6.4. Other Finance Questions

Besides option pricing problems, a variety of other quantitative finance problems are tested in quantitative interviews as well. Many of these problems tend to be position-specific. For example, if you are applying for a risk management job, prepare to answer questions about VaR; for fixed-income jobs, get ready to answer questions about interest rate models. As I explained in Chapter 1, it always helps if you grasp the basic knowledge before the interview. In this section, we use several examples to show some typical interview problems.

Portfolio optimization

You are constructing a simple portfolio using two stocks $A$ and $B$. Both have the same expected return of 12%. The standard deviation of $A$’s return is 20% and the standard deviation of $B$’s return is 30%; the correlation of their returns is 50%. How will you allocate your investment between these two stocks to minimize the risk of your portfolio?

Solution: Portfolio optimization has always been a crucial topic for investment management firms. Harry Markowitz’s mean-variance portfolio theory is by far the most well-known and well-studied portfolio optimization model. The essence of the mean-variance portfolio theory assumes that investors prefer (1) higher expected returns for a given level of standard deviation/variance and (2) lower standard deviations/variances for a given level of expected return. Portfolios that provide the minimum standard deviation for a given expected return are termed efficient portfolios. The expected return and the variance of a portfolio with $N$ assets can be expressed as

where $w_i,\ \forall i=1,\cdots,N$, is the weight of the $i$-th asset in the portfolio; $\mu_i,\ \forall i=1,\cdots,N$, is the expected return of the $i$-th asset; $\sigma_i^2$ is the variance of $i$-th asset’s return; $\sigma_{ij}=\rho_{ij}\sigma_i\sigma_j$ is the covariance of the returns of the $i$-th and the $j$-th assets and $\rho_{ij}$ is their correlation; $w$ is an $N\times1$ column vector of $w_i$’s; $\mu$ is an $N\times1$ column vector of $\mu_i$’s; $\Sigma$ is the covariance matrix of the returns of $N$ assets, an $N\times N$ matrix.

Since the optimal portfolio minimizes the variance of the return for a given level of expected return, the efficient portfolio can be formulated as the following optimization problem:

$\begin{aligned}\min_w\quad&w^T\Sigma w\\s.t.\quad&w^T\mu=\mu_p,\ w^Te=1\end{aligned}$, where $e$ is an $N\times1$ vector with all elements equal to 1.¹³

For this specific problem, the expected returns are 12% for both stocks. So $\mu_p$ is always 12% no matter what $w_A$ and $w_B$ ($w_A+w_B=1$) are. The variance of the portfolio is

Taking the derivative of $\operatorname{var}(r_p)$ with respect to $w_A$ and setting it to zero, we have

So we should invest 6/7 of the money in stock $A$ and 1/7 in stock $B$.

Value at risk

Briefly explain what VaR is. What is the potential drawback of using VaR to measure the risk of derivatives?

Solution: Value at Risk (VaR) and stress test—or more general scenario analysis—are two important aspects of risk management. In the Financial Risk Manager Handbook,¹⁴ VaR is defined as the following: VAR is the maximum loss over a target horizon such that there is a low, pre-specified probability that the actual loss will be larger.

Given a confidence level $\alpha\in(0,1)$, the VaR can be implicitly defined as $\alpha=\displaystyle\int_{-VaR}^{\infty}xf(x)dx$, where $x$ is the dollar profit (loss) and $f(x)$ is its probability density function. In practice, $\alpha$ is often set to 95% or 99%. VaR is an extremely popular choice in financial risk management since it summarizes the risk to a single dollar number. 165 Mathematically, it is simply the (negative) first or fifth percentile of the profit distribution.

As a percentile-based measure on the profit distribution, VaR does not depend on the shape of the tails before (and after) probability $1-\alpha$, so it does not describe the loss on the left tail. When the profit/loss distribution is far from a normal distribution, as in the cases of many derivatives, the tail portion has a large impact on the risk, and VaR often does not reflect the real risk.¹⁵ For example, let’s consider a short position in a credit default swap. The underlying asset is bond $A$ with a $1M notional value. Further assume that $A$ has a 3% default probability and the loss given default is 100% (no recovery). Clearly we are facing the credit risk of bond $A$. Yet if we use 95% confidence level, $VaR(A)=0$ since the probability of default is less than 5%.

