1 . 1 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction o f Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("Bonds", " Risk Free ", "Hedge" ) ,← -v a l u e s = c(" blue "," green ", " red ") ) +

l a b s(t i t l e=" ( a ) Risk Aversion = 2") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") ,

panel.g r i d. = _ ( ) ,

a x i s.t e x t.y=e l e m e n t_t e x t(s i z e=9) , a x i s.t i t l e.y=e l e m e n t_t e x t(s i z e=9) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 0 . 3 4 5 , 0 . 7 5 ) , legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") )

#Combine p l o t s

g r i d.a r r a n g e(p l o t.risk2, p l o t.risk10, n c o l=2)

#Bond a l l o c a t i o n , Constant and s t o c h a s t i c model , Figure 5 . 4

#Load e x c e l f i l e

s t o c h a s.df < data.frame(S t o c h a s t i c_a n a l y s i s)

#View data h e a d(s t o c h a s.df)

#Bond a l l o c a t i o n f o r constant model

b o n d.c o n s t a n t < g g p l o t(s t o c h a s.df, aes(x = Gamma, y = Con_B o n d) ) + g e o m_l i n e(aes(c o l = " blue ") , s i z e = 1 . 5 ) +

c o o r d_c a r t e s i a n(x l i m = c( 0 , 2 0 ) , y l i m = c( 0 . 7 5 , 1 . 1 ) ) + s c a l e_x_c o n t i n u o u s(" Risk Aversion ") +

s c a l e_y_c o n t i n u o u s(" Fraction o f Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("Bonds") , v a l u e s = c(" blue ") ) ← -+

l a b s(t i t l e=" ( a ) Constant Model") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 0 . 3 , 0 . 8 7 ) , legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") )

#Bond a l l o c a t i o n f o r s t o c h a s t i c model

b o n d.s t o 2 5 < g g p l o t(s t o c h a s.df, aes(x =Gamma, y = B o n d_2 . 5 ) ) + g e o m_l i n e(aes(c o l = " blue ") , s i z e = 1 . 5 ) +

g e o m_l i n e(aes(y = B o n d_5 , c o l = " darkred ") , s i z e = 1 . 5 , l i n e t y p e = 2) + c o o r d_c a r t e s i a n(x l i m = c( 0 , 2 0 ) , y l i m = c( 0 . 7 5 , 1 . 1 ) ) +

s c a l e_x_c o n t i n u o u s(" Risk Aversion ") + s c a l e_y_c o n t i n u o u s(" Fraction o f Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("Bond , T=2.5", "Bond , T=5") , ← -v a l u e s = c(" blue "," darkred ") ) +

l a b s(t i t l e=" ( b ) S t o c h a s t i c Model") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 0 . 4 , 0 . 8 2 ) , legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") )

#Combine p l o t s

g r i d.a r r a n g e(b o n d.c o n s t a n t, b o n d.sto25, n c o l=2)

#( i v ) g g p l o t o f cash a l l o c a t i o n , constant and s t o c h a s t i c model , Figure 5 . 5

#Cash a l l o c a t i o n f o r constant model

c a s h.c o n s t a n t < g g p l o t(s t o c h a s.df, aes(x = Gamma, y = Con_C a s h) ) + g e o m_l i n e(aes(c o l = " green ") , s i z e = 1 . 5 ) +

c o o r d_c a r t e s i a n(x l i m = c( 0 , 2 0 ) , y l i m = c( 0 . 7 5 , 1 . 1 ) ) + s c a l e_x_c o n t i n u o u s(" Risk Aversion ") +

s c a l e_y_c o n t i n u o u s(" Fraction o f Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("Cash") , v a l u e s = c(" green ") ) + l a b s(t i t l e=" ( a ) Constant Model") +

t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 0 . 3 , 0 . 8 5 ) , legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") )

g e o m_l i n e(aes(y = C a s h_5 , c o l = " darkgreen ") , s i z e = 1 . 5 ) + c o o r d_c a r t e s i a n(x l i m = c( 0 , 2 0 ) , y l i m = c( 0 . 7 5 , 1 . 1 ) ) + s c a l e_x_c o n t i n u o u s(" Risk Aversion ") +

s c a l e_y_c o n t i n u o u s(" Fraction o f Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("Cash , T=2.5", "Cash , T=5") , ← -v a l u e s = c(" green "," darkgreen ") ) +

l a b s(t i t l e=" ( b ) S t o c h a s t i c Model") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 0 . 4 , 0 . 8 ) , legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") )

#Combine p l o t s

g r i d.a r r a n g e(c a s h.c o n s t a n t, c a s h.sto25, n c o l=2)

#############################################################################

## PROGRAM 6 MAIN PROGRAM FOR ALLOCATION RESULTS, PLOTS AND LOSS FUNCTION #

#############################################################################

#CONTENT

#( i ) PORTFOLIO MODEL FOR BOTH MODEL 1 AND MODEL 2

#( i i ) TABLE 6 . 1 Bond decomposition

#( i i i ) PLOTS f o r Figure 6 . 1 , Figure 6 . 2 , cOMPARE MODEL1 AND MODEL 2

#( i v ) Loss FUNCTIONS BETWEEN CONSTANT MODEL AND MODEL 1

#### Required Packages #####

r e q u i r e(g g p l o t 2) r e q u i r e(s t a r g a z e r) ; r e q u i r e(g r i d E x t r a)

#############################################################################

#### I n t e r e s t Rate Simulation ####

## Define model parameters r0 < 0 . 0 1

r_bar < 0 . 0 1 #theta kappa < 0.4965 #k s i g m a_r < 0 . 0 5 #beta

## s i m u l a t e s h o r t r a t e paths n < 1 # MC s i m u l a t i o n t r i a l s T < 30 # t o t a l time

m < 30 # s u b i n t e r v a l s

dt < T/m # d i f f e r e n c e i n time each s u b i n t e r v a l

r < matrix( 0 ,m+1,n) # matrix to hold s h o r t r a t e paths r[ 1 , ] < r0

f o r(j in 1 :n) { f o r(i in 2 : (m+1) ) {

dr < kappa∗(r_bar r[i 1 ,j] )∗ dt + s i g m a_r∗ s q r t(dt)∗rnorm( 1 , 0 , 1 ) r[i,j] < r[i 1 ,j] + dr

} }

## p l o t paths t < seq( 0 , T, dt)

rT.e x p e c t e d < r_bar + (r0 r_bar)∗exp( kappa∗ t)

rT.s t d e v < s q r t( s i g m a_r^2/(2∗kappa)∗( 1 exp( 2∗kappa∗ t) ) ) matplot(t, r[ , 1 ] , t y p e=" l ", lty=1, y l a b=" r t ")

a b l i n e(h=r_bar, c o l=" red ", lty=2) l i n e s(t, rT.e x p e c t e d, lty=2)

l i n e s(t, rT.e x p e c t e d + 2∗rT.stdev, lty=2) l i n e s(t, rT.e x p e c t e d 2∗rT.stdev, lty=2) p o i n t s( 0 ,r0)

#### PORTFOLIO MODEL ####

#Function begin g a m m a L i s t = l i s t( )

g a m m a V e c = c( 0 . 5 , 1 , 2 , 5 , 10 , 20)

f o r(g in 1 :l e n g t h(g a m m a V e c) ) { gamma = g a m m a V e c[g]

#Parameters

l a m b d a_b a r 2 < 0.3650 #0.3650 #0.15 l a m b d a_t i l d e 2 < 0 . 0 6 #0 . 0 6 #8.925 rho < 0 . 2

s i g m a_s < 0.202

l a m b d a_b a r 1 < 0.109 #0.109 #0.05 l a m b d a_t i l d e 1 < 0.067 #0.067 #2.48 tau < t

b_tau < (1/kappa)∗( 1 exp( kappa∗ t) ) s i g m a_b < 0 . 1

mu_s < 0.087 mu_b < 0.021

#r < rep ( 0 . 0 1 , 3 1 )

#### #Model with s t o c h a s t i c i n t e r e s t and a f f i n e market p r i c e o f r i s k ####

#ODE' s

v < 2∗ s q r t( ( kappa + ( (gamma 1 )/gamma)∗s i g m a_r∗l a m b d a_t i l d e 1) ^2 + ( (gamma1 )/← -gamma)∗( (l a m b d a_t i l d e 1^2+l a m b d a_t i l d e 2^2)/gamma)∗s i g m a_r^2)

