Our focus is on bond portfolio strategies. In this chapter we explain how to measure a portfolio’s risk profile. Because we focus on bond portfolio strategies, our interest is on performance relative to a bond market index. We refer to the bond market index as the benchmark index. So the process of managing funds relative to a benchmark index requires that a manager thoroughly understand the portfolio’s risk profile relative to that of the benchmark index. The difference between the risk profile of the portfolio and the benchmark index results in performance differences. The best way to construct and control the risk profile of a portfolio relative to a benchmark index is to quantify all of the important risks. It is important to understand that a trade can reduce one type of risk while simultaneously increasing another.

One of the problems with using the variance as a measure of dispersion is that the variance is in terms of squared units of observed returns. Consequently, the square root of the variance, which is called the standard deviation, is used as a more understandable measure of the degree of dispersion.

There are some important qualifications to be aware of when using the standard deviation as a risk measure. In many applications of probability theory, it is assumed that the underlying probability distribution is a normal distribution. Exhibit 1(a) shows the graph of a normal probability distribution. The average value is also called the expected value. The normal distribution has the following properties:

For a normal distribution, the expected value and the standard deviation are all the information needed to make statements about the probabilities of realizing certain outcomes. In order to apply the normal distribution to make statements about probabilities, it is necessