Fixed Income Portfolio Management
Tomasz R. Bielecki and Stanley R. Pliska
Abstract. This paper presents an application of risk sensitive control theory in financial decision making. Specifically, we develop optimal, risk-sensitive in-vestment strategies for a long-term investor who is interested in optimal allo-cation of her/his capital between cash, equities and fixed income instruments. The long-term fixed income instruments used are so called rolling-horizon bonds. In order to construct the optimal risk-sensitive control policies rele-vant for the present application we advance the risk sensitive control theory developed in our previous papers.
1. Introduction
Beginning with the pioneering work by Merton [19, 20, 21] and continuing through the recent books by Karatzas and Shreve [16] and Korn [18], some very sophisti-cated stochastic control methods have been applied to portfolio management. But most of these applications have been concerned with, at least implicitly, the man-agement of equities. In spite of an abundance of well known mathematical interest rate models, exemplified by the classical models of Vasicek [28] and Heath, Jarrow, and Morton [13], one can find very few applications of modern control theory to fixed income management.
There are at least two possible explanations for this deficiency in the litera-ture. First, most fixed income assets have finite lives, so they cannot be modeled by simple stochastic processes such as the infinitely-lived geometric Brownian mo-tions that are commonly used for equities. Moreover, the maturing of bonds before the planning horizon causes modeling difficulties by forcing discontinuities in the trading strategy. Another possible explanation for the limited number of control JEL Classification.C61, C63, G11, E21.
1991Mathematics Subject Classification. 60J20, 90A09, 90C40, 93E20.
Key words and phrases. risk sensitive control, portfolio management, fixed income management, intertemporal capital asset pricing model.
The research of the authors was partially supported by NSF Grant DMS-9971307, and NSF Grant DMS-9971424, respectively. We appreciate helpful comments by Marek Musiela and Marek Rutkowski as well as by participants at the December 1998 control theory workshop at the University of Southern California.
theory applications to fixed income management is that, unlike equities, the risk and return characteristics of fixed income assets explicitly depend upon underlying exogenous factors, namely, levels of interest rates. Consequently, sensible trading strategies will need to be explicit functions of these interest rates, and so these interest rate factors need to be explicitly incorporated in any mathematical model. Of course a stochastic control model having a general form suitable for fixed income management was formulated more than 25 years ago by Merton [20]. His famous intertemporal capital asset pricing model (ICAPM) features both assets and the exogenous factors which affect them, all of which are modeled as diffusion processes. For the problem of maximizing expected utility of consumption and/or terminal wealth, he then derived and provided economic interpretations of the cor-responding Hamilton-Jacobi-Bellman equation. It is straightforward, in principle, to adapt his model to fixed income management, namely, by letting the factors be interest rates in accordance with a desired interest rate model and then letting the assets be securities like zero coupon, discount bonds. Unfortunately, this approach is unlikely to be successful because, except for a very few special cases, the ICAPM is intractable.
In one successful special case Kim and Omberg [17] derived explicit solutions of the ICAPM where the interest rate is constant and the Sharpe ratio of the only available risky asset follows an Ornstein-Uhlenbeck process. Canestrelli [8] and Canestrelli and Pontini [9] obtained explicit solutions where the short interest rate is described by an Ornstein-Uhlenbeck process and there is also a single risky asset. In all these cases the only fixed income asset is the bank account which earns interest at the short, locally riskless rate.
There have also been a few studies of the ICAPM where one or more of the risky assets are taken to be fixed income securities. Brennan and Schwartz [7] and Brennan, Schwartz, and Lagnado [6] used numerical methods to solve the Hamilton-Jacobi-Bellman partial differential equation for the optimal trad-ing strategy, but they were hard pressed to derive a solution even though there were only three factors and a similar number of assets. Explicit results on the application of the ICAPM to fixed income management were recently obtained by Bajeux-Besnainou, Jordan, and Portait [1] and Sørensen [27]. Apparently working independently, they studied the same special case. They both assumed there is one factor, an Ornstein-Uhlenbeck process for the short, locally riskless interest rate, and there are three assets. The first is a bank account where money grows according to the short rate. The second is the zero coupon bond which matures at a fixed planning horizonT. With the Brownian motion driving the short rate being the sole Brownian motion affecting the term structure of interest rates, the zero coupon bond thus evolves according to the well-known model due to Vasicek [28]. The third asset is a stock which evolves according to a process that is a geometric Brownian motion except that the appreciation rate equals the short interest rate plus a constant; the Brownian motion driving this process is correlated with the one driving the interest rate term structure.
Both studies focus on the problem of maximizing expected utility of wealth at a finite planning horizon T, where the utility function is of the form u(w) =
w1−γ/(1−γ) andγ > 0 is the parameter of constant relative risk aversion. For
γ= 1 one actually has as a special limiting caseu(w) = ln(w), giving rise to what is sometimes called the growth optimal or numeraire portfolio. Note that increasing values of the parameterγ correspond to increasing levels of risk aversion for the investor.