_ < (2∗( ( 1/gamma)∗( _ ^2+ _ ^2) )∗(exp( ∗ ) 1 ) )/( (v←

-( -( 1/gamma)∗(l a m b d a_t i l d e 1^2+l a m b d a_t i l d e 2^2) )∗A2_tau +(2/v)∗((1+(1/gamma)∗(← -l a m b d a_b a r 1∗l a m b d a_t i l d e 1+l a m b d a_b a r 2∗l a m b d a_t i l d e 2) )∗

(2∗(kappa+ ( ( 1 gamma)/gamma)∗s i g m a_r∗l a m b d a_t i l d e 1) )+

2∗( (kappa∗r_bar ( 1 gamma)/gamma)∗s i g m a_r∗l a m b d a_b a r 1)∗

( ( 1/gamma)∗(l a m b d a_t i l d e 1^2+l a m b d a_t i l d e 2^2) ) )∗ ( (exp( (v∗tau)/2) 1 ) ^2)/ ( (v+(2∗(← -kappa+ ( ( 1gamma)/gamma)∗s i g m a_r∗l a m b d a_t i l d e 1) ) )∗

(exp(v∗tau) 1 ) +2∗v)

A_f u n c t i o n s < data.frame(A1_tau, A2_tau, v)

#Stock Weight

pi_s2 < (1/gamma)∗( (l a m b d a_b a r 2+l a m b d a_t i l d e 2∗r)/(s q r t( 1 rho^2)∗s i g m a_s) )

#Bond Weight

pi_b2 < (1/(gamma∗s i g m a_b) )∗(l a m b d a_b a r 1+l a m b d a_t i l d e 1∗r rho/(s q r t( 1 rho^2) )←

-∗(l a m b d a_b a r 2+l a m b d a_t i l d e 2∗r) ) +(((gamma 1 )/gamma) ∗ (s i g m a_r/s i g m a_b) )∗(← -A1_tau +A2_tau∗r)

pi_b2. 1t e r m < (1/(gamma∗s i g m a_b) )∗(l a m b d a_b a r 1+l a m b d a_t i l d e 1∗r) mean(pi_b2. 1t e r m)

pi_b2. 2t e r m < (1/(gamma∗s i g m a_b) )∗( rho/(s q r t( 1 rho^2) )∗(l a m b d a_b a r 2+l a m b d a_← -t i l d e 2∗r) )

mean(pi_b2. 2t e r m)

#Hedge part

pi_b2_h e d g e < ( ( (gamma 1 )/gamma) ∗ (s i g m a_r/s i g m a_b) )∗(A1_tau +A2_tau∗r)

#Mean value o f p i_s2

mean_pi_s2 < rep(mean(pi_s2) , 31)

#### Model only s t o c h a s t i c i n t e r e s t r a t e #####

#Parameters :

l a m b d a 1 < (l a m b d a_b a r 1+l a m b d a_t i l d e 1∗mean(r) ) #0 . 1 1 l a m b d a 2 < (l a m b d a_b a r 2+l a m b d a_t i l d e 2∗mean(r) ) #0.3666

#Stock weight

pi_s1 < (1/gamma)∗(l a m b d a 2/(s q r t( 1 rho^2)∗s i g m a_s) ) s t o c k_weights_old < data.frame(pi_s1)

#Bond weight

pi_b1 < (1/gamma)∗( ( (l a m b d a 1)/s i g m a_b) ( (rho∗(l a m b d a 2) )/(s q r t( 1 rho^2)∗s i g m a← -_b) ) ) +((gamma 1 )/gamma)∗(s i g m a_r/s i g m a_b)∗b_tau

b o n d_weights_old < data.frame(pi_b1)

pi_b1_h e d g e < ( (gamma1 )/gamma)∗(s i g m a_r/s i g m a_b)∗b_tau

#F i l l a v e c t o r the constant s t o c k weight vac < rep(pi_s1, 3 1 )

#Hedging importance

h e d g e 1 < pi_b1 pi_b1_h e d g e h e d g e 2 < pi_b2 pi_b2_h e d g e

#Stock Bond Ratio r a t i o_1 < pi_b1/pi_s1 r a t i o_2 < pi_b2/pi_s2

r a t i o.df < data.frame(r a t i o_1 ,r a t i o_2)

#Cash f r a c t i o n

c a s h_1 < 1 pi_s1 pi_b1 check_1 < pi_s1 + pi_b1 + c a s h_1 c a s h_2 < 1 pi_s2 pi_b2 check_2 < pi_s2 + pi_b2 + c a s h_2

#Dataframe f o r s t o c k s

s t o c k_weights < data.frame(pi_s2)

#Dataframe f o r bonds

b o n d_weights < data.frame(pi_b2)

h e d g e_weights < data.frame(pi_b2_h e d g e)

b o n d_terms < data.frame(pi_b2. 1term,pi_b2. 2term, pi_b2_hedge, A1_tau,A2_tau)

### LOSS FUNCTION ###

g a i n_model < (1/gamma 1)∗( 1 gamma)/2 ∗ s i g m a_r^2 ∗ ( (kappa∗tau ( 1 exp( kappa∗tau) ) )/(kappa^3)

( 1 exp( kappa∗tau) ) ^2/(2∗kappa^3) ) l o s s_model < exp(g a i n_model∗r0) 1

#a s s i g n ( paste0 ("G" , g ) , s t o c k_weights )

g a m m a L i s t[ [p a s t e 0("G", g a m m a V e c[g] ) ] ] < s t o c k_weights g a m m a L i s t[ [p a s t e 0("A", g a m m a V e c[g] ) ] ] < s t o c k_weights_old g a m m a L i s t[ [p a s t e 0("B", g a m m a V e c[g] ) ] ] < b o n d_weights g a m m a L i s t[ [p a s t e 0("O", g a m m a V e c[g] ) ] ] < b o n d_weights_old g a m m a L i s t[ [p a s t e 0("H", g a m m a V e c[g] ) ] ] < h e d g e_weights g a m m a L i s t[ [p a s t e 0("T", g a m m a V e c[g] ) ] ] < b o n d_terms g a m m a L i s t[ [p a s t e 0("V", g a m m a V e c[g] ) ] ] < A_f u n c t i o n s g a m m a L i s t[ [p a s t e 0("L", g a m m a V e c[g] ) ] ] < l o s s_model g a m m a L i s t[ [p a s t e 0("C", g a m m a V e c[g] ) ] ] < c a s h_1 }

##############################################

#Producing output

##############################################

#S e t t i n g up dataframes

a v e r s i o n.df.s2 < data.frame(g a m m a L i s t$G0. 5 ,g a m m a L i s t$G1,g a m m a L i s t$G2,g a m m a L i s t$← -G5,g a m m a L i s t$G10,g a m m a L i s t$G20)

a v e r s i o n.df.s1 < data.frame(g a m m a L i s t$A0. 5 ,g a m m a L i s t$A1,g a m m a L i s t$A2,g a m m a L i s t$← -A5,g a m m a L i s t$A10,g a m m a L i s t$A20)

a v e r s i o n.df.b2 < data.frame(g a m m a L i s t$B0. 5 ,g a m m a L i s t$B1,g a m m a L i s t$B2,g a m m a L i s t$← -B5,g a m m a L i s t$B10,g a m m a L i s t$B20)

a v e r s i o n.df.b1 < data.frame(g a m m a L i s t$O0. 5 ,g a m m a L i s t$O1,g a m m a L i s t$O2,g a m m a L i s t$← -O5,g a m m a L i s t$O10,g a m m a L i s t$O20)

