duration mismatch for credit securities can be longer than 1 year and hedging with coupon securities will not offset the duration mismatch. Using modified duration, the duration mismatch will be even larger, since the duration is scaled by the yield, which is larger for zero coupon bonds in a steep yield curve environment than the combined durations of the security.
1.4 KEY RATE DURATIONS
Key rate duration, proposed by Ho [5], is an attempt to measure the risks of a portfolio across the yield curve. The key rate is usually referred to a very liquid security such as an on-the-run treasury. Usually the yield curve is broken into about 10 duration buckets, each representing one key rate. The most widely used key rates are the 6 month, 1 year, 2, 3, 4, 5, 7, 10, 20, and 30 year points on the curve. To calculate the exposure of a security to a key rate, the yield of the security is shifted at the respective key rate by a small amount, while maintaining all other key rates constant, and the impact of the changes in the price of the security is calculated. The yield shift is linearly interpolated between the key rate and the preceding or following key rates. For example, to calculate the 5-year key rate duration of a security, if we shift the 5-year yield by 10 bps, the shift for a cash flow that is at 5.50 years will be
While key rates address the exposure of a portfolio to different parts of the curve, they have many shortcomings for complex securities and some derivatives. Additionally, they do not address the incorrect duration calculation that was mentioned in the previous section. Table 1.2 lists the KRDs of the combined security that was shown in Table 1.1 along with the correct KRDs based on individual securities.
Table 1.2 Key rate duration of a portfolio
Key rate duration is a good measure of the risk of a security or a portfolio to interest rates. However, the curve exposure of credit and inflation linked securities and many derivatives is either not measured by key rates or the exposure cannot be used effectively. For example, knowing ten key rate credit durations of a security is not useful for hedging its risks.
One of the biggest shortcomings of the KRD is its relation to performance attribution and return calculation. To calculate the performance of a security from its KRDs, we need to multiply the changes in the key rates by the respective durations. This implies that we must have an unbiased measurement of the change in the yields of the key rates. On-the-run securities, due to their liquidity, are often very rich relative to the rest of the treasury market and cannot be used as unbiased indicators of the level of interest rates at a given maturity. As on-the-run securities season, they underperform the rest of the market. For example, if the yield of a newly issued 10-year bond with 8 years of duration is 5 bps below the previously issued 10-year bond, over time, as its yield normalizes, it will have a negative cumulative performance of 0.0005 × 8 = 0.4 %. Thus, a security whose yield does not change will benefit from an apparent gain in performance as a result of the underperformance of the key rate security. Therefore, the use of KRDs requires calculation of a smoothed curve for the treasury market. The smoothed curve has to be derived from a pool of treasuries that do not have a liquidity premium and thus cannot include on-the-run securities.
Positive relative KRDs can also be an issue for a portfolio that cannot use derivatives. If the relative exposure of a portfolio at the 10-year part of the curve is –0.2 years, it can be hedged by buying 0.2 years of the 10-year zero coupon bond. However, if the portfolio is long duration, then selling 0.2 years of the 10-year zero coupon bond may not be practical if it is not already part of the portfolio, without changing the structure of the portfolio.
Since KRDs are based on localized changes in the yield curve, it is very difficult to compare competing trades that have similar goals. For example, there is no way to compare a barbell trade that is overweight 10-year, underweight 2-year treasuries with a similar trade that is overweight 20-year, underweight 2-year treasuries. Similarly, there is no way to compare a 2–5–10-year butterfly trade with a 2–10–30-year butterfly. Hedging the risks of credit securities where only a few bonds are available is not practical by using key rate credit exposures. Some derivative securities can have interest rate exposures that require treasuries that are longer than 30 years for their effective hedging. Likewise, long dated inflation linked bonds have a small exposure to nominal rates due to the inflation lag which may require longer than 30-year treasuries for their hedging.
Linear time interpolation of key rates can be a source of overestimation or underestimation of duration at some key rates. The correlation between 2-year and 4-year rates (2, 4) is significantly lower than the correlation between (18, 20). For a constant difference in the maturity of two key rates, the longer their maturity, the higher their correlation is. Historical correlation of (18, 20) rates is very similar to correlations of (2, 2.25) year maturities. Similarly, the correlation between (18, 20) is higher than the correlations between (10, 12), (12, 14), (14, 16), and (16, 18) maturities. We now return to the above example and calculate the 20-year KRD of the combined security in Table 1.2. We assumed that for a change of 5 bps in the yield of the 20-year key rate, the yield of the 18-year cash flow changed proportionally by
Chapter 2
Term Structure of Rates
In the previous chapter, we briefly discussed some of the shortcomings of the traditional measurements of risk and return in the treasury markets. Analysis of more complex fixed income instruments such as options and futures, credit products and mortgages requires more elaborate mathematical analysis and cannot be handled using the simple price/yield formulas. As we discussed previously, the result of yield or duration calculation of a portfolio was path dependent, that is, the calculated yield and duration were different if we treated all cash flows as one security or calculated the yield and duration for each cash flow separately and then combined the results. The primary reason for this path dependency was the use of different discount yields in one path versus another. Our primary objective in this chapter is to develop a term structure of interest rates (TSIR) model that provides a basis for discounting all cash flows at the correct discount yield. We will then provide examples of market derived yield curves based on our methodology.
2.1 LINEAR AND NON-LINEAR SPACE
Perhaps the most important issue in developing a TSIR model is the choice of reference frame. To motivate the development of a logical reference frame, we will compare an investment instrument to a pedestrian.
Consider a fixed income instrument that pays or receives a constant cash flow of c at regular intervals such as a fixed rate bond or a home mortgage loan. Likewise, consider a
