The State Price Matrix

Using the previously estimated parameters for our model and the estimation procedure for state prices described in section 2.7.1, we can estimate the state price vectors for the respective market states, which are aggregated into a state price matrix. We start by using our return bounds from table 5.3 to calculate the strike prices for claims on the underlying asset. We note that as we define the states of the markets in terms of returns, the level of the market portfolio

CHAPTER 5. THE PROPOSED MODEL

is irrelevant. Hence, we normalize all strike prices to a current level on the market portfolio of 1. If the current market level is 1, all strike prices can be calculated as K = exp(B), where B is a return bound. The resulting strike prices are

Low Mid High

[0.2931, 0.9617] (0.9617, 1.0967] (1.0967, 3.5928]

Table 5.7: Strike prices (absolute states)

From equation (2.35), we have that the value of a state-contingent claim on state s is v_s = exp(−rt)

(1−Φ(d^HK₂ ^s)−(1−Φ(d^LK₂ ^s)

Where LK_s and HK_s stand for the lower and upper bounds of state s respectively, and d₂ is d^K₂^s = ln(1/K_s) + (r−¹₂σ²_s0)t

σ_s0√ t

Here K_s is any of the two strikes corresponding to state s. Note that the volatility is also state-contingent, which is indicated by the s₀ subscript. However, note that the 0 subscript indicates that the volatility depends on the state we are currently in. Calculating the claims on the states from table 5.7 three times, each time with a different volatility corresponding to the estimations from table 5.5, we obtain three state price vectors, which we aggregate into the following state price matrix.

1 year (V) Low Mid High Low 0.4798 0.2418 0.2841 Mid 0.4276 0.3679 0.2101 High 0.4228 0.3802 0.2026 Table 5.8: 1-Year state price matrix

As one would expect, investors are willing to pay more for state-contingent assets giving them a certain payoff of 1 unit in low states of the market portfolio. Furthermore, as shown in section 2.7.3, one can simply take the matrix power of the state price matrix to obtain the state prices for periods further into the future. Below we report the state price matrices for 2 and 5 years (state price matrices for 10 consecutive years can be seen in appendix .3).

2 year (V²) Low Mid High Low 0.4536 0.3130 0.2446 Mid 0.4513 0.3186 0.2413 High 0.4511 0.3192 0.2410 Table 5.9: 2-Year state price matrix

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5 year (V⁵) Low Mid High Low 0.4599 0.3216 0.2469 Mid 0.4599 0.3216 0.2469 High 0.4599 0.3216 0.2469 Table 5.10: 5-Year state price matrix

We note that the row-sums are in fact increasing when we take powers of the matrices that use a negative real risk-free rate. However, this is only allowed for in the forecasting period.

In addition to the regular state price matrices, we also estimate a perpetuity matrix. The perpetuity matrix is used for calculating the value of any project in the terminal period, and requires a positive risk-free rate. We will now show how it is derived and thereafter report the estimated perpetual values.

Derivation of the Perpetuity Matrix

We start of by giving an intuitive statement describing the perpetuity matrix

The perpetuity matrix is the matrix of prices an investor is willing to pay to receive 1 unit every year in perpetuity in a specific state, when standing in a specific state today

Hence, we can easily deduce that the perpetuity matrix must be the sum of all the state price matrices in perpetuity. In other words, it must be the sum of an infinite series. Let V∞ = V+V²+· · · denote this infinite series. Furthermore, let S∞ denote the infinite series I+V∞, such that

S∞ =I+V+V²+· · · If we multiply this series by V we get

VS∞=VI+V²+V³+· · ·=V+V²+V³+· · ·=V∞ (5.6) Which when subtracted from S∞ gives

S_∞−VS_∞= (I+V+V²+· · ·)−(V+V²+V³+· · ·) =I We can rewrite this as

(I−V)S∞=I Equivalently

S∞= (I−V)⁻¹ (5.7)

Furthermore, it is crucial that the infinite sumS∞converges for (5.7) to hold. S∞will converge as long as the series sums to less than infinity. I.e. it will converge as long as powers of

n

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approaches ∞. Formally, for the partial sum of the infinite series,S_n =I+V+· · ·+Vⁿ, we have

S_n−VS_n= (I+· · ·+Vⁿ)−(V+· · ·+Vⁿ⁺¹) = I−Vⁿ⁺¹ Furthermore, since S_n−VS_n= (I−V)S_n, we have

(I−V)Sn=I−Vⁿ⁺¹

IfVⁿ⁺¹ →0asn → ∞, then(I−V)S_n→Iasn → ∞. Which is the same as S_n→(I−V)⁻¹ as n→ ∞. Furthermore, we know that Vⁿ⁺¹ →0as n→ ∞ if the spectral radius of V is less than one, i.e. if ρ(V) = sup{|λ₁|,|λ₂|,|λ₃|} < 1, where λ_i denotes the eigenvalues of V. This can bee seen from the following

Assuming that V is an 3×3diagonalizable matrix, then V can be written as V=PDP⁻¹

Where the columns of Pare 3 linearly independent eigenvectors of V, and the diagonal entries of D are the corresponding eigenvalues. We note that this is in fact the case for V. Since it can be shown that in general, for n ≥1

Vⁿ=PDⁿP⁻¹ we have

Vⁿ⁺¹ =P











λⁿ⁺¹₁ 0 0 0 λⁿ⁺¹₂ 0 0 0 λⁿ⁺¹₃









 P⁻¹

From this we can easily see that if for all eigenvalues |λ| < 1, then D will become the zero matrix asn→ ∞. Since the eigenvalues of the 1-year matrix used for calculating the perpetuity matrix are {λ₁ = 0.9900, λ₂ = 0.0614, λ₃ = −0.0008} (see appendix .8 for matrix), we have that Vⁿ⁺¹ goes to zero in the limit, which as stated above implies that S_n converges. It then follows that S∞converges. Note that we use the one year matrix V with 1% interest rate since this is the interest rate to be used in the perpetuity matrix. Returning to equation (5.6), we can now write the expression for the perpetuity matrix as

V∞=VS∞=V(I−V)⁻¹ (5.8)

Which is the same expression for the perpetuity matrix that Banz and Miller (1978) provide.

Note: The single number, or 1×1 matrix, familiar result is that the seriesP∞

i=0Kⁱ converges to _1−K^K if |K| <1, where K is a discount factor. This leads to the well-known result that the perpetuity value of a simple cash flow, CF, with discount rate k, can simply be calculated as

CF k .

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Estimated perpetuity matrix

Using equation (5.8) we estimate the following perpetuity matrix.

(V∞) Low Mid High

Low 40.3551 32.3719 26.7738 Mid 40.2835 32.5107 26.7066 High 40.2768 32.5245 26.6995 Table 5.11: Perpetuity state price matrix

Now that we have derived the state price matrices, we will present a stylized example on how to use them in a valuation.

CHAPTER 5. THE PROPOSED MODEL