.df. < data.frame( $ . 5 , $ , $ ,←

-a v e r s i o n.df.A_f u n c t i o n s < data.frame(g a m m a L i s t$V0. 5 ,g a m m a L i s t$V1,g a m m a L i s t$V2, ← -g a m m a L i s t$V5,g a m m a L i s t$V10,g a m m a L i s t$V20)

a v e r s i o n.df.c a s h.m1 < data.frame(g a m m a L i s t$C0. 5 ,g a m m a L i s t$C1,g a m m a L i s t$C2, ← -g a m m a L i s t$C5,g a m m a L i s t$C10,g a m m a L i s t$C20)

new.df.A_f u n c t i o n s_g a m m a 2 < data.frame(a v e r s i o n.df.A_f u n c t i o n s$A1_tau. 2 ,a v e r s i o n← -.df.A_f u n c t i o n s$A2_tau. 2 ,a v e r s i o n.df.A_f u n c t i o n s$v. 2 )

new.df.b2 < data.frame(a v e r s i o n.df.b2)

# S e l e c t i n g time p e r i o d s

new.SW < a v e r s i o n.df.s2[c( 2 , 3 0 ) , 1 : 6 ] new.BW < a v e r s i o n.df.b2[c( 2 , 3 0 ) , 1 : 6 ] new.HE < a v e r s i o n.df.h e d g e[c( 2 , 3 0 ) , 1 : 6 ]

new.BT < a v e r s i o n.df.b o n d t e r m s[ (c( 1 , 1 0 , 2 0 , 3 0 ) ) , 1 : 5 ]

### NEW THINGS TABLE MODEL 1 ###

a v e r s i o n.df.b1[c( 2 , 3 0 ) , 1 : 6 ] a v e r s i o n.df.s1[c( 2 , 3 0 ) , 1 : 6 ] a v e r s i o n.df.c a s h.m1[c( 2 , 3 0 ) , 1 : 6 ]

# D e f i n i n g new dataframes RA.SW < new.SW[ 1 : 2 , ] RA.BW < new.BW[ 1 : 2 , ] RA.HE < new.HE[ 1 : 2 , ] RA.BT < new.BT[ 1 : 4 , ]

#Creating a Bond Table with decomposed terms BT. 0 5 < a v e r s i o n.df.b o n d t e r m s[ (c( 2 , 3 0 ) ) , 1 : 5 ] BT. 1 < a v e r s i o n.df.b o n d t e r m s[ (c( 2 , 3 0 ) ) , 6 : 1 0 ] BT. 2 < a v e r s i o n.df.b o n d t e r m s[ (c( 2 , 3 0 ) ) , 1 1 : 1 5 ] BT. 5 < a v e r s i o n.df.b o n d t e r m s[ (c( 2 , 3 0 ) ) , 1 6 : 2 0 ] BT. 1 0 < a v e r s i o n.df.b o n d t e r m s[ (c( 2 , 3 0 ) ) , 2 1 : 2 5 ] BT. 2 0 < a v e r s i o n.df.b o n d t e r m s[ (c( 2 , 3 0 ) ) , 2 6 : 3 0 ]

colnames(BT. 0 5 ) < c(" f i r s t ","2nd", " hedge ", "A1", "A2") colnames(BT. 1 ) < c(" f i r s t ","2nd", " hedge ", "A1", "A2") colnames(BT. 2 ) < c(" f i r s t ","2nd", " hedge ", "A1", "A2") colnames(BT. 5 ) < c(" f i r s t ","2nd", " hedge ", "A1", "A2") colnames(BT. 1 0 ) < c(" f i r s t ","2nd", " hedge ", "A1", "A2") colnames(BT. 2 0 ) < c(" f i r s t ","2nd", " hedge ", "A1", "A2")

#########################################################

#( i i ) TABLE 6 . 1 Bond decomposition

BT.t a b l e < rbind(BT. 0 5 ,BT. 1 ,BT. 2 ,BT. 5 ,BT. 1 0 ,BT. 2 0 )

# S t a r g a z e r t a b l e output

BT.t a b l e.l a t e x < s t a r g a z e r(BT.table, summary = F A L S E)

########################################################

#Creating a t a b l e f o r comparison between the models

#S e t t i n g up the t a b l e

dim(RA.SW) = c(l e n g t h(RA.SW) , 1) dim(RA.BW) = c(l e n g t h(RA.BW) , 1) dim(RA.HE) = c(l e n g t h(RA.HE) , 1)

SW.df < t(RA.SW) BW.df < t(RA.BW) HE.df < t(RA.HE)

SW.t1.df < data.frame(SW.df[ , 1 ] )

SW.t30.df < data.frame(SW.df[ , 2 ] ) colnames(SW.t1.df) < c(" Stocks ") colnames(SW.t30.df) < c(" Stocks ")

BW.t1.df < data.frame(BW.df[ , 1 ] ) BW.t30.df < data.frame(BW.df[ , 2 ] ) colnames(BW.t1.df) < c("Bonds") colnames(BW.t30.df) < c("Bonds")

HE.t1.df < data.frame(HE.df[ , 1 ] ) HE.t30.df < data.frame(HE.df[ , 2 ] ) colnames(HE.t1.df) < c("Hedge") colnames(HE.t30.df) < c("Hedge")

SW.rbind < rbind(SW.t1.df,SW.t30.df) BW.rbind < rbind(BW.t1.df,BW.t30.df) HE.rbind < rbind(HE.t1.df,HE.t30.df) C a s h.df < 1 SW.rbind BW.rbind colnames(C a s h.df) < c("Cash")

check.df < C a s h.df + SW.rbind + BW.rbind

new.t a b l e.PF < cbind(SW.rbind,BW.rbind,HE.rbind,C a s h.df)

################################################

#( i i i ) Table 6 . 2 f o r comparison new.t a b l e.PF.df < new.t a b l e.PF

#Table S o l u t i o n

PF.t a b l e.l a t e x < s t a r g a z e r(new.t a b l e.PF, summary = F A L S E)

##########################################################

#Labeling to p r i n t

rownames(BW.df) = colnames(RA.BW) rownames(SW.df) = colnames(RA.BW) rownames(HE.df) = colnames(RA.HE) rownames(SW.df) = c( 0 . 5 , 1 , 2 , 5 , 1 0 , 2 0 ) rownames(BW.df) = c( 0 . 5 , 1 , 2 , 5 , 1 0 , 2 0 ) rownames(HE.df) = c( 0 . 5 , 1 , 2 , 5 , 1 0 , 2 0 )

#Print

SW.df # Stocks at T= 1 and T=30 BW.df # Bonds

HE.df # Hegde term

########################################################

#( i i i ) PLOTS f o r Figure 6 . 1 , Figure 6 . 2 ###############

########################################################

h e a d(a v e r s i o n.df.b1) h e a d(a v e r s i o n.df.b2)

# Bond Plot f o r RA = 1 #

b o n d.p l o t. 1 < g g p l o t(a v e r s i o n.df.b1, aes(x = t, y = g a m m a L i s t$O1, c o l = " blue ") ,← -s i z e = 1 . 2 ) +

_ ( ) +

c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) , y l i m = c( 0 . 2 5 , 0 . 7 5 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction Of Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("Bonds M1", "Bonds M2" ) , ← -v a l u e s = c(" blue ", " green ") ) +

l a b s(t i t l e=" ( a ) Risk Aversion = 1") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

legend.t i t l e = e l e m e n t_b l a n k( ) , a x i s.t e x t.x=e l e m e n t_t e x t(s i z e=8) , a x i s.t i t l e.x=e l e m e n t_t e x t(s i z e=8) , a x i s.t e x t.y=e l e m e n t_t e x t(s i z e=8) , a x i s.t i t l e.y=e l e m e n t_t e x t(s i z e=8) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= "none", # c ( 0 . 4 , 0 . 8 ) , legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") ) b o n d.p l o t. 1

# Bond Plot f o r RA = 2 #

b o n d.p l o t. 2 < g g p l o t(a v e r s i o n.df.b1, aes(x = t, y = g a m m a L i s t$O2, c o l = " blue ") ,← -s i z e = 1 . 2 ) +

g e o m_l i n e( ) +

g e o m_l i n e(aes(y = g a m m a L i s t$B2, c o l = " green ") , s i z e = 1 . 2 ) +

g e o m_r i b b o n(aes(x = t, y m a x=g a m m a L i s t$B2, y m i n=g a m m a L i s t$O2, l i n e t y p e = NA) , ← -f i l l=" pink ", a l p h a=.3) +

c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) , y l i m = c( 0 . 2 5 , 0 . 7 5 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction Of Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("Bonds M1", "Bonds M2" ) , ← -v a l u e s = c(" blue ", " green ") ) +

l a b s(t i t l e=" ( b ) Risk Aversion = 2") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") ) b o n d.p l o t. 2

