2.4 Model
2.4.1 Demand For Long-Dated Swaps by Pension Funds
whether it will vanish within a much shorter and practical horizon. To benefit from negative swap spreads arbitrage a high amount of leverage is required and arbitraging negative swap spreads can therefore be seen as a case of “picking up Nickels in front of a steamroller”
(Duarte, Longstaff, and Yu, 2007).13
where we assume CRRA utility with risk-aversion coefficient β > 1 : u(y) = yβ and where the fund optimally chooses its funding rate and asset allocation between interest rate swaps and safe assets to reach a fully funded status at a future point.14 Once the plan is fully funded, it simply dedicates the cash flows from its portfolio to meet its liabilities.
We define a swap as one in which the pension plan receives a fixed dollar amount of 1 per unit time, and pays a floating rate of rt per unit time. The derivatives dealer, who is the counterparty to this IRS, will be introduced in the following section. We abstract from credit risk, which implies that the fixed rate of the swap can be funded at the risk-free floating rate. The value of this swap is P −1, where P is the value of a perpetuity which pays $1 forever. The value of floating payments {rs, s≥ t} is simply 1.15 This is a stylized representation of an interest rate swap, which differs from a newly minted interest rate swap, which will always be valued at zero. Our stylized representation of the swap provides tractable and simple closed form solutions.16 The swap can be regarded as a seasoned swap, which the fund enters into for duration matching purposes. The pension funds can buy m swaps which cost m[P −1]< At < P V[L], where At is the value of the assets held by the pension funds at time t. The remaining fundsAt−m[P −1] are invested in rtat each time.
It should be noted the fund will pay the intermediation costs,δ, per unit time, which affects the cash flows. The cash flows from the swap position is: m(1−r−δ) per unit time.
The term structure model is a simple one-factor model where the instantaneous interest rate r follows a diffusion process, as in Constantinides and Ingersoll (1984):17
dr=αr2dt+sr3/2dw1 (2.2)
For this process, the consol bond price P is:
P = 1
(1 +α−s2)r =H/r, (2.3)
where H ≡ 1+α−s1 2.We can derive the dynamics of the pension fund’s asset value as:
dA= [Ar+m(1−δ) +y−L]dt−mP s√
rdw1. (2.4)
The dynamic problem facing the pension sponsor is specified below: we choose A to be
14We have also solved the model under the alternative assumption that the pension fund can invest in a perpetual bond by borrowing short-term. The results for this assumption are available upon request.
15See, Cox, Ingersoll, and Ross, 1985 for a proof of this assertion.
16Modeling a swap that is zero valued is feasible, but may likely require a numerical approach in the context of our model, where the fund has to also choose asset allocation.
17In Appendix 2.7.1, we briefly characterize the term structure of zero coupon yields implied by this model.
our state variable, and formulate the HJB equation associated with the funds’ optimization problem next:
0 = inf
yt,m
"
yβ−ρG+GA[m(1−δ) +rA+y−L] +1
2GAAm2P2s2r
#
. (2.5)
The fund will close out the swaps, payoff any loans, and stop contributing when the assets are sufficient to meet the present value of the liabilities. Note that whenm =y = 0, the dynamics of assets are:
dA= [Ar−L]dt=r[A− L
r]dt=r[A−LP(1 +α−s2)]dt (2.6) When A ↑ A∗ where A∗ = LP(1 +α−s2), note that A∗r =L, and the fund can meet its liabilities from its assets, and the value function goes to zero. This leads to the following boundary condition for the HJB equation above:
G(A↑A∗) = 0. (2.7)
The above condition follows from the fact that the cost of funding goes to zero when the assets are sufficient to meet the liabilities. Let us define Ψ≡A∗−A.We can now characterize the demand functions of the pension fund and the optimal funding policy.
Demand Functions
Proposition 2. The sponsor’s optimal contribution and optimal asset allocation are given as:18
y =
rβ−ρ− 12β
(1−δ)2 (β−1)P2s2r
(β−1) Ψ (2.8)
and
m∗ = (1−δ)(1 +α−s2)Ψ
(β−1)P s2 . (2.9)
The proof of Proposition 2 can be found in the appendix. Note that ∂m∂δ∗ <0 and ∂m∂P∗ >0, which makes intuitive sense. The fund’s demand for IRS falls when the intermediation costs (negative swap spreads) are higher. As interest rates go down, P increases, and this leads to a higher demand for IRS.
18We require that rβ−ρ−12β
(1−δ)2 (β−1)P2s2r
>0. In addition, the intermediation costs represented by δ cannot be too high, i.e.,δ <1.
When the underfunding is high, the fund uses more IRS and funds more aggressively:
this is in fact the basic implication of our model. Asβ increases, the pension fund increases y much more and reduces its positions in IRS: this suggests that funding requirements imposed by regulators may have beneficial implications for the way in which pension assets are managed by the sponsors.
Model with Stocks, Swaps and Safe Assets
We now extend our model to allow the pension fund to additionally invest in a risky asset.
To keep the model simple, we introduce a generic risky asset which can be interpreted as stock portfolio. The price of the stock portfolio follows a geometric Brownian motion:
dS =Sµdt+Sσdw2. (2.10)
We allow the processes{w1, w2}to be correlated with each other with correlation coefficient Rand introduce the notationσ12:=sσR.The fund invests innshares of the stock portfolio, m swaps and places the remainder in risk-free asset.
Proposition 3. The sponsor’s optimal contribution and optimal asset allocation are given as:
y=gβ−11 Ψ, (2.11)
nS =
"
µ−r σ2
+Pσσ122
√r
1−δ P2s2r
#
(β−1)
1− sσ2212σ2
Ψ≡λ1Ψ, (2.12)
and
m =
1−δ P2s2r
+P sσ212√r
µ−r σ2
(β−1)
1−sσ212σ22
Ψ≡λ2Ψ, (2.13)
where g is given as:
g = 1
1−β
(ρ+β(λ1(µ−r) +λ2(1−δ)−r)
− 1
2β(β−1) λ21σ2+λ22P2s2r−2λ1λ2P√ rσ12
β−1
. (2.14)
UFR (%)
0 10 20 30 40 50 60 70 80 90 100
Optimal Stock Holdings (% of wealth)
0 5 10
Optimal Swap Holdings (% of wealth)
0 1 2 3 Stocks
Swaps
Figure 2.3: Pension funds’ optimal holdings of stocks and swaps. UFR is computed as (L/r−A)/A. Parameter choices are: β = 10, ρ= 0.05, µ= 0.06, σ= 0.2, α= 0.1, s= 0.3, R= 0, P = 50,andδ = 50·10−4.
The proof of Proposition 3 can be found in the appendix. We now illustrate the pension fund’s optimal holdings of stocks and swaps in a numerical example. We define the pension fund’s underfunded ratio as: U F R = L/r−AA and choose the following parameters for our illustrations: β = 10, ρ = 0.05, µ = 0.06, σ = 0.2, α = 0.1, s = 0.3, R = 0, P = 50, and δ= 50·10−4,corresponding to a negative swap spread of 50 basis points.
As we can see from Figure 2.3, both, risky asset holdings and the amount of swaps held increase with UFR. ForU F R↓0 the fund closes out his risky asset and swap holdings and pays off any loans. It is important to note that the pension fund increase both the exposure to swaps and risky assets as the fund gets more underfunded: the increase in risky asset is due to a desire to get out of the underfunded status in the future. On the other hand, the increase in swap position is to manage the duration risk to prevent future losses arising from interest rate changes.