Using the risk neutral computational approach introduced by Pliska [24, 25], both studies derived the same general form for the optimal trading strategy: at every point in time hold the fraction 1/γ of one’s wealth in the growth optimal portfolio and invest the rest in the zero coupon bond. This makes some intuitive sense, because the zero coupon bond is the riskless asset for this problem, and the bigger the value ofγ, the bigger the proportion in this asset. Unfortunately, however, their solutions leave unresolved a troublesome issue. Although their pro-portions in the stock are constants, their propro-portions in the zero coupon bond are deterministic functions of time which are unbounded in every neighborhood of the planning horizonT. This difficulty is associated with the fact that the zero coupon bond’s volatility converges to zero as time approaches maturity.
Research in mathematical control theory has progressed to the point where it is likely that soon there will be explicit solutions of much more general versions of Merton’s ICAPM. This will be very important, both for fixed income management as well as many other kinds of consumption/investment problems. However, in this paper we pursue a variation of the ICAPM where explicit, general solutions can be obtained right now. Retaining the same diffusion process model for assets and factors, our approach calls for changing from Merton’s finite horizon expected utility objective to the infinite horizon objective of maximizing the risk adjusted growth rate. As explained fully in our earlier work [2, 3, 4, 5], this criterion can be viewed as being analogous to the classical Markowitz single-period model except that instead of trading off single-period criteria we are trading off the long run growth rate (which by itself is maximized by the growth optimal portfolio) versus the average volatility of the portfolio. Our risk adjusted growth rate objective emerges naturally from the application of recent mathematical results on risk sensitive control theory. The principal benefit of this objective is that it is an infinite horizon criterion and therefore, as with most control problems in general, the ICAPM is more tractable than if a finite horizon criterion is used.
Our initial work in this direction developed a model with Gaussian factors and with assets having constant volatilities and appreciation rates that are affine functions of the factor levels. The theoretical foundations of this model are found in [2], while its applications to asset allocation problems are explored in [5]. But while this model fully allows for there to be correlations between asset returns and movements of the factor levels, it has a critical shortcoming: the partial correlations between asset returns and movements of the factor levels must be zero, that is, the residuals of the asset returns must be independent of the residuals of the factors. While this assumption is reasonable for applications where the only interest rate
asset is the bank account, it is unacceptable for models which include both interest rate factors and additional fixed income assets.
In this paper we develop a rather general model featuring our risk sensitive criterion and, as in the ICAPM, exogenous factors which explicitly affect the dif-fusion process dynamics of the assets. The details and main results are provided in the following section. A special case of our model was recently studied by Fleming and Sheu [12]. They obtained sub-optimal investment strategies for small values of the risk sensitivity parameter.
In section 3 we turn to the special case of fixed income management. By utilizing the concept of rolling-horizon bonds, a concept introduced by Rutkowski [26], we are able to incorporate fixed income assets having infinite lives. These assets can be viewed, roughly, as mutual funds of zero coupon bonds, all of which mature at about the same fixed distance in the future; these bonds are rolled over in a self financing manner so this same fixed distance is preserved through the course of time. We provide explicit theoretical results for a model having three kinds of assets: the bank account, rolling horizon bonds, and (possibly) equities; all the factors are various interest rates. Proofs and additional results will be provided in a future, extended version of this paper.
2. Formulation of the Optimal Risk Sensitive Asset Management
Problem and the Main Results
In this section we formulate a general optimal dynamic asset management problem and we provide the characterization of optimal investment strategies. We consider a market consisting ofn≥2 securities andm+1,m≥0, economic factors. The set of securities may include stocks, certain fixed income assets, cash and derivative securities, as in Brennan, Schwartz and Lagnado [6] and Bajeux-Besnainou, Jor-dan, and Portait [1] for example. The set of factors may include dividend yields, price-earning ratios, short-term interest rate, yields on various bonds, the rate of inflation, etc., as in Pesaran and Timmermann [22, 23] for example.
Let (Ω,{Ft}t≥0,F,P) be the underlying probability space. Denoting bySi(t)
the price of thei-th security and byXj(t) the level of thej-th factor at timet, we
consider the following market model for the dynamics of the security prices and factors: dSi(t) Si(t) = (a+AX(t))idt+ N X k=0 σikdWk(t), Si(0) =si, i= 1,2,· · · , n (1) dX(t) = (b+BX(t))dt+ ΛdW(t), X(0) =x , (2) whereW(t) = (W0(t), W1(t), . . . , WN)0 is aRN+1valued standard Brownian
mo-tion process, X(t) = (X0(t), X1(t), . . . , Xm)0 is the Rm+1 valued factor process,
the market parameters a, A, Σ := [σij], b, B, Λ := [Λij] are matrices of
It is well known that a unique, strong solution exists for (1), (2), and that the processesSi(t) are positive with probability 1 (see e.g. [15], chapter 5).