# Bond Plot f o r RA = 5 #

b o n d.p l o t. 5 < g g p l o t(a v e r s i o n.df.b1, aes(x = t, y = g a m m a L i s t$O5, c o l = " blue ") ,← -s i z e = 1 . 2 ) +

g e o m_l i n e( ) +

g e o m_l i n e(aes(y = g a m m a L i s t$B5, c o l = " green ") , s i z e = 1 . 2 ) +

g e o m_r i b b o n(aes(x = t, y m a x=g a m m a L i s t$B5, y m i n=g a m m a L i s t$O5, l i n e t y p e = NA) , ← -f i l l=" pink ", a l p h a=.3) +

c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) , y l i m = c( 0 . 5 , 1 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction Of Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("Bonds M1", "Bonds M2" ) , ← -v a l u e s = c(" blue ", " green ") ) +

l a b s(t i t l e=" ( c ) Risk Aversion = 5") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n="none", # c ( 0 . 4 , 0 . 8 ) , legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") ,

legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") ) b o n d.p l o t. 5

# Bond Plot f o r RA = 10 #

b o n d.p l o t. 1 0 < g g p l o t(a v e r s i o n.df.b1, aes(x = t, y = g a m m a L i s t$O10, c o l = " blue "← -) , s i z e = 1 . 2 ) +

g e o m_l i n e( ) +

g e o m_l i n e(aes(y = g a m m a L i s t$B10, c o l = " green ") , s i z e = 1 . 2 ) +

g e o m_r i b b o n(aes(x = t, y m a x=g a m m a L i s t$B10, y m i n=g a m m a L i s t$O10, l i n e t y p e = NA) , ← -f i l l=" pink ", a l p h a=.3) +

c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) , y l i m = c( 0 . 5 , 1 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction Of Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("Bonds M1", "Bonds M2" ) , ← -v a l u e s = c(" blue ", " green ") ) +

l a b s(t i t l e=" ( d ) Risk Aversion = 10") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") ,

panel.g r i d. = _ ( ) ,

a x i s.t e x t.y=e l e m e n t_t e x t(s i z e=8) , a x i s.t i t l e.y=e l e m e n t_t e x t(s i z e=8) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") ) b o n d.p l o t. 1 0

# Bond Plot f o r RA = 20 #

b o n d.p l o t. 2 0 < g g p l o t(a v e r s i o n.df.b1, aes(x = t, y = g a m m a L i s t$O20, c o l = " blue "← -) , s i z e = 1 . 2 ) +

g e o m_l i n e( ) +

g e o m_l i n e(aes(y = g a m m a L i s t$B20, c o l = " green ") , s i z e = 1 . 2 ) +

g e o m_r i b b o n(aes(x = t, y m a x=g a m m a L i s t$B20, y m i n=g a m m a L i s t$O20, l i n e t y p e = NA)← -, f i l l=" pink ", a l p h a=.5) +

c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) , y l i m = c( 0 . 7 5 , 1 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction Of Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("Bonds M1", "Bonds M2" ) , ← -v a l u e s = c(" blue ", " green ") ) +

l a b s(t i t l e=" ( d ) Risk Aversion = 20") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= "none", #c ( 0 . 4 , 0 . 8 ) , legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") )

#A l l p l o t s

g r i d.a r r a n g e(b o n d.p l o t. 2 , b o n d.p l o t. 5 , nrow = 1 , n c o l = 2) g r i d.a r r a n g e(b o n d.p l o t. 1 0 , b o n d.p l o t. 2 0 , nrow = 1 , n c o l = 2)

g r i d.a r r a n g e(b o n d.p l o t. 1 , b o n d.p l o t. 2 , b o n d.p l o t. 5 , b o n d.p l o t. 1 0 , nrow = 2 , n c o l ←

-= 2)

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### STOCKS ##############################################

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#RA = 2

s t o c k.p l o t. 2 < g g p l o t(a v e r s i o n.df.s2, aes(x = t, y = g a m m a L i s t$G2) ) + g e o m_l i n e(aes(c o l = " red ") , s i z e = 1) +

g e o m_l i n e(aes(y = g a m m a L i s t$A2, c o l = " darkred ") , s i z e = 1 , l i n e t y p e = 2) +

c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction o f Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c(" Stocks M1", " Stocks M2" ) , ← -v a l u e s = c(" red ", " darkred ") ) +

l a b s(t i t l e=" ( a ) Risk Aversion = 2") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

legend.t i t l e = e l e m e n t_b l a n k( ) , a x i s.t e x t.x=e l e m e n t_t e x t(s i z e=9) , a x i s.t i t l e.x=e l e m e n t_t e x t(s i z e=9) , a x i s.t e x t.y=e l e m e n t_t e x t(s i z e=8) , a x i s.t i t l e.y=e l e m e n t_t e x t(s i z e=8) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 0 . 4 , 0 . 6 5 ) , legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") )

#RA = 5

s t o c k.p l o t. 5 < g g p l o t(a v e r s i o n.df.s2, aes(x = t, y = g a m m a L i s t$G5) ) + g e o m_l i n e(aes(c o l = " red ") , s i z e = 1) +

g e o m_l i n e(aes(y = g a m m a L i s t$A5, c o l = " darkred ") , s i z e = 1 , l i n e t y p e = 2) + c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) ) +

s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction o f Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c(" Stocks M1", " Stocks M2" ) , ← -v a l u e s = c(" red ", " darkred ") ) +

l a b s(t i t l e=" ( b ) Risk Aversion = 5") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 0 . 4 , 0 . 6 5 ) , legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") )

g e o m_l i n e(aes(y = g a m m a L i s t$A10, c o l = " darkred ") , s i z e = 1 , l i n e t y p e = 2) + c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) ) +

s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction o f Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c(" Stocks M1", " Stocks M2" ) , ← -v a l u e s = c(" red ", " darkred ") ) +

l a b s(t i t l e=" ( c ) Risk Aversion = 10") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 0 . 4 , 0 . 6 5 ) , legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") )

## RA = 20

s t o c k.p l o t. 2 0 < g g p l o t(a v e r s i o n.df.s2, aes(x = t, y = g a m m a L i s t$G20) ) + g e o m_l i n e(aes(c o l = " red ") , s i z e = 1) +

g e o m_l i n e(aes(y = g a m m a L i s t$A20, c o l = " darkred ") , s i z e = 1 , l i n e t y p e = 2) + c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) ) +

s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction o f Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c(" Stocks M1", " Stocks M2" ) , ← -v a l u e s = c(" red ", " darkred ") ) +

l a b s(t i t l e=" ( d ) Risk Aversion = 20") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 0 . 4 , 0 . 6 5 ) , legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") )

g r i d.a r r a n g e(s t o c k.p l o t. 2 , s t o c k.p l o t. 5 , s t o c k.p l o t. 1 0 , s t o c k.p l o t. 2 0 , nrow = 2 , ← -n c o l = 2)

###############################################################

### ( i v ) Loss FUNCTIONS BETWEEN CONSTANT MODEL AND MODEL 1 ##

###############################################################

l o s s.f u n c t i o n < data.frame(g a m m a L i s t$L0. 5 ,g a m m a L i s t$L1,g a m m a L i s t$L2, g a m m a L i s t$← -L5,g a m m a L i s t$L10,g a m m a L i s t$L20)

### Risk Aversion = 2 ###

p l o t.l o s s.rv2 < g g p l o t(l o s s.function , aes(x = t, y = l o s s.f u n c t i o n $g a m m a L i s t.L2)← -) +

g e o m_l i n e(c o l= " red ", s i z e = 1) + c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Welfare Loss (%)") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c(" Loss " ) , v a l u e s = c(" red ") ) + l a b s(t i t l e=" ( a ) Risk Aversion = 2") +

t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

legend.t i t l e = e l e m e n t_b l a n k( ) , a x i s.t e x t.x=e l e m e n t_t e x t(s i z e=7) , a x i s.t i t l e.x=e l e m e n t_t e x t(s i z e=9) , a x i s.t e x t.y=e l e m e n t_t e x t(s i z e=7) , a x i s.t i t l e.y=e l e m e n t_t e x t(s i z e=9) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=8, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 1 . 0 2 4 9 , 0 . 8 4 5 ) , legend.key.s i z e = u n i t( 0 . 3 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 6) , legend.d i r e c t i o n = " h o r i z o n t a l ",