Furthermore, VaR is not sub-additive and is not a coherent measure of risk, which means that when we combine two positions $A$ and $B$ to form a portfolio $C$, we do not always have $VaR(C)\le VaR(A)+VaR(B)$. For example, if we add a short position in a credit default swap on bond $B$ with a $1M notional value. $B$ also has a 3% default probability independent of $A$ and the loss given default is 100%. Again we have $VaR(B)=0$. When $A$ and $B$ form a portfolio $C$, the probability that at least one bond will default becomes $1-(1-3\%)(1-3\%)\approx5.9\%$. So $VaR(C)=\$1\text{M}>VaR(A)+VaR(B)$. Lack of sub-additivity directly contradicts the intuitive idea that diversification reduces risk. So it is a theoretical drawback of VaR.

(Sub-additivity is one property of a coherent risk measure. A risk measure $\rho(X)$ is considered coherent if the following conditions holds: $\rho(X+Y)\le\rho(X)+\rho(Y);\ \rho(aX)=a\rho(X),\ \forall a>0;\ \rho(X)\le\rho(Y),$ if $X\le Y$; and $\rho(X+k)=\rho(X)-k$ for any constant $k$. It is defined in Coherent Measure of Risk by Artzner, P., et al., Mathematical Finance, 9 (3):203–228. Conditional VaR is a coherent risk measure.)

Duration and convexity

The duration of a bond is defined as $D=-\dfrac1P\dfrac{dP}{dy}$, where $P$ is the price of the bond and $y$ is yield to maturity. The convexity of a bond is defined as $C=\dfrac1P\dfrac{d^2P}{dy^2}$. Applying Taylor’s expansion, $\dfrac{\Delta P}P\approx-D\Delta y+\dfrac12C\Delta y^2$. when $\Delta y$ is small, $\dfrac{\Delta P}P\approx-D\Delta y$.

For a fixed-rate bond with coupon rate $c$ and time-to-maturity $T$:

Another important concept is dollar duration: $\$D=-\dfrac{dP}{dy}=P\times D$. Many market participants use a concept called DV01: $DV01=-\dfrac{dP}{10,000\times dy}$, which measures the price change when the yield changes by one basis point. For some bond derivatives, such as swaps, dollar duration is especially important. A swap may have value $P=0$, in which case dollar duration is more meaningful than duration.

When $n$ bonds with values $P_i,\ i=1,\cdots,n$, and Durations $D_i$ (convexities $C_i$) form a portfolio, the duration of the portfolio is the value-weighted average of the durations of the components: $D=\displaystyle\sum_{i=1}^n\dfrac{P_i}PD_i$ ($C=\displaystyle\sum_{i=1}^n\dfrac{P_i}PC_i$), where $P=\displaystyle\sum_{i=1}^nP_i$. The dollar duration of the portfolio is simply the sum of the dollar durations of the components: $\$D=\displaystyle\sum_{i=1}^n\$D_i$.

What are the price and duration of an inverse floater with face value $100 and annual coupon rate $30\%-3r$ that matures in 5 years? Assume that the coupons are paid semiannually and the current yield curve is flat at 7.5%.

Solution: The key to solving basic fixed-income problems is cash flow replication. To price a fixed-income security with exotic structures, if we can replicate its cash flow using a portfolio of fundamental bond types such as fixed-rate coupon bonds (including zero-coupon bonds) and floating-rate bonds, no-arbitrage arguments give us the following conclusions:

Price of the exotic security = Price of the replicating portfolio
Dollar duration of the exotic security = Dollar duration of the replicating portfolio

To replicate the described inverse floater, we can use a portfolio constructed by shorting 3 floating rate bonds, which is worth $100 each, and longing 4 fixed-rate bonds with a 7.5% annual coupon rate, which is worth $100 each as well. The coupon rate of a floating-rate bond is adjusted every 0.5 years payable in arrear: the coupon rate paid at $t+0.5y$ is determined at $t$. The cash flows of both positions and the whole portfolio are summarized in the following table. It is apparent that the total cash flows of the portfolio are the same as the described inverse floater. So the price of the inverse float is the price of the replicating portfolio: $P_{inverse}=\$100$.