Let Gt :=σ((S(s), X(s)),0 ≤s ≤t), where S(t) = (S1(t), S2(t), . . . , Sn(t))
is the security price process. Leth(t) denote anRn valued investment process (or
strategy) whose componentshi(t), i= 1,2, . . . , n, represent the time-tproportions
of wealth in the corresponding assets.
Definition 2.1. An investment processh(t)isadmissibleif the following conditions are satisfied:
(i) Pni=1hi(t) = 1,
(ii) h(t)is measurable andGt-adapted,
(iii) for every θ >0 there exists a probability measurePh,θ on(Ω,F)so that
dPh(·),θ dP Ft =ηt(h(·), θ), where ηt(h(·), θ) =exp{−θ 2 8 Z t 0 k h0(s)Σk2ds−θ2 Z t 0 h0(s)ΣdWs},
(iv) lim supt→∞t−1lnEh(·),θexp{
θ
2
(X0
tK1(θ)Xt+K2(θ)Xt} ≤0, where the
matrixK1(θ)andK2(θ)are matrices defined later in the text, andEh(·),θ
denotes expectation under Ph(·),θ.
The class of admissible investment strategies will be denoted byH.
Let nowh(t) be an admissible investment process. Then there exists a unique, strong, and almost surely positive solutionV(t) to the following equation:
dV(t) = m X i=1 hi(t)V(t)µi(X(t))dt+ N X k=0 σikdWk(t), V(0) =v >0, (3)
whereµi(x) is thei-th coordinate of the vector a+Axforx∈Rm+1. The process
V(t) represents the investor’s capital at timet,andhi(t) represents the proportion
of capital that is invested in security i, so that hi(t)V(t)/Si(t) represents the
number of shares invested in securityi, just as in, for example, Section 3 of [14]. In this paper we consider the following family of risk sensitized optimal in-vestment problems, labeled asPθ:
forθ∈(0,∞), maximize the risk sensitized expected growth rate Jθ(v, x;h(·)) := lim inf
t→∞ (−2/θ)t
−1lnEh(·) [e−(θ/2)lnV(t)|V(0) =v, X(0) =x] (4)
over the class of all admissible investment processesh(·), subject to (1) and (2) ,
where Eh(·) is the expectation with respect to P. The notation Eh(·) emphasizes that the expectation is evaluated for process V(t) generated by (3) under the investment strategyh(t).
Remark 2.2. The positive value of the risk sensitivity parameter θ corresponds to a risk averse investor. The techniques used in this paper can also be used to study problemsPθ for negative values ofθ, corresponding to risk seeking investors. The
risk null case, for θ = 0, can be studied independently or as the limit of the risk averse situation when the risk sensitivity parameter θ goes to zero. However, we shall not consider the case ofθ≤0 here.
We make the following standing assumptions:
Assumption (A1)The spectrum of the matrixB is contained in the left half plane [i.e. the matrixB is stable].
Assumption (A2)(a) The matrix ΣΣ0 is positive definite; (b) The matrix ΛΛ0 is
positive definite.
Remark 2.3. It is worth emphasizing at this point that we do not assume indepen-dence of random perturbations driving the price and the factor dynamics. That is, we do not assume thatΣΛ0 = 0. Such an assumption was made in[2].
Let us consider now the following continuous algebraic Riccati equation (CARE) K0R1(θ)K+K0R2(θ) +R2(θ)0K+R3(θ) = 0, (5) where R1(θ) := θ 2 2 θ 2+ 1 −1 ΛΣ0ΨΣΛ0−θΛΛ0, R2(θ) :=− θ 2 θ 2+ 1 −1 ΛΣ0ΨA+B , R3(θ) := (1/2)( θ 2 + 1 −1 A0ΨA , and Ψ := (ΣΣ0)−1−Γ, Γ := (ΣΣ0)− 1110(ΣΣ0)−1 10(ΣΣ0)−11 , 1= (1,1, . . . ,1)0.
In order to formulate our main results we need one more assumption:
Assumption (A3) For every θ > 0 the equation (5) admits a unique, positive semi-definitive solution, say K1(θ). Moreover, the matrix G(θ) defined below is
stable:
G(θ) :=R2(θ) +R1(θ)K1(θ). (6)
Remark 2.4. Standard controllability and observability conditions imposed on the parameters of problems Pθ will guarantee that Assumption (A3) is satisfied. See
In view of the Assumption (A3), for every θ > 0 we may define a matrix K2(θ) as follows: K2(θ) := G(θ)0−1 ( A−θΣΛ0K1(θ) 0h (ΣΣ0)−11 10(ΣΣ0)−11−( θ 2+1 −1 Ψai−2K1(θ)b0 ) . (7) Theorems 2.5 and 2.6 below contain the main results of this paper. Theorem 2.5 characterizes optimal, risk-sensitive investment strategies. Theorem 2.6 character-izes the optimal value of the objective criterion.