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=a l p h a(' white ', 0 . 4 ) ) ) p l o t.l o s s.rv2

### Risk Aversion = 5 ###

p l o t.l o s s.rv5 < g g p l o t(l o s s.f unct ion , aes(x = t, y = l o s s.f u n c t i o n $g a m m a L i s t.L5)← -) +

g e o m_l i n e(c o l= " red ", s i z e = 1) + c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Welfare Loss (%)") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c(" Loss " ) , v a l u e s =← -c(" red ") ) +

l a b s(t i t l e=" ( b ) Risk Aversion = 5") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5)← -,

a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = "← -dotted ") ,

panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

legend.t i t l e = _ ( ) ,

a x i s.t i t l e.y=e l e m e n t_t e x t(s i z e=9) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=8, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=a l p h a(' white ', 0 . 4 ) ) )

### Risk Aversion = 10 ###

p l o t.l o s s.r v 1 0 < g g p l o t(l o s s.function, aes(x = t, y = l o s s.f u n c t i o n $g a m m a L i s t.← -L10) ) +

g e o m_l i n e(c o l= " red ", s i z e = 1) + c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Welfare Loss (%)") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c(" Loss " ) , v a l u e s = c(" red ") ) + l a b s(t i t l e=" ( c ) Risk Aversion = 10") +

t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=8, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=a l p h a(' white ', 0 . 4 ) ) )

### Risk Aversion = 20 ###

p l o t.l o s s.r v 2 0 < g g p l o t(l o s s.f unct ion, aes(x = t, y = l o s s.f u n c t i o n $g a m m a L i s t.← -L20) ) +

g e o m_l i n e(c o l= " red ", s i z e = 1) + c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Welfare Loss (%)") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c(" Loss " ) , v a l u e s = c(" red ") ) + l a b s(t i t l e=" ( d ) Risk Aversion = 20") +

t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=8, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=a l p h a(' white ', 0 . 4 ) ) )

# A l l p l o t s

g r i d.a r r a n g e(p l o t.l o s s.rv2, p l o t.l o s s.rv5,p l o t.l o s s.rv10,p l o t.l o s s.rv20, nrow = ← -2 , n c o l = 2)

######################################################

#### Program 7 A l t e r n a t i v e S p e c i f i c a t i o n o f MPR ####

######################################################

#Content#

#( i ) Simulation o f models with a constant i n t e r e s t rate , i f r < rep ( 0 . 0 1 , 3 1 ) i s ← -a c t i v e .

#( i i ) Simulation o f models with a s t o c h a s t i c i n t e r e s t rate , i f r < rep ( 0 . 0 1 , 3 1 ) ← -i s d e a c t -i v e .

#( i i i ) Figure 6 . 3 and Fiure 6 . 4

#### Required Packages #####

r e q u i r e(g g p l o t 2) r e q u i r e(s t a r g a z e r) ; r e q u i r e(g r i d E x t r a)

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#### I n t e r e s t Rate Simulation ####

## Define model parameters r0 < 0 . 0 1

r_bar < 0 . 0 1 #theta kappa < 0.4965 #k s i g m a_r < 0 . 0 5 #beta

## s i m u l a t e s h o r t r a t e paths n < 1 # MC s i m u l a t i o n t r i a l s T < 30 # t o t a l time

m < 30 # s u b i n t e r v a l s

dt < T/m # d i f f e r e n c e i n time each s u b i n t e r v a l

r < matrix( 0 ,m+1,n) # matrix to hold s h o r t r a t e paths r[ 1 , ] < r0

f o r(j in 1 :n) { f o r(i in 2 : (m+1) ) {

dr < kappa∗(r_bar r[i 1 ,j] )∗ dt + s i g m a_r∗ s q r t(dt)∗rnorm( 1 , 0 , 1 ) r[i,j] < r[i 1 ,j] + dr

} }

## p l o t paths

matplot(t, r[ , 1 ] , t y p e=" l ", lty=1, y l a b=" r t ") a b l i n e(h=r_bar, c o l=" red ", lty=2)

l i n e s(t, rT.e x p e c t e d, lty=2)

l i n e s(t, rT.e x p e c t e d + 2∗rT.stdev, lty=2) l i n e s(t, rT.e x p e c t e d 2∗rT.stdev, lty=2) p o i n t s( 0 ,r0)

#### PORTFOLIO MODEL ####

#Function begin g a m m a L i s t = l i s t( )

g a m m a V e c = c( 0 . 5 , 1 , 2 , 5 , 10 , 20)

f o r(g in 1 :l e n g t h(g a m m a V e c) ) { gamma = g a m m a V e c[g]

#Parameters

l a m b d a_b a r 2 < 0.301 #Base : 0.365 #A l t e r n a t i v e : 0.301 l a m b d a_t i l d e 2 < 6 . 5 #Base : 0 . 0 6 #A l t e r n a t i v e : 6 . 5 rho < 0 . 2

s i g m a_s < 0.202

l a m b d a_b a r 1 < 0 . 0 6 #Base : 0.109 #A l t e r n a t i v e : 0 . 0 6 l a m b d a_t i l d e 1 < 5 #Base : 0.067 #A l t e r n a t i v e : 5 tau < t

b_tau < (1/kappa)∗( 1 exp( kappa∗ t) ) s i g m a_b < 0 . 1

mu_s < 0.087 mu_b < 0.021

#r < rep ( 0 . 0 1 , 3 1 ) ######## IMPORTANT: i f ACTIVE: Simulation with constant ← -i n t e r e s t r a t e ####

#### #Model with s t o c h a s t i c i n t e r e s t and a f f i n e market p r i c e o f r i s k ####

#ODE' s

v < 2∗ s q r t( ( kappa + ( (gamma 1 )/gamma)∗s i g m a_r∗l a m b d a_t i l d e 1) ^2 + ( (gamma1 )/← -gamma)∗( (l a m b d a_t i l d e 1^2+l a m b d a_t i l d e 2^2)/gamma)∗s i g m a_r^2)

A2_tau < (2∗( ( 1/gamma)∗(l a m b d a_t i l d e 1^2+l a m b d a_t i l d e 2^2) )∗(exp(v∗tau) 1 ) )/( (v← -+(2∗(kappa + ( ( 1 gamma)/gamma)∗s i g m a_r∗l a m b d a_t i l d e 1) ) )∗(exp(v∗tau) 1 ) +2∗v)

A1_tau < (1+(1/gamma)∗(l a m b d a_b a r 1∗l a m b d a_t i l d e 1+l a m b d a_b a r 2∗l a m b d a_t i l d e 2) )/

( ( 1/gamma)∗(l a m b d a_t i l d e 1^2+l a m b d a_t i l d e 2^2) )∗A2_tau +(2/v)∗ ((1+(1/gamma)∗(l a m b d a_b a r 1∗l a m b d a_t i l d e 1+l a m b d a_b a r 2∗l a m b d a_←

-t i l d e 2) )∗

(2∗(kappa+ ( ( 1gamma)/gamma)∗s i g m a_r∗l a m b d a_t i l d e 1) )+

2∗( (kappa∗r_bar ( 1 gamma)/gamma)∗s i g m a_r∗l a m b d a_b a r 1)∗ ( ( 1/gamma)∗(l a m b d a_t i l d e 1^2+l a m b d a_t i l d e 2^2) ) )∗

( (exp( (v∗tau)/2) 1 ) ^2)/

( (v+(2∗(kappa+ ( ( 1gamma)/gamma)∗s i g m a_r∗l a m b d a_t i l d e 1) ) )∗(exp(v∗← -tau) 1 ) +2∗v)

A_f u n c t i o n s < data.frame(A1_tau, A2_tau, v)

#Stock Weight

pi_s2 < (1/gamma)∗( (l a m b d a_b a r 2+l a m b d a_t i l d e 2∗r)/(s q r t( 1 rho^2)∗s i g m a_s) )