The dollar duration of the inverse floater is the same as the dollar duration of the portfolio as well: $\$D_{inverse}=4\times\$D_{fixed}-3\times\$D_{floating}$. Since the yield curve is flat, $r_0=7.5\%$ and the floating-rate bond is always worth $103.75 (after the payment of $3.75, the price of the floating-rate bond is $100) at year 0.5, and the dollar duration¹⁶ is

The price of a fixed-rate bond is $P=\displaystyle\sum_{t=1}^{2T}\dfrac{c/2}{(1+y/2)^t}+\dfrac{100}{(1+y/2)^{2T}}$, where $T$ is the maturity of the bond. So the dollar duration of the fixed-rate bond is

So $\$D_{inverse}=4\times\$D_{fixed}-3\times\$D_{floating}=1498$ and the duration of the inverse floater is $D_{inverse}=\$D_{inverse}/P_{inverse}=14.98$.

Forward and futures

What's the difference between futures and forwards? If the price of the underlying asset is strongly positively correlated with interest rates, and the interest rates are stochastic, which one has higher price: futures or forwards? Why?

Solution: Futures contracts are exchange-traded standardized contracts; forward contracts are over-the-counter agreements so they are more flexible. Futures contracts are marked-to-market daily; forwards contacts are settled at the end of the contract term.

If the interest rate is deterministic, futures and forwards have the same theoretical price: $F=Se^{(r+u-y)\tau}$, where $u$ represents all the storage costs and $y$ represents dividend yield for investment assets, convenience yield for commodities and foreign risk-free interest rate for foreign currencies.

The mark-to-market property of futures makes their values differ from forwards when interest rates vary unpredictably (as they do in the real world). As the life of a futures contract increases, the differences between forward and futures contracts may become significant. If the futures price is positively correlated with the interest rate, the increases of the futures price tend to occur the same time when interest rate is high. Because of the mark-to-market feature, the investor who longs the futures has an immediate profit that can be reinvested at a higher rate. The loss tends to occur when the interest rate is low so that it can be financed at a low rate. So a futures contract is more valuable than the forward when its value is positively correlated with interest rates and the futures price should be higher.

Interest rate models

Explain some of the basic interest rate models and their differences.

Solution: In general, interest rate models can be separated into two categories: short-rate models and forward-rate models. The short-rate models describe the evolution of the instantaneous interest rate $R(t)$ as stochastic processes, and the forward rate models (e.g., the one- or two-factor Heath-Jarrow-Morton model) capture the dynamics of the whole forward rate curve. A different classification separates interest rate models into arbitrage-free models and equilibrium models. Arbitrage-free models take the current term structure—constructed from most liquid bonds—and are arbitrage-free with respect to the current market prices of bonds. Equilibrium models, on the other hand, do not necessarily match the current term structure.

Some of the simplest short-rate models are the Vasicek model, the Cox-Ingersoll-Ross model, the Ho-Lee model, and the Hull-White model.

Equilibrium short-rate models

Vasicek model: $dR(t)=a(b-R(t))dt+\sigma dW(t)$

When $R(t)>b$, the drift rate is negative; when $R(t)<b$, the drift rate is positive. So the Vasicek model has the desirable property of mean-reverting towards long-term average $b$. But with constant volatility, the interest rate has positive probability of being negative, which is undesirable.

Cox-Ingersoll-Ross model: $dR(t)=a(b-R(t))dt+\sigma\sqrt{R(u)}dW(t)$

The Cox-Ingersoll-Ross model keeps the mean-reversion property of the Vasicek model. But the diffusion rate $\sigma\sqrt{R(u)}$ addresses the drawback of Vasicek model by guaranteeing that the short rate is positive.

No-arbitrage short-rate models

Ho-Lee model: $dr=\theta(t)dt+\sigma dz$

The Ho-Lee model is the simplest no-arbitrage short-rate model where $\theta(t)$ is a time-dependent drift. $\theta(t)$ is adjusted to make the model match the current rate curve.

Hull-White model: $dR(t)=a(b(t)-R(t))dt+\sigma dW(t)$

The Hull-White model has a structure similar to the Vasicek model. The difference is that $b(t)$ is a time-dependent variable in the Hull-White model to make it fit the current term structure.