Theorem 2.5. Assume (A1), (A2) and (A3). Fix θ >0 and consider the process hθ(t)defined as hθ(t) := (θ 2 + 1 −1 (ΣΣ0)−1hβ(X t) +λ(Xt)1 i , (8) where β(x) :=A−θΣΛ0K1(θ)x+a−θ 2ΣΛ 0K 2(θ), (9) λ(x) := ( θ 2+ 1 −10(ΣΣ0)−1β(x) 10(ΣΣ0)−11 , (10)
the matrixK1(θ) is as in Assumption (A3), and the corresponding matrixK2(θ)
is defined in (7). Then for θ sufficiently small the investment process hθ(t) is
optimal for problem (Pθ), that is,
Jθ(v, x;h(·))≤Jθ(v, x;hθ(·))
holds for allh(·)∈ H,v >0,x∈Rm+1.
Theorem 2.6. Assume (A1), (A2) and (A3), fix θ > 0 sufficiently small, and consider the problem Pθ. Let hθ(t) be as in theorem 2.5. Then
(a) For allv >0 andx∈Rm+1 we have
Jθ(v, x;hθ(·)) = lim
t→∞(−2/θ)t
−1lnEhθ(·) [e−(θ/2)lnV(t)
|V(0) =v, X(0) =x] =:ρ(θ).
(b) The constantρ(θ)in (a) is given by the formula ρ(θ) =b0K2(θ) + (1/2) h (−θ/2)K2(θ)0ΛΛ0K2(θ) + 2trΛ0K1(θ)Λ i + (1/2)θ 2+ 1 −1 a−(θ/2)ΣΛ0K2(θ) 0 Ψa−(θ/2)ΣΛ0K2(θ) − 1 0(ΣΣ0)−1 10(ΣΣ0)−11(a−(θ/2)ΣΛ0K2(θ) +(1/2) θ 2+ 1 10(ΣΣ0)−11 , (11)
where the matrix K1(θ) is as in Assumption (A3) and the corresponding
Remark 2.7. The key point of the first equality in(a)is, of course, that the optimal objective value is given by an ordinary lim rather than thelim inf as in (4). The key point of the second equality in(a)is that the optimal objective value does not depend on either the initial amount of the investor’s capital (v) or on the initial values of the underlying economic factors (x), although it depends, of course, on the investor’s attitude towards risk (encoded in the value ofθ).
Remark 2.8. It appears that the results of theorems 2.5 and 2.6 are stronger than the results of Section 7 in Fleming and Sheu[12], where only suboptimal investment strategies are discussed.
Remark 2.9. Even though we require in theorems 2.5 and 2.6 that the parameter θ is sufficiently small [the case that we can prove at the moment], we conjecture that the theorems remain true for allθ positive.
Remark 2.10. Theorems 2.5 and 2.6 are demonstrated in the Appendix.
2.1. The case where one of the assets is “risk free”
Let us consider the market (1) and (2) provided with an additional instantaneously risk-free asset, sayS0(t), whose dynamics are given as
dS0(t)
S0(t)
= (a0+A0X(t))dt, S0(0) =s0. (12) Leth(t) denote anRn+1-valued investment process or strategy whose components are hi(t), i= 0,1,2, . . . , n, and which satisfies conditions analogous to the ones
specified in definition 2.1. It is convenient to partitionh(t) as
h(t) = (h0(t),˜h(t)). Define
˜
A:=A−1A0, ˜a:=a−1a0, Ψ := (ΣΣ˜ 0)−1.
In order to characterize the optimal investment strategyhθ(t) in this case we need
to consider the CARE (5) and the formulas (6) and (7) withA a, and Ψ replaced with ˜A, ˜aand ˜Ψ, respectively. Then we have that an optimal investment strategy is given as hθ(t) = (hθ0(t),˜hθ(t)), (13) where ˜ hθ(t) = (θ/2 + 1)−1Ψ˜A˜−θΣΛ0K˜1(θ) Xt+ ˜a−(θ/2)ΣΛ0K˜2(θ) , (14) hθ0(t) = 1−10˜hθ(t), (15) ˜
K1(θ) is the unique solution to (5) withA and Ψ replaced with ˜Aand ˜Ψ, respec-tively, and ˜ K2(θ) =− ˜ G(θ)0 −1 (θ/2)(θ/2 + 1)−1A˜−θΣΛ0K˜1(θ)0Ψ˜˜a+A0 0+ 2 ˜K1(θ)0b ! . (16)
In the above formula the matrix ˜G(θ) is defined just asG(θ) withAand Ψ replaced with ˜Aand ˜Ψ, respectively.
