Robust Bond Portfolio Construction via Convex–Concave Saddle Point Optimization

Abstract

The minimum (worst case) value of a long-only portfolio of bonds, over a convex set of yield curves and spreads, can be estimated by its sensitivities to the points on the yield curve. We show that sensitivity based estimates are conservative, i.e., underestimate the worst case value, and that the exact worst case value can be found by solving a tractable convex optimization problem. We then show how to construct a long-only bond portfolio that includes the worst case value in its objective or as a constraint, using convex–concave saddle point optimization.

Appendices

Worst Case Analysis CVXPY Code

Explicit Dual Portfolio Construction CVXPY Code

Derivation of Dual Form

We now derive the dual form of the worst case portfolio construction problem for the case where the uncertainty set in polyhedral. We note that the worst case $\log$ change in portfolio value for a fixed h, $\varDelta ^\text {wc}(h)$ is given by the optimal value of the optimization problem

$$\begin{aligned} \begin{array}{ll} \text{ minimize } &{} \log \left( \sum _{i=1}^n\sum _{t=1}^Th_ic_{i,t}\exp (-t(y_t+s_i))\right) -\log (p^Th)\\ \text{ subject } \text{ to } &{} A(y,s)\preceq b. \end{array} \end{aligned}$$

We have written this problem with (y, s) explicitly, instead of with x, to emphasize the objective’s dependence on each component. We note that due to our budget constraint, $\log (V(y,s))=\log (p^Th)$.

In order to obtain a closed form dual, we introduce a new variable $z\in {\text{ R }}^{nT}$, where we think of $z_{i,t}$ as corresponding to $y_t+ s_i$. This is a very general formulation which allows each bond to be associated with its own yield curve $y_i\in {\text{ R }}^T$, with $z_{i,t}$ corresponding to the t’th entry of the i’th bond’s yield curve. Since we model each bond as having its own yield curve, this formulation generalizes the earlier treatment with yields and spreads handled separately. We can recover the original structure with the linear constraints

$$\begin{aligned} z_{i,t} = y_t+s_i,\qquad t=1,\ldots ,T,\quad i=1,\ldots ,n, \end{aligned}$$

(16)

which are representable as $z=Fx$ for an appropriate $F\in {\text{ R }}^{nT\times (T+n)}$. As such, the problem is equivalent to

$$\begin{aligned} \begin{array}{ll} \text{ minimize } &{} \log \left( \sum _{i=1}^n\sum _{t=1}^Th_ic_{i,t}\exp (-tz_{i,t})\right) -\log (p^Th) \\ \text{ subject } \text{ to } &{} Ax \preceq b,\quad z=Fx. \end{array} \end{aligned}$$

(17)

Strong duality tells us that $\varDelta ^\text {wc}(h)$ is equal to the optimal value of the dual problem of (17) [3, §5.2].

Dual Problem We derive the dual of the problem

$$\begin{aligned} \begin{array}{ll} \text{ minimize } &{} \log \left( \sum _{i=1}^n\sum _{t=1}^Th_ic_{i,t}\exp (-tz_{i,t})\right) -\log (p^Th) \\ \text{ subject } \text{ to } &{} Ax \preceq b,\quad z=Fx. \end{array} \end{aligned}$$

(18)

First, with f the log-sum-exp function $f(x)=\log \left( \sum \exp x_i\right)$, we observe that our problem can be rewritten as

$$\begin{aligned} \begin{array}{ll} \text{ minimize } &{} f(Cz+d)-\log (p^Th) \\ \text{ subject } \text{ to } &{} Ax \preceq b,\quad z=Fx. \end{array} \end{aligned}$$

We define C to be the diagonal matrix with $C_{i,t}=-t$, where we are using unwound vectorized indexing for z, and $d\in {\text{ R }}^{nT}$ to be the vector with $d_{i,t} = \log (c_{i,t}h_i)$. Then, the Lagrangian is given by

$$\begin{aligned} L(z,x,\mu ,\nu ) = f(Cz+d) +\mu ^T(Ax-c)+\nu ^T(z-Fx)-\log (p^Th). \end{aligned}$$

The Lagrange dual function is given by

$$\begin{aligned} g(\mu ,\nu ){} & {} =\inf _{z,x} L(z,x,\mu ,\nu ) = \inf _z\left( f(Cz+d)+\nu ^Tz\right) +\inf _x\left( \mu ^TAx-\nu ^TFx\right) \\{} & {} \quad -\mu ^Tc-\log (p^Th). \end{aligned}$$

The second term is equal to $-\infty$ unless $A^T\mu +F^T\nu =0$, so this condition will implicitly restrict the domain of g. Now, note that with $g(z)=f(Cz+d)$, the first term can be rewritten as

$$\begin{aligned} \inf _z\left( f(Cz+d)+\nu ^Tz\right)= & {} \inf _z\left( g(z)+\nu ^Tz\right) \\= & {} -\max _z \left( -\nu ^Tz-g(z)\right) \\= & {} -g^*(-\nu ), \end{aligned}$$

where $g^*(y)=\max {y}y^Tz-g(z)$ is the conjugate of g. [3, §3.3.1].

We now use two facts from [3, §3.3.2]. First, in general the conjugate of the linear precomposition $\phi (z)=\rho (Cz+d)$ can be written in terms of the conjugate of $\rho$ as $\phi ^*(y) = \rho ^*(C^{-T}y)-d^TC^{-T}y$. Second, the dual of the log-sum-exp function f is

$$\begin{aligned} f^*(y) = \left\{ \begin{array}{ll} \sum _iy_i\log (y_i) &{} \text { if } y\ge 0,\quad \textbf{1}^Ty = 1\\ \infty &{} \text { otherwise.} \end{array}\right. \end{aligned}$$

Combining these two facts, and expanding terms, we find that

$$\begin{aligned} g(\mu ,\nu ) =\left\{ \begin{array}{ll} -\log (p^Th)-\mu ^Tb-\sum \limits _{i=1}^n\sum \limits _{t=1}^T\zeta (c_{i,t}h_i,\frac{\nu _{i,t}}{t}) &{} \text {if } A^T\mu -F^T\nu = 0,\ \textbf{1}^T\nu = 1,\ \nu \ge 0\\ -\infty &{} \text {otherwise,} \end{array}\right. \end{aligned}$$

with

$$\begin{aligned} \zeta (x,t) = -t\log (x/t) = t\log (t/x) = t\log (t)-t\log (x). \end{aligned}$$

Thus the robust bond portfolio problem can be written as

$$\begin{aligned} \begin{array}{ll} \text{ minimize } &{} \phi (h) -\lambda \max \limits _{\mu ,\nu }~g(\mu ,\nu )\\ \text{ subject } \text{ to } &{} \mu \ge 0,\quad A^T\mu -F^T\nu = 0,\quad \textbf{1}^T\nu = 1,\quad \nu \ge 0\\ &{} h\in \mathcal {H},\\ \end{array} \end{aligned}$$

where we have moved the implicit constraints in the definition of g to explicit constraints in the optimization problem. Note this equivalent optimization problem has new variables $\mu$ and $\nu$. By using that $-\lambda \max _{\mu ,\nu }g(\mu ,\nu )= \min _{\mu ,\nu }-\lambda g(\mu ,\nu )$ and collecting the minimization over h, $\mu$, and $\nu$, we obtain the form in Sect. 5.2.