#Bond Weight

pi_b2 < (1/(gamma∗s i g m a_b) )∗(l a m b d a_b a r 1+l a m b d a_t i l d e 1∗r rho/(s q r t( 1 rho^2) )←

-∗(l a m b d a_b a r 2+l a m b d a_t i l d e 2∗r) ) +(((gamma 1 )/gamma) ∗ (s i g m a_r/s i g m a_b) )∗(← -A1_tau +A2_tau∗r)

pi_b2. 1t e r m < (1/(gamma∗s i g m a_b) )∗(l a m b d a_b a r 1+l a m b d a_t i l d e 1∗r) mean(pi_b2. 1t e r m)

pi_b2. 2t e r m < (1/(gamma∗s i g m a_b) )∗( rho/(s q r t( 1 rho^2) )∗(l a m b d a_b a r 2+l a m b d a_← -t i l d e 2∗r) )

mean(pi_b2. 2t e r m)

#Hedge part

pi_b2_h e d g e < ( ( (gamma 1 )/gamma) ∗ (s i g m a_r/s i g m a_b) )∗(A1_tau +A2_tau∗r)

#Mean value o f pi_s2

mean_pi_s2 < rep(mean(pi_s2) , 31)

#### Model only s t o c h a s t i c i n t e r e s t r a t e #####

#Parameters :

l a m b d a 1 < (l a m b d a_b a r 1+l a m b d a_t i l d e 1∗mean(r) ) #0 . 1 1 l a m b d a 2 < (l a m b d a_b a r 2+l a m b d a_t i l d e 2∗mean(r) ) #0.3666

#Stock weight

pi_s1 < (1/gamma)∗(l a m b d a 2/(s q r t( 1 rho^2)∗s i g m a_s) )

s t o c k_weights_old < data.frame(pi_s1)

#Bond weight

pi_b1 < (1/gamma)∗( ( (l a m b d a 1)/s i g m a_b) ( (rho∗(l a m b d a 2) )/(s q r t( 1 rho^2)∗s i g m a← -_b) ) ) +((gamma 1 )/gamma)∗(s i g m a_r/s i g m a_b)∗b_tau

b o n d_weights_old < data.frame(pi_b1)

pi_b1_h e d g e < ( (gamma1 )/gamma)∗(s i g m a_r/s i g m a_b)∗b_tau

#F i l l a v e c t o r the constant s t o c k weight vac < rep(pi_s1, 3 1 )

#Hedging importance

h e d g e 1 < pi_b1 pi_b1_h e d g e h e d g e 2 < pi_b2 pi_b2_h e d g e

#Stock Bond Ratio r a t i o_1 < pi_b1/pi_s1 r a t i o_2 < pi_b2/pi_s2

r a t i o.df < data.frame(r a t i o_1 ,r a t i o_2)

#Cash f r a c t i o n

c a s h_1 < 1 pi_s1 pi_b1

check_1 < _ + _ + _1

#Dataframe f o r s t o c k s

s t o c k_weights < data.frame(pi_s2)

#Dataframe f o r bonds

b o n d_weights < data.frame(pi_b2)

h e d g e_weights < data.frame(pi_b2_h e d g e)

b o n d_terms < data.frame(pi_b2. 1term,pi_b2. 2term, pi_b2_hedge, A1_tau,A2_tau)

### LOSS FUNCTION ###

g a i n_model < (1/gamma 1)∗( 1 gamma)/2 ∗ s i g m a_r^2 ∗ ( (kappa∗tau ( 1 exp( kappa←

-∗tau) ) )/(kappa^3) ( 1 exp( kappa∗tau) ) ^2/(2∗kappa^3) ) l o s s_model < exp(g a i n_model∗r0) 1

#a s s i g n ( paste0 ("G" , g ) , s t o c k_weights )

############################################################

#Producing output

############################################################

#S e t t i n g up dataframes

a v e r s i o n.df.s2 < data.frame(g a m m a L i s t$G0. 5 ,g a m m a L i s t$G1,g a m m a L i s t$G2,g a m m a L i s t$← -G5,g a m m a L i s t$G10,g a m m a L i s t$G20)

a v e r s i o n.df.s1 < data.frame(g a m m a L i s t$A0. 5 ,g a m m a L i s t$A1,g a m m a L i s t$A2,g a m m a L i s t$← -A5,g a m m a L i s t$A10,g a m m a L i s t$A20)

a v e r s i o n.df.b2 < data.frame(g a m m a L i s t$B0. 5 ,g a m m a L i s t$B1,g a m m a L i s t$B2,g a m m a L i s t$← -B5,g a m m a L i s t$B10,g a m m a L i s t$B20)

a v e r s i o n.df.b1 < data.frame(g a m m a L i s t$O0. 5 ,g a m m a L i s t$O1,g a m m a L i s t$O2,g a m m a L i s t$← -O5,g a m m a L i s t$O10,g a m m a L i s t$O20)

a v e r s i o n.df.h e d g e < data.frame(g a m m a L i s t$H0. 5 ,g a m m a L i s t$H1,g a m m a L i s t$H2,← -g a m m a L i s t$H5,g a m m a L i s t$H10,g a m m a L i s t$H20)

a v e r s i o n.df.b o n d t e r m s < data.frame(g a m m a L i s t$T0. 5 ,g a m m a L i s t$T1,g a m m a L i s t$T2, ← -g a m m a L i s t$T5,g a m m a L i s t$T10,g a m m a L i s t$T20)

a v e r s i o n.df.A_f u n c t i o n s < data.frame(g a m m a L i s t$V0. 5 ,g a m m a L i s t$V1,g a m m a L i s t$V2, ← -g a m m a L i s t$V5,g a m m a L i s t$V10,g a m m a L i s t$V20)

new.df.b2 < data.frame(a v e r s i o n.df.b2)

# S e l e c t i n g time p e r i o d s

new.SW < a v e r s i o n.df.s2[c( 2 , 3 0 ) , 1 : 6 ] new.BW < a v e r s i o n.df.b2[c( 2 , 3 0 ) , 1 : 6 ] new.HE < a v e r s i o n.df.h e d g e[c( 2 , 3 0 ) , 1 : 6 ]

new.BT < a v e r s i o n.df.b o n d t e r m s[ (c( 1 , 1 0 , 2 0 , 3 0 ) ) , 1 : 5 ]

# D e f i n i n g new dataframes RA.SW < new.SW[ 1 : 2 , ] RA.BW < new.BW[ 1 : 2 , ] RA.HE < new.HE[ 1 : 2 , ] RA.BT < new.BT[ 1 : 4 , ]

BT.t a b l e.l a t e x < s t a r g a z e r(BT.table, summary = F A L S E)

#Creating a t a b l e f o r comparison between the models dim(RA.SW) = c(l e n g t h(RA.SW) , 1)

dim(RA.BW) = c(l e n g t h(RA.BW) , 1) dim(RA.HE) = c(l e n g t h(RA.HE) , 1)

SW.df < t(RA.SW) BW.df < t(RA.BW) HE.df < t(RA.HE)

SW.t1.df < data.frame(SW.df[ , 1 ] ) SW.t30.df < data.frame(SW.df[ , 2 ] ) colnames(SW.t1.df) < c(" Stocks ") colnames(SW.t30.df) < c(" Stocks ")

BW.t1.df < data.frame(BW.df[ , 1 ] ) BW.t30.df < data.frame(BW.df[ , 2 ] ) colnames(BW.t1.df) < c("Bonds") colnames(BW.t30.df) < c("Bonds")

HE.t1.df < data.frame(HE.df[ , 1 ] ) HE.t30.df < data.frame(HE.df[ , 2 ] ) colnames(HE.t1.df) < c("Hedge") colnames(HE.t30.df) < c("Hedge")

SW.rbind < rbind(SW.t1.df,SW.t30.df) BW.rbind < rbind(BW.t1.df,BW.t30.df) HE.rbind < rbind(HE.t1.df,HE.t30.df) C a s h.df < 1 SW.rbind BW.rbind colnames( .df) < c("Cash")

new.t a b l e.PF < cbind(SW.rbind,BW.rbind,HE.rbind,C a s h.df) new.t a b l e.PF.df < new.t a b l e.PF