Finally, in this case we have
ρ(θ) =(1/2)(θ/2 + 1)−1 ˜ a−(θ/2)ΣΛ0K˜2(θ) !0 ˜ Ψ ˜ a−(θ/2)ΣΛ0K˜2(θ) ! + (1/2) " −(θ/2) ˜K2(θ)0ΛΛ0K˜2(θ) + 2trΛ0K˜1(θ)Λ # +b0K˜2(θ). (17)
In section 4 we shall apply the general results in this section to a fixed income management problem where there are a money market account, some fixed income assets, and possibly some equities.
3. Rolling-Horizon Bonds and Gaussian Yield-Factor Models
In the finance literature one can find studies where the authors assume infin-itely-lived geometric Brownian motion models of fixed income securities, but this choice of model is never justified. In particular, no arguments can be found which start with a specific interest rate model together with a specific fixed income asset and then derive the stochastic dynamics of this asset’s price process, showing this process to be a geometric Brownian motion. A purpose of this section is to make such an argument.
The appropriate asset is called a rolling-horizon bond, a concept introduced by Rutkowski [26]. To understand this it is convenient to assume the so-called Heath-Jarrow-Morton framework for interest rate models, namely, that for any maturity date T the instantaneous forward rate f(·, T) satisfies the stochastic differential equation
df(t, T) =α(t, T)dt+ν(t, T)·dWt, (18)
where α(·, T) and ν(·, T) are adapted stochastic processes, W is a standard (possibly vector valued) Brownian motion defined on a filtered probability space (Ω,{Ft}t≥0,F,P), and P is the actual, real-world probability measure. Letting
P(t, T) denote the time-t price of a zero-coupon (i.e., pure discount) bond that pays one unit of cash at timeT, it is well known (see Heathet al. [13]) that under a martingale measureP∗ the dynamics ofP are given by
dP(t, T) =P(t, T)(rtdt+b(t, T)·dWt∗), (19)
whereW∗is a standard (possibly vector valued) Brownian motion underP∗,r
t:=
f(t, t) is the short-term, locally riskless interest rate, andb(t, T) :=−RtTν(t, u)du
is the bond’s volatility.
Now consider a self-financing trading strategy in which the total wealth is continuously reinvested in zero-coupon bonds which mature exactlyT time units in the future. This strategy is a bit abstract, but it can be viewed as the limit as δ→0 of the self-financing strategy which everyδ time units rolls over all the
funds from the bonds that mature inT−δtime units into bonds that mature inT
time units. The wealth process denotedU(·, T) of such a strategy is called a rolling horizon bond. As reported by Rutkowski [26], the dynamics ofU(·, T) under the martingale measureP∗ are
dU(t, T) =U(t, T)(rtdt+b(t, t+T)·dWt∗). (20)
Meanwhile, under the actual probability measurePthe rolling-horizon bond process will have the same volatilityb(t, t+T), while its drift will depend in a simple way upon the market price of risk (i.e., the Girsanov transformation relating P
andP∗). In particular, and this is the case that will be studied in this paper, for the Gaussian HJM model, where the diffusion coefficient ν in the forward rate SDE is nonrandom, the volatility b(t, t+T) of the rolling-horizon bond is also a deterministic function of time. Hence the rolling-horizon bond can be a geometric Brownian motion whose appreciation rate depends upon the market price of risk. We now turn to the interest rate model that is the subject of this paper. We fix
ktimesT1< T2<· · ·< Tk, whereT1>0, and consider the yieldsYt1, Yt2, . . . , Ytk
of the corresponding sliding bonds (this is Rutkowski’s [26] terminology)P(t, t+
Ti), i= 1,2, . . . , k.In other words,
Yti=−
1
Ti
lnP(t, t+Ti), i= 1,2, . . . , k . (21)
There is also the usual short-term, locally riskless interest ratert, the yield
corre-sponding to a zero coupon bond having a maturity of zero, if you will. Thesek+ 1 interest rates will be the factors affecting and thus characterizing our interest rate model. We assume they satisfy, under the actual, real-world probability measure
P, the system of stochastic differential equations:
drt= (b0+B00rt+B01Yt1+· · ·+B0kYtk)dt+ N X j=0 λ0jdWj(t), (22) dYti= (bi+Bi0rt+Bi1Yt1+· · ·+BikYtk)dt+ N X j=0 λijdWj(t), i= 1,2, . . . , k . (23)
Herebi, Bij,andλil are all fixed constants for alli, j= 0,1, . . . , k,l= 0,1, . . . , N
andW = (W0, W1, . . . , WN) is a standard (N + 1)-dimensional Brownian motion
underP, as before. Note that the (k+1)-dimensional factor process (r, Y1, . . . , Yk)
is Markovian with drift coefficients that are affine functions of the factor levels and with constant diffusion coefficients.
We now make an important
Assumption (A4).The market price of risk is an affine function of thek+1interest rate factors.
