Sensitivity Analysis

Reading on, you will notice that a large part of deals with uncertainty. Uncertainty comes in many forms:

• We might be unsure about the value of input data we use in making decisions.
• We might be unsure about the future values of some relevant exogenous variable.

One way to deal with uncertainty is to rely on the tools of probability theory and statistics. The main limitation of these tools is that they may require a lot of past data to characterize uncertainty, assuming that past data do tell us something useful about the future. Alternatively, we may analyze how changes in the input data affect our decision, in order to check its robustness. This may be more appropriate for the first situation above, which is more linked to ignorance than to genuine randomness. This fundamental process is called sensitivity analysis and we illustrate the concept with a few examples.

Example 2.32 (Sensitivity analysis of EOQ model) In the EOQ model we rely on a few pieces of information that in practice must be estimated: the fixed ordering charge A, the holding cost h, and the demand rate d. If we knew these quantities exactly, we would be sure that the optimal order quantity and the ensuing average cost per unit time are

respectively (if the assumptions behind the model are satisfied). In the total cost function we have neglected the total purchasing cost which does not really depend on the order size and just contributes a constant term. But if we make some mistake in estimating the input data, we will end up with some other order quantity Q, which will be suboptimal: C_tot(Q) > C_tot(Q*). What is the effect of our mistake in economic terms? We can measure this by taking the ratio of the two costs:

where the last step relies on the expression of Q*. To see the value of this relationship, let us assume that we make a rather gross mistake and apply an order quantity that is twice the right one, i.e., Q = 2Q*. Plugging this into the ratio above yields

In practice, if we make a 100% mistake in setting the order quantity, the total cost will increase by just 25%. This illustrates a robustness property of the EOQ formula. Of course, this tells us something useful only as far as the assumptions behind the model hold, at least approximately.

The example above is, in a sense, an example of sensitivity analysis “in the large.” We have not actually examined the effect of a change in a single factor, such as h, A, or d; rather, we have put everything together, analyzing the overall effect of making a mistake in setting the order quantity. However, one could ask what the effect of a small change in, say, the ordering cost A, is on both cost and the resulting EOQ. When we analyze the effect of a small change in a single factor, Taylor’s expansions come into play. We illustrate with an example from finance.

Example 2.33 (Bond duration) We got acquainted with bonds in Example 2.7. The price of a bond is essentially the present value of the cash flows up to bond maturity, using yield to maturity as the discount rate. Most long-term bonds offer the periodic payment of a coupon, in addition to repayment of the face value, but there are bonds that do not. A zero-coupon bond is a bond that promises only repayment of the face value F at maturity.

If we hold a zero-coupon bond maturing in 5 years, are we safe? When asked this question, students typically start mentioning a few risks that may be associated with such a bond:

• One obvious risk is default. A default occurs when bond issuers do not repay their debt. Default risk is rather high for bonds issued by distressed corporations or governments of unstable countries.
• Another possible risk is inflation. Indeed, with a long-term bond, even if we are repaid, the real value of the money we get back might have been significantly eroded.

What most students fail to mention is the fact that the general level of interest rates may change. This happens as central banks adjust interest rates in order to control inflation or to help economy against recession. Another important point is: Why are we holding that bond in our portfolio? If we want to keep the bond until maturity, and there is neither default nor inflation risk, we are perfectly safe. But if we want to sell it before, because we need immediate cash, an unfavorable change in the interest rate might affect bond value.

Actually, all of the risk factors that we mentioned are somehow captured by the required bond yield y.¹⁹ This is typically larger than the current level of risk-free rates, as it reflects different kinds of risk premia. Reasonably, when holding a risky security, an investor expects to be compensated for the risk she is bearing, and this implies a rate of return higher than that of a risk-free asset. The price of a zero-coupon bond with face value F, maturing in T years, is given by

If, for whatever reason, the required yield y goes up, the bond price will go down. As an illustration, consider the price of a zero-coupon bond, with face value $1000, maturing in 10 years, when yield to maturity is y = 5%:

If the required yield increases by 100 basis points,²⁰ i.e., the new yield is y = 0.05 + 0.01 = 0.06, the new price turns out to be P(0.05 + 0.01) = 558.39. The resulting loss is

It would be very nice to have a sensitivity measure telling us the effect of a small variation δy of the required yield, possibly applying to coupon-bearing bonds as well.

Before doing so for a general bond, let us check the quality of a Taylor expansion of the yield–price relationship for the zero-coupon bond. This requires taking first- and second-order derivatives with respect to yield in (2.16):

The first-order approximation gives $555.45 as an estimate of the new price, whereas the second-order approximation gives $558.51. We see that the approximations are rather good. Indeed, the change in the required yield cannot be a huge number, since we are talking about interest rates. Now, let us explore the concept further for a general coupon-bearing bond. The price–yield relationship is

where C is the coupon paid every period; note again that the last cash flow at bond maturity t = T includes both payment of last coupon and refund of face value F. For the sake of simplicity, we assume that one coupon is paid per year, so time subscripts t = 1, …, T correspond to years. The sensitivity of price with respect to yield is associated with the first-order derivative of Eq. (2.17):

where d_t is the discounted cash flow for time period t:

Not surprisingly, the derivative is negative, as an increase in yield implies a drop in bond price. We can rewrite the relationship in a more informative way by noting that the bond price is just the sum of the discounted cash flows d_t:

The ratio of the two sums can be regarded as a weighted average of the time instants t at which cash flows occur, where weights are just the discounted cash flows d_t. The ratio

is called Macaulay duration. In order to get rid of the annoying 1 + y factor,²¹ we introduce modified duration

which allows us to write the following relationships:

This is just a rewriting of Eq. (2.18) based on the finite-difference approximation²² P′(y) ≈ δP/δy. The second relationship states that the percentage change in bond price can be approximated by the product of modified duration times the change in yield. The minus sign accounts for the fact that bond prices and yields are inversely related. Equation (2.20) shows that duration is a risk measure for bonds. The larger the duration, the larger the sensitivity of bond prices to unexpected changes in the required yield.

Other things being equal, the longer the bond maturity, the larger the bond duration. Hence, to stay safe, one should invest in short-term bonds. By the way, not surprisingly, bond duration for a zero-coupon bond is just bond maturity. However, other factors may affect bond duration. Since this is a weighted average of the times at which cash flows occur, we may also see that a large coupon C reduces duration. This happens because the weights d_t for periods t = 1, 2, …, T − 1 are relatively larger when C is significant with respect to face value F.

Bond duration is a useful concept for bond portfolio management, but it is rich in shortcomings. For instance, it assumes that there is only one catchall risk factor y, whereas in practice one should account, e.g., for relative variations in the short- versus the long-term interest rates. Moreover, it is associated with a first-order approximation that is likely to give a poor approximation unless the change δy in required yield is small. To ease the last difficulty, one can resort to a second-order Taylor expansion

The term C, which is just the second order derivative P″(y) is called bond convexity.