#Table S o l u t i o n

PF.t a b l e.l a t e x < s t a r g a z e r(new.t a b l e.PF, summary = F A L S E)

#Labeling to p r i n t

#Print

SW.df # Stocks at T= 1 and T=30 BW.df # Bonds

HE.df # Hegde term

######################################################

####### ( i i i ) Figure 6 . 3 and Fiure 6 . 4 ###############

######################################################

h e a d(a v e r s i o n.df.b1) h e a d(a v e r s i o n.df.b2)

# Bond Plot f o r RA = 1 #

b o n d.p l o t. 1 < g g p l o t(a v e r s i o n.df.b1, aes(x = t, y = g a m m a L i s t$O1, c o l = " blue ") ,← -s i z e = 1 . 2 ) +

g e o m_l i n e( ) +

g e o m_l i n e(aes(y = g a m m a L i s t$B1, c o l = " darkblue ") , s i z e = 1 . 2 ) +

g e o m_r i b b o n(aes(x = t, y m a x=g a m m a L i s t$B1, y m i n=g a m m a L i s t$O1, l i n e t y p e = NA) , ← -f i l l=" pink ", a l p h a=.3) +

c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) , y l i m = c( 0 . 2 5 , 0 . 7 5 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction Of Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("Bonds M1", "Bonds M2" ) , ← -v a l u e s = c(" blue ", " darkblue ") ) +

l a b s(t i t l e=" ( a ) Risk Aversion = 1") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") ) b o n d.p l o t. 1

# Bond Plot f o r RA = 2 #

b o n d.p l o t. 2 < g g p l o t(a v e r s i o n.df.b1, aes(x = t, y = g a m m a L i s t$O2, c o l = " blue ") ,← -s i z e = 1 . 2 ) +

g e o m_l i n e( ) +

g e o m_l i n e(aes(y = g a m m a L i s t$B2, c o l = " darkblue ") , s i z e = 1 . 2 ) +

g e o m_r i b b o n(aes(x = t, y m a x=g a m m a L i s t$B2, y m i n=g a m m a L i s t$O2, l i n e t y p e = NA) , ← -f i l l=" pink ", a l p h a=.3) +

c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) , y l i m = c( 1 . 5 , 4 . 5 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction Of Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("Bonds M1", "Bonds M2" ) , ← -v a l u e s = c(" blue ", " darkblue ") ) +

l a b s(t i t l e=" ( a ) Risk Aversion = 2") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") ) b o n d.p l o t. 2

# Bond Plot f o r RA = 5 #

b o n d.p l o t. 5 < g g p l o t(a v e r s i o n.df.b1, aes(x = t, y = g a m m a L i s t$O5, c o l = " blue ") ,← -s i z e = 1 . 2 ) +

g e o m_l i n e( ) +

g e o m_l i n e(aes(y = g a m m a L i s t$B5, c o l = " darkblue ") , s i z e = 1 . 2 ) +

g e o m_r i b b o n(aes(x = t, y m a x=g a m m a L i s t$B5, y m i n=g a m m a L i s t$O5, l i n e t y p e = NA) , ← -f i l l=" pink ", a l p h a=.3) +

c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) , y l i m = c( 1 . 5 , 4 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction Of Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("Bonds M1", "Bonds M2" ) , ← -v a l u e s = c(" blue ", " darkblue ") ) +

l a b s(t i t l e=" ( b ) Risk Aversion = 5") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

legend.t i t l e = _ ( ) ,

a x i s.t i t l e.y=e l e m e n t_t e x t(s i z e=8) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n="none", # c ( 0 . 4 , 0 . 8 ) , legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") ,

legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") ) b o n d.p l o t. 5

# Bond Plot f o r RA = 10 #

b o n d.p l o t. 1 0 < g g p l o t(a v e r s i o n.df.b1, aes(x = t, y = g a m m a L i s t$O10, c o l = " blue "← -) , s i z e = 1 . 2 ) +

g e o m_l i n e( ) +

g e o m_l i n e(aes(y = g a m m a L i s t$B10, c o l = " darkblue ") , s i z e = 1 . 2 ) +

g e o m_r i b b o n(aes(x = t, y m a x=g a m m a L i s t$B10, y m i n=g a m m a L i s t$O10, l i n e t y p e = NA) , ← -f i l l=" pink ", a l p h a=.3) +

c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) , y l i m = c( 1 . 5 , 4 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction Of Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("Bonds M1", "Bonds M2" ) , ← -v a l u e s = c(" blue ", " darkblue ") ) +

l a b s(t i t l e=" ( c ) Risk Aversion = 10") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") ) b o n d.p l o t. 1 0

# Bond Plot f o r RA = 20 #

b o n d.p l o t. 2 0 < g g p l o t(a v e r s i o n.df.b1, aes(x = t, y = g a m m a L i s t$O20, c o l = " blue "← -) , s i z e = 1 . 2 ) +

g e o m_l i n e( ) +

g e o m_l i n e(aes(y = g a m m a L i s t$B20, c o l = " darkblue ") , s i z e = 1 . 2 ) +

g e o m_r i b b o n(aes(x = t, y m a x=g a m m a L i s t$B20, y m i n=g a m m a L i s t$O20, l i n e t y p e = NA) , ← -f i l l=" pink ", a l p h a=.5) +

c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) , y l i m = c( 1 . 5 , 4 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction Of Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("Bonds M1", "Bonds M2" ) , ← -v a l u e s = c(" blue ", " darkblue ") ) +

l a b s(t i t l e=" ( d ) Risk Aversion = 20") +

t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= "none", #c ( 0 . 4 , 0 . 8 ) , legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") ) b o n d.p l o t. 2 0

#A l l p l o t s

g r i d.a r r a n g e(b o n d.p l o t. 2 , b o n d.p l o t. 5 , nrow = 1 , n c o l = 2) g r i d.a r r a n g e(b o n d.p l o t. 1 0 , b o n d.p l o t. 2 0 , nrow = 1 , n c o l = 2)

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#( i i i ) Figure 6 . 3 and Fiure 6 . 4

g r i d.a r r a n g e(b o n d.p l o t. 2 , b o n d.p l o t. 5 , b o n d.p l o t. 1 0 , b o n d.p l o t. 2 0 , nrow = 2 , n c o l←

-= 2)

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### STOCKS ###

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#RA = 2

s t o c k.p l o t. 2 < g g p l o t(a v e r s i o n.df.s2, aes(x = t, y = g a m m a L i s t$G2) ) + g e o m_l i n e(aes(c o l = " red ") , s i z e = 1) +

g e o m_l i n e(aes(y = g a m m a L i s t$A2, c o l = " darkred ") , s i z e = 1 , l i n e t y p e = 2) + c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) ) +

s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction o f Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c(" Stocks M1", " Stocks M2" ) , ← -v a l u e s = c(" red ", " darkred ") ) +

l a b s(t i t l e=" ( a ) Risk Aversion = 2") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e= _t e x t( =9, =" black ") ,

legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") )

#RA = 5

s t o c k.p l o t. 5 < g g p l o t(a v e r s i o n.df.s2, aes(x = t, y = g a m m a L i s t$G5) ) + g e o m_l i n e(aes(c o l = " red ") , s i z e = 1) +

g e o m_l i n e(aes(y = g a m m a L i s t$A5, c o l = " darkred ") , s i z e = 1 , l i n e t y p e = 2) + c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) ) +

s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction o f Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c(" Stocks M1", " Stocks M2" ) , ← -v a l u e s = c(" red ", " darkred ") ) +

l a b s(t i t l e=" ( b ) Risk Aversion = 5") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 0 . 4 , 0 . 6 5 ) , legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") )

#RA = 10

s t o c k.p l o t. 1 0 < g g p l o t(a v e r s i o n.df.s2, aes(x = t, y = g a m m a L i s t$G10) ) + g e o m_l i n e(aes(c o l = " red ") , s i z e = 1) +

g e o m_l i n e(aes(y = g a m m a L i s t$A10, c o l = " darkred ") , s i z e = 1 , l i n e t y p e = 2) + c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) ) +

s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction o f Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c(" Stocks M1", " Stocks M2" ) , ← -v a l u e s = c(" red ", " darkred ") ) +

l a b s(t i t l e=" ( c ) Risk Aversion = 10") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 0 . 4 , 0 . 6 5 ) ,

legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") )