The market price of risk, of course, provides the connection via Girsanov between the real-world probability measure P and the martingale measure P∗. Therefore, as a consequence of our assumption (or “equivalently”, if you prefer),
underP∗the dynamics of thek+ 1 interest rate factors will have exactly the same form as underP. In other words, under P∗ the SDE’s describing the interest rate dynamics will have drift coefficients that are affine functions of the interest rate levels and diffusion coefficients that are constants.
We thus have specified what is a special case of the affine yield-factor model developed by Duffie and Kan [10]. Accordingly, there exist deterministic, real-val-ued functionsα(·), β0(·), . . . , βk(·) on [0,∞) such that the price of any sliding bond
is given by
P(t, t+T) =exp[α(T) +β0(T)rt+β1(T)Yt1+· · ·+βk(T)Ytk], ∀t≥0. (24)
These k+ 1 deterministic functions can be computed from the parameters in equations (22) and (23) together with the market price of risk parameters, but doing so will not be necessary for our purposes. Instead, it suffices to observe that by taking T =Ti for i = 1,2, . . . , k and using equation (21) we must have (see
also (5.2) of Duffie and Kan [10])
α(Ti) =βj(Ti) = 0, j6=i;βi(Ti) =−Ti, i, j= 1, . . . , k . (25)
We shall use this to derive the real-world dynamics of the rolling-horizon bonds. Applying Ito’s formula to the above expression for the sliding bond we obtain the following real-world dynamics:
dP(t, t+T)/P(t, t+T) = (. . .)dt+β(T)Λ·dW(t), (26) where β(T) := (β0(T), β1(T), . . . , βk(T)) is a row vector and Λ := (λij) is the
(k+ 1)×(N+ 1) matrix of diffusion coefficients (as in (22), (23) as well as the corresponding risk neutral SDE’s for the interest rates); the drift coefficient here does not matter, so we omit the details.
In particular, whenT =Ti for anyi= 1, . . . , k we can apply (25) and thus
obtain
dP(t, t+Ti)/P(t, t+Ti) = (. . .)dt−Tiλi0dW0(t)− · · · −TiλiNdWN(t). (27)
As explained by Rutkowski [26], it thus follows that our bond price volatilities
b(t, T) satisfy
b(t, t+Ti) =−Ti(λi0, λi1, . . . , λiN), i= 1, . . . , k . (28)
Hence substituting this in (20) and using again our assumption that the market price of risk is an affine function of the k+ 1 interest rate factors, we obtain the following characterization of the real-world dynamics of thek rolling-horizon bonds corresponding to thekspecified maturities and interest rate factors:
dU(t, Ti)/U(t, Ti) = (αi+Ai0rt+Ai1Yt1+· · ·+AikYtk)dt −Ti N X j=0 λijdWtj, i= 1. . . , k . (29)
Hereαi andAij fori= 1, . . . , kandj= 0,1, . . . , kare all fixed constants that can
U(t, Ti) is an asset price which under the martingale measure has drift coefficient
rtU(t, Ti) and the same diffusion coefficient as under the real-world probability
measure). Alternatively, and this is more convenient for implementation purposes, one can estimate these drift parameters directly, ignoring the market price of risk parameters.
In the following section we incorporate this Gaussian yield-factor model and the corresponding rolling-horizon bonds as part of the general risk sensitive asset management model that was developed in section 2.
4. Optimal Dynamic Risk-Sensitive Management of Cash, Equities
and Bonds
In this section we consider a specification of the general model of section 2.1. Specifically, we consider the case where [in the notation of sections 2 and 3]:
X(t) = (rt, Yt1, . . . , Ytk)0. (30)
That is to say, the economic factors considered are the short ratertand the yields
Yi
t. As the tradable assets we take the bank account S0(t) whose dynamics are given as
dS0(t)
S0(t)
=rtdt, S0(0) =s0, (31)
¯
m stocks and stock indices Si(t), i = 1, . . . ,¯n whose dynamics are given by (1),
andk rolling bondsSn¯+j(t) :=U(t, Tj), j = 1, . . . k whose dynamics are given by
(29).
