Bond Pricing Portfolio Management

Styles of Bond Funds Bond funds are usually divided along the dimension of the two major risks that bond holders face – interest rate risk and credit risk. With reference to interest rate risk, funds typically choose a target duration. –For example, PIMCO has a “PIMCO Low Duration” fund that has a target maturity of 1-3 years average duration. With reference to credit (default) risk, fund may specialize in investing in US Treasury (“zero” credit risk), in investment grade corporate bonds, or high yield (“junk bonds”) corporate bonds.

Managing a Fixed Duration Fund As an application of duration and convexity, consider the management of a fixed duration treasury fund. Suppose you have a target duration for your fund equal to D. You can achieve this target duration by buying a number of bonds. Which ones should you buy? Consider an example. Suppose you want to manage a fund with duration equal to 6.7. Given the bonds available to you for investment (see next page), how should you allocate your money?

Bond Universe MaturityCouponBidAskYieldDuration 10/31/ :2496: /15/ :2696: /15/ :25106:

Optimizing a Fund: Case for Convexity If all else is equal (this is important), then a fund with a higher convexity will be safer. Suppose Fund A has a lower convexity than Fund B but equal durations. Then: –If interest rates increase, Fund B will lose less money than Fund A. –If interest rates decrease, Fund B will gain more money than Fund A. Thus, one way of determining the allocation across bonds, given the target duration, is to find the allocation across bonds such that the convexity of the portfolio is maximized. We can do this using “solver”.

An Example Let Portfolio A be a fund with 100% investment in Bond 2. –As computed in the spreadsheet, the bond and fund have a duration equal to We find Portfolio B by searching for the weights that maximize the convexity. The resultant weights are: 60.23% in Bond 1, 0% in Bond 2, and 39.77% in Bond 3. –Note that the maximum convexity portfolio has zero investment in Bond 2. –This is to be expected – in general, a portfolio of a target duration formed by bonds of lower and higher duration than the target duration will have a higher convexity.

Comparing the performance If all else works as assumed, Portfolio B with investments in the short-term and long- term bond is safer than a portfolio of 100% in the medium-term bond. Of course, it is possible that you may pay different amounts for these two portfolios (that is, “convexity” may be priced in the market).

Limitations (1/2) In actual practice, bond fund management is more complex. There are two reasons for this. First, our analysis of duration and convexity works only if all yields change by the same amount. Implicitly, we are assuming a parallel shift of yield curve. But in reality, it is unlikely that yields change by the same amount. If so, we cannot say which portfolio will perform better. Currently, the yield curve (see next slide) is downward sloping. It is unlikely that there will be parallel shifts in the yield curve.

Limitations (2/2) Second, even if yield changes are parallel, we may not prefer Portfolio B. This is because we only looked at risk for forming our portfolio, and not return. We may not prefer Portfolio B if the expected yield on Portfolio B is much lower than the yield on Portfolio A, even though Portfolio B is safer. It turns out that Portfolio B has a weighted average yield of 3.82% as compared with a yield of 4.72% of Portfolio A. Thus, you are “receiving” about 10 bps to invest in the lower risk portfolio! (This should, of course, make you ask whether this is indeed the lower risk portfolio.)

Credit Risk: Basics There are a number of measured used to keep track of and measure credit risk. We will consider three: 1. Probability of Default and Loss Given Default –The new Basel II standards require all banks to estimate these numbers for their loan portfolios. 2. Rating Transition –Bond ratings are provided by Moody’s S&P, and Fitch. 3. Spread Risk

Credit Risk in Bond Portfolios: Probability of Default and Loss Given Default Risk of default: The fundamental risk in holding a non-Treasury bond is that it may default on its obligations to pay coupons/principal. Thus, the risk is that you will not receive all promised payments. To evaluate this risk of default, we normally focus on two quantities: Probability of default, and the loss given default.

Probability of Default The probability of default is hard to measure. In general, it will depend on the leverage, and the volatility of the firm’s asset returns. –The higher the leverage, the higher the probability of default. –The higher the volatility, the higher the probability of default.

Loss Given Default The loss given default is the loss of the value of the bond, given default occurs. The loss given default can vary depending on the value of the collateral and the seniority of the bond. –It turns out that loss given default also depends on the state of the economy.

Credit Risk in Bond Portfolios: Rating Transition Given the difficulty of estimating both the probability of default, and the loss given default, many funds measure risk in terms of “rating transition” risk. When the probability of default of a bond changes by a significant amount, the bond’s rating will change. –Example: On 9 November, MRK’s long-term rating was downgraded by Moodys from Aaa to Aa2. A problem with using rating transitions is that rating agencies tend to lag the market – they change the rating much after the market has already re-priced the bond.

Credit Risk in Bond Portfolios: Spread Risk Spread risk: Just like how interest rates may change, the credit spread for a bond of a specific rating also changes. Spreads can change because (a) the probability of default changes, (b) the loss given default changes, (c ) investor risk aversion changes. Even if the probability of default and the loss given default have not changed, investors may require a greater or lower spread because they are more or less willing to take risk. Thus, spread risk encapsulates the total risk of all these three components.