BASICS OF INTEREST RATE RISK MANAGEMENT
Fact 3.6 Let dr have a normal distribution with mean μ and standard deviation σ. Then Equation 3.28 implies that dP has a normal distribution with mean and standard deviation
given by:
μP = −DP× P × μ and σP = DP× P × σ. (3.29)
DURATION 87
That is:
dr∼ N(μ, σ2) =⇒ dP ∼ N(μP, σ2P) (3.30) The 95% VaR is then given by
95% VaR =−(μP− 1.645 × σP) (3.31)
where−1.645 corresponds to the 5-th percentile of the standard normal distribution, that is, if x ∼ N (0, 1) then P rob(x < −1.645) = 5%. The 99% VaR is computed as in Equation 3.31 except that the number “1.645” is substituted by “2.326.”
This result of course relies on Equation 3.28, which is only an approximation. If dr is not normal, Equation 3.31 does not hold. The next example illustrates one popular approach to dealing with this latter case.
EXAMPLE 3.7
A portfolio manager has $100 million invested in a bond portfolio with duration DP = 5. What is the 95% one-month Value-at-Risk of the portfolio?
1. Historical Distribution Approach. We can use the past changes in the level of interest rates dr as a basis to evaluate the potential changes in a portfolio value dP . Panel B of Figure 3.1 shows the historical observations of the level of interest rates up to 2005. Panel A of Figure 3.3 shows the monthly changes in the level of interest rates, while Panel B makes a histogram of these variations. As we can see large increases and decreases are not very likely, but they do occur occasionally. We can now multiply each of these changes dr observed in the plot by−DP×P = −5×100 million to obtain the variation in dP . Panel C of Figure 3.3 plots the histogram of the changes in the portfolio i.e., the portfolio profits and losses (P&L).4 Given this distribution, we can compute the maximum loss that can occur with 95% probability.
We can start from the left-hand side of the distribution, and move rightward until we count 5% of the observations. That number is the 95% monthly VaR computed using the historical distribution approach. In this case, we find it equal to $3 million.
That is, there is only 5% probability that the portfolio losses will be higher than $3 million.
2. Normal Distribution Approach. From Fact 3.6, a normal distribution assumptin on dr translates into a normal distribution on dP . Using the data plotted in Panel A of Figure 3.3, we find that the monthly change in interest rate has mean μ = 6.5197× 10 − 006 and stadard deviation σ = .4153%. Therefore, μP = −5 × 100 × μ =
−.0033 and σP = 5×100×σ = 2.0767. The standard normal distributionis plotted along with the (renormalized) histogram in Panel C of Figure 3.3. The 95% VaR is then equal to 95% VaR =−(μP − 1.645 × σP) = $3.4194 million.
4We renormalized the histogram to make it comparable with the normal distribution case, discussed in the next point.
88 BASICS OF INTEREST RATE RISK MANAGEMENT
Figure 3.3 Changes in the Level of Interest Rates: 1965 - 2005
1965 1970 1975 1980 1985 1990 1995 2000 2005
−4
−2 0 2 4
A: Monthly Changes in the Level of Interest Rates
Interest Rate (%)
−4 −3 −2 −1 0 1 2 3 4
0 10 20 30 40
Interest Rate (%)
B: Histogram of Monthly Changes in the Level of Interest Rates
Occurrences
−200 −15 −10 −5 0 5 10 15 20
0.1 0.2 0.3 0.4
C: Probability Distribution of Portfolio P/L
Millions of Dollar
Probability Density
Data Source: CRSP.
3.2.8.1 Warnings It is worth emphasizing immediately a few problems with the Value-at-Risk measure of risk, as well as some potential pitfalls:
1. VaR is a statistical measure of risk, and as with any other statistical measure, it depends on distributional assumptions and the sample used for the calculation. The difference can be large. For instance, in Example 3.7 the VaR varies depending on whether we use the normal distribution approach or the historical distribution approach.
2. The duration approximation in Equation 3.28 is appropriate for small parallel changes in the level of interest rates. However, by definition, VaR is concerned with large changes. Therefore, the duration approximation method is internally incosistent.
DURATION 89
The problem turns out to be especially severe for portfolios that include derivative securities, either implicitly or explicitly. We will return to this issue in later chapters.
3. The VaR measures the maximum loss with 95% probability. However, it does not say anything about how large the losses could be if they do occur. The tails of the probability distribution matter for risk. For instance, in Example 3.7 the 99% VaR using the historical distribution approach is $5.52 million, while this figure is only
$4.83 million using the normal distribution assumption. The tails of the normal distribution are thin, in the sense that they give an extremely low probability to large events, which instead in reality occur with some frequency.
4. The VaR formula used in Equation 3.31 includes the expected change in the portfolio μP = −DP×P ×E[dr]. The computation on the expected change E[dr] is typically very imprecise, and standard errors are large. Such errors can generate a large error in the VaR computation. For this reason, it is often more accurate to consider only the unexpected VaR, that is, consider only the 95% loss compared to the expected P&L μP. Practically, we simply need to set μP = 0 in Equation 3.31.
3.2.9 Duration and Expected Shortfall
Some of the problems with VaR can be solved by using a different measure of risk, called the expected shortfall. This measure of risk answers the following question: How large can we expect the loss of a portfolio to be when it is higher than VaR? As mentioned in point 3 in the above Subsection 3.2.8.1, the VaR measure does not say anything about the tails of the statistical distribution. This is an especially important problem when the underlying risk factor has a fat-tailed distribution, as shown in Figure 3.3, or when the portfolio contains highly nonlinear derivative securities, as we will see in later chapters.
Definition 3.7 The expected shortfall is the expected loss on a portfolio P over the horizon