Thus, we are considering the model of section 2.1 with the following param-eterization: n= ¯n+k , m=k , (32) a0= 0, A0= (1,0, . . . ,0)1×(1+k), (33) a¯n+j =αj, An¯+j,l=Aj,l, j, l= 1, . . . , k , (34) Σi,l=σi,l, i= 1, . . . ,¯n , l= 0,1, . . . , N , (35) Σi,l=−Ti−¯nλi−n,l¯ , i= ¯n+ 1, . . . ,n¯+k , l= 0,1, . . . , N , (36) Λi,l=λi,l, i= 0,1, . . . , k , l= 0,1, . . . , N , (37)
Remark 4.1. For example, letting ¯n=k= 1 andN= 3 implies
n= 2, m= 1 (38) W(t) = (W0(t), W1(t), W2(t), W3(t))0, (39) S2(t) =U(t, T1), X0(t) =rt, X1(t) =Yt1, (40) A0= (1,0), a= a1 α1 , A= A1,0 A1,1 A1,0 A1,1 , (41)
Σ = σ1,0 σ1,1 σ1,2 σ1,3 −T1λ1,0 −T1λ1,1 −T1λ1,2 −T1λ1,3 , (42) b= b0 b1 , B= B00 B01 B10 B11 , (43) Λ = λ0,0 λ¯0,1 λ0,2 λ0,3 λ1,0 λ1,1 λ1,2 λ1,3 . (44)
On the other hand, letting n¯= 1, k= 2andN = 2 implies
n= 3, m= 2 (45) W(t) = (W0(t), W1(t), W2(t))0, (46) S2(t) =U(t, T1), S3(t) =U(t, T2), X0(t) =rt, X1(t) =Yt1, X2(t) =Yt2 (47) A0= (1,0,0), a= αa11 α2 , A= AA11,,00 AA11,,11 AA11,,22 A2,0 A2,1 A2,2 , (48) Σ = −Tσ11λ,01,0 −Tσ11λ,11,1 −Tσ11λ,21,2 −T2λ2,0 −T2λ2,1 −T2λ2,2 , (49) b= bb01 b2 , B = BB0010 BB0111 BB0212 B20 B21 B22 , (50) Λ = λλ01,,00 λλ10,,11 λλ10,,22 λ2,0 λ2,1 λ2,2 . (51)
Formulas of section 2.1 may now be applied to compute optimal investment strategies and the optimal value of the objective criterion for the problem (4) specified to the present setting. Economic interpretations of illustrative numerical examples based on historical data will be provided in the extended version of this paper.
5. Appendix: Proofs of The Main Results
We fixθ >0. Also, without any loss of generality, we may and we do assume that
V(0) =v= 1.
Step 1: The BHJ Equation.We begin with writing down the Bellman-Hamilton-Jacobiequation corresponding to the problemPθ:
ρ= Φ0 x(x)Bx+b0Φx(x) + 1 2 h −θ 2 Φ0 x(x)ΛΛ0Φx(x) +trΛ0Φxx(x)Λ i − inf {h∈Rm+1,10h=1} h1 2 θ 2 + 1 h0ΣΣ0h−h0Ax+a+θ 2h 0ΣΛ0Φ x(x) i , ∀x∈Rm+1. (52)
The equation (52) needs to be solved for the pair (ρ,Φ) whereρis a real constant, and Φ :Rm+1→Ris a function. We denote the gradient and the Hessian of Φ by Φxand Φxxrespectively.
Step 2: A Solution of the BHJ Equation.Consider a function Φθ of the quadratic
form
Φθ(x) :=x0K1(θ)x+K20(θ)x, ∀x∈Rm+1, (53) where the matrixK1(θ) is as in Assumption (A3) and the corresponding matrix
K2(θ) is defined in (7). Consider also the constant ρ(θ) defined in (11). Direct calculations verify that the pair (ρ(θ),Φθ) satisfies the equation (52).
Step 3: Minimal Selector in the BHJ Equation.It is easy to verify that the function
Hθ(x) := (θ 2 + 1
−1
(ΣΣ0)−1hβ(x) +λ(x)1i, (54) whereβ(x) andγ(x) are given by the formulas (9) and (10), is the minimal selector in the BHJ equation (52).
Step 4: Representations of the Wealth Process V(·).Here we provide some repre-sentations ofV(·)−θ/2, whereV(·) is the wealth process (3). It easily follows [using the It´o formula] that
V(t)−θ/2= expn−θ/2Z t o h0(s)(AXt+a)ds+ θ/4θ/2 + 1Z t 0k h0(s)Σk2ds −θ2/8 Z t 0 k h0(s)Σk2ds−θ/2 Z t 0 h0(s)ΣdWs o , (55) wherek · kdenotes the Euclidean norm. Now, using the above representation (55), with h(t) substituted with hθ(t) =Hθ(X
t), and invoking the BHJ equation (52)
combined with the It´o formula, we conclude that V(t)−θ/2= expn−θ/2ρ(θ)t+θ/2Φθ(Xt)− θ/2Φθ(X0) −θ2/8 Z t 0 k ηsk2ds− θ/2 Z t 0 ηsdWs o , (56) whereηt:= Φθx(Xt) 0 Λ +hθ(t)0Σ.
Step 5: Girsanov Change of Measure. Let now Pθ be a probability measure on
(Ω,F) such that dPθ dP Ft =ξt, ∀t≥0, (57) where ξt:= exp n −θ2/8 Z t 0 k ηsk2ds− θ/2 Z t 0 ηsdWs o .