## RA = 20

s t o c k.p l o t. 2 0 < g g p l o t(a v e r s i o n.df.s2, aes(x = t, y = g a m m a L i s t$G20) ) + g e o m_l i n e(aes(c o l = " red ") , s i z e = 1) +

g e o m_l i n e(aes(y = g a m m a L i s t$A20, c o l = " darkred ") , s i z e = 1 , l i n e t y p e = 2) + c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) ) +

s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Fraction o f Wealth") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c(" Stocks M1", " Stocks M2" ) , ← -v a l u e s = c(" red ", " darkred ") ) +

l a b s(t i t l e=" ( d ) Risk Aversion = 20") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 0 . 4 , 0 . 6 5 ) , legend.key.s i z e = u n i t( 0 . 3 5 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 7) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.key.s i z e = u n i t( 2 , "cm") ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=" t r a n s p a r e n t ") )

g r i d.a r r a n g e(s t o c k.p l o t. 2 , s t o c k.p l o t. 5 , s t o c k.p l o t. 1 0 , s t o c k.p l o t. 2 0 , nrow = 2 , ← -n c o l = 2)

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### Program 8 Loss f u n c t i o n f o r constant investment o p p o r t u n i t i e s ##

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#Content

#( i ) Loss f u n c t i o n f o r r i s k a v e r s i o n f o r Figure 7 . 1

#( i i ) Loss f u n c t i o n f o r time h o ri zo n f o r Figure 7 . 1

#Packages

r e q u i r e(g g p l o t 2)

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#Loss f u n c t i o n Gamma

l o s s.f u n c t i o n.gamma.df < data.frame(l o s s_f u n c t i o n_c o n s t a n t 1) ( .f u n c t i o n.gamma.df)

g e o m_l i n e(aes(x = pi, y = gamma_2 , c o l = " red ") , s i z e = 1) + g e o m_l i n e(aes(x = pi, y = gamma_3 , c o l = " green ") , s i z e = 1) + g e o m_l i n e(aes(x = pi, y = gamma_6 , c o l = " purple ") , s i z e = 1) + s c a l e_x_c o n t i n u o u s(" Fraction o f Risky Assets ") +

s c a l e_y_c o n t i n u o u s(" Welfare Loss (%)") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("RRA = 1", "RRA = 2","RRA = 3","← -RRA = 6") , v a l u e s = c(" blue ", " red ", " green ", " purple ") ) +

l a b s(t i t l e=" ( a ) Time = 10") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 0 . 1 4 , 0 . 5 ) , legend.key.s i z e = u n i t( 0 . 3 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 9) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=a l p h a(' white ', 0 . 4 ) ) )

#Loss f u n c t i o n Time

l o s s.f u n c t i o n.time.df < data.frame(l o s s_f u n c t i o n_c o n s t a n t 2) h e a d(l o s s.f u n c t i o n.time.df)

l o s s.con.time < g g p l o t(l o s s.f u n c t i o n.time.df) +

g e o m_l i n e(aes(x = pi, y = time_1 , c o l = " blue ") , s i z e = 1) + g e o m_l i n e(aes(x = pi, y = time_10 , c o l = " red ") , s i z e = 1) + g e o m_l i n e(aes(x = pi, y = time_20 , c o l = " green ") , s i z e = 1) + g e o m_l i n e(aes(x = pi, y = time_30 , c o l = " purple ") , s i z e = 1) + c o o r d_c a r t e s i a n(x l i m = c( 0 , 2 ) , y l i m = c( 0 , 1 ) ) +

s c a l e_x_c o n t i n u o u s(" Fraction o f Risky Assets ") + s c a l e_y_c o n t i n u o u s(" Welfare Loss (%)") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c("T = 1", "T = 10","T = 20","T = ← -30") , v a l u e s = c(" blue ", " red ", " green ", " purple ") ) +

l a b s(t i t l e=" ( b ) Risk Aversion = 2") + t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=9, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.p o s i t i o n= c( 0 . 1 3 , 0 . 5 ) , legend.key.s i z e = u n i t( 0 . 3 ,"cm") , legend.t e x t = e l e m e n t_t e x t(s i z e = 9) , legend.d i r e c t i o n = " v e r t i c a l ",

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=a l p h a(' white ', 0 . 4 ) ) ) g r i d.a r r a n g e(l o s s.con.gamma, l o s s.con.time, nrow = 2 , n c o l =1)

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### Program 9 Loss f u n c t i o n between model 1 and model 2##

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#Content

#( i ) Figure 7 . 2 Loss f u n c t i o n f o r model and model 2

#Packages

r e q u i r e(g g p l o t 2) r e q u i r e(g r i d E x t r a)

#Load Excel f i l e

l o s s.f u n c t i o n.df < data.frame(l o s s_f u n c t i o n_2) h e a d(l o s s.f u n c t i o n.df)

t a i l(l o s s.f u n c t i o n.df)

#Plot f o r RA=2

p l o t.l o s s.rv2 < g g p l o t(l o s s.f u n c t i o n.df, aes(x = time, y = l o s s_g2) ) + g e o m_l i n e(c o l= " red ", s i z e = 1) +

c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) , y l i m = c( 0 , 8 0 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Welfare Loss (%) ") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c(" Loss " ) , v a l u e s = c(" red ") ) + l a b s(t i t l e=" ( a ) Risk Aversion = 2") +

t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=8, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=a l p h a(' white ', 0 . 4 ) ) )

#Plot f o r RA=5

p l o t.l o s s.rv5 < g g p l o t(l o s s.f u n c t i o n.df, aes(x = time, y = l o s s_g5) ) + g e o m_l i n e(c o l= " red ", s i z e = 1) +

c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) , y l i m = c( 0 , 5 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e__ (" Welfare Loss (%)") +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=8, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=a l p h a(' white ', 0 . 4 ) ) )

#Plot f o r RA=10

p l o t.l o s s.r v 1 0 < g g p l o t(l o s s.f u n c t i o n.df, aes(x = time, y = l o s s_g10) ) + g e o m_l i n e(c o l= " red ", s i z e = 1) +

c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) , y l i m = c( 0 , 1 . 5 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Welfare Loss (%)") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c(" Loss " ) , v a l u e s = c(" red ") ) + l a b s(t i t l e=" ( c ) Risk Aversion = 10") +

t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

p l o t.t i t l e=e l e m e n t_t e x t(s i z e=8, c o l o r=" black ") , legend.j u s t i f i c a t i o n=c( 1 , 0 ) ,

legend.b a c k g r o u n d = e l e m e n t_r e c t(f i l l=a l p h a(' white ', 0 . 4 ) ) )

#Plot f o r RA=20

p l o t.l o s s.r v 2 0 < g g p l o t(l o s s.f u n c t i o n.df, aes(x = time, y = l o s s_g20) ) + g e o m_l i n e(c o l= " red ", s i z e = 1) +

c o o r d_c a r t e s i a n(x l i m = c( 0 , 3 0 ) , y l i m = c( 0 , 1 . 5 ) ) + s c a l e_x_c o n t i n u o u s(" Years ") +

s c a l e_y_c o n t i n u o u s(" Welfare Loss (%)") +

s c a l e_c o l o r_m a n u a l("Legend T i t l e \n",l a b e l s = c(" Loss " ) , v a l u e s = c(" red ") ) + l a b s(t i t l e=" ( d ) Risk Aversion = 20") +

t h e m e_bw( ) +

t h e m e(panel.b a c k g r o u n d = e l e m e n t_r e c t(c o l o u r = " black ", s i z e=1.5) , a x i s.t i c k s = e l e m e n t_l i n e(c o l o r = " black ") ,

panel.g r i d.m a j o r = e l e m e n t_l i n e(c o l o r = " grey ", l i n e t y p e = " dotted ") , panel.g r i d.m i n o r = e l e m e n t_b l a n k( ) ,

legend.t i t l e = e l e m e n t_b l a n k( ) ,

In document Optimal Allocation under a Stochastic Interest Rate and the Costs from Suboptimal Allocation (Sider 129-172)