Remark 5.1. Existence of such measure Pθ will be discussed in a later version of
the manuscript.
Step 6: Dynamics of the Factor Process Under the MeasurePθ.The processWθ t :=
Wt+
θ/2 R0tη0
sds is a standard Brownian motion under the measure Pθ. A
straightforward algebra yields the following representation for the dynamics of the factor processX under the measure Pθ:
dXt=G(θ)Xtdt+g(θ)dt+ ΛdWtθ (58)
whereG(θ) is the matrix defined in the Assumption (A3), andg(θ) is a remainder term that only depends on θ [and on the other parameters of the problem Pθ.]
Recall that the matrix G(θ) is stable. Thus, under the measure Pθ the process
X is a strongly ergodic Gaussian diffusion. This means in particular that ifF(x) is any function which is integrable w.r.t. the invariant measure for X [under the measurePθ] then we have
lim t→∞EPθ h F(Xt) X0=x i = Z F dµθ, ∀x∈Rm+1, (59) where byµθwe denote the corresponding invariant measure. The invariant measure
µθis in fact a Gaussian measure onRm+1with the covariance operatorC(θ) given as
C(θ) = Z ∞
0
etG(θ)ΛΛ0etG0(θ)dt . (60) Observe that the matrixC(θ) is positive definite and that it solves the Lyapunov equation
G(θ)C(θ) +C(θ)G0(θ) + ΛΛ0= 0. (61)
Step 7: Representation of the Risk Sensitive Criterion Under the Measure Pθ.It
follows from (56) and (57) that for everyx∈Rm+1 we have [recall that we have assumed thatV(0) = 1] lim inf t→∞ (−2/θ)t −1ln Ehθ( ·)[e−(θ/2) lnV(t) |X(0) =x] =ρ(θ) + lim inf t→∞ (−2/θ)t −1ln E Pθ [e(θ/2)Φ θ(Xt) |X(0) =x]. (62) Similar considerations combined with the condition (iv) of Definition 2.1 imply that for every admissible investment strategyh(·) we have
lim inf t→∞ (−2/θ)t −1ln Eh(·)[e−(θ/2) lnV(t) |X(0) =x] ≤ρ(θ) + lim inf t→∞ (−2/θ)t −1ln Eh(·),θ [e(θ/2)Φθ(Xt)|X(0) =x]≤ρ(θ). (63) In order to conclude the proof of theorems 2.5 and 2.6 it is thus sufficient to verify that the second term on the right hand side of (62) is zero forθsufficiently small. This will be done in the remaining step.
Step 8: Conclusion of the Proof. It follows from (61) that the matrix C−1(θ) satisfies the reciprocal equation
G(θ)C−1(θ) +C−1(θ)G0(θ) +C−1(θ)ΛΛ0C−1(θ) = 0. (64) Substituting forG(θ) we thus obtain
C−1(θ)R 2(θ) +C−1(θ)R1(θ)K1(θ) +R02(θ)C−1(θ) +K1(θ)R1(θ)C−1(θ) +C−1(θ)ΛΛ0C−1(θ) = 0. (65) Consequently, C−1(θ)−θK1(θ)R2(θ) + (C−1(θ)−θK1(θ)R1(θ)K1(θ) +R02(θ) C−1(θ)−θK1(θ)+R1(θ)K1(θ)C−1(θ)−θK1(θ) +θhK1(θ)R2(θ) +K1(θ)R1(θ)K1(θ) +R02(θ)K1(θ) +R3(θ) i +θK1(θ)R1(θ)K1(θ)−θR3(θ) +C−1(θ)ΛΛ0C−1(θ) = 0, (66) so that (C−1(θ)−θK 1(θ) G(θ) +G0(θ)C−1(θ)−θK 1(θ) +θK1(θ)R1(θ)K1(θ)−θR3(θ) +C−1(θ)ΛΛ0C−1(θ) = 0. (67) Now, observe that for θ sufficiently close to 0 the matrix θK1(θ)R1(θ)K1(θ)−
θR3(θ) +C−1(θ)ΛΛ0C−1(θ) is positive definite. This implies that the matrix (C−1(θ)−θK
1(θ)), is positive definite, and thus the matrix (θK1(θ)−C−1(θ)) is negative definite. Consequently, the function F(x) =e
θ/2Φθ(x)
is integrable w.r.t.µθ which implies that
lim inf t→∞ (−2/θ)t −1ln E Pθ [e(θ/2)Φ θ (Xt)|X(0) =x] = 0. (68) This completes the proof of theorems 2.5 and 2.6.
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T. R. Bielecki
Department of Mathematics, The Northeastern Illinois University, 5500 North St. Louis Avenue, Chicago, IL 60625-4699 USA
E-mail address:[email protected] S. R. Pliska
Department of Finance,
University of Illinois at Chicago, 601 S. Morgan Street,
Chicago, IL 60607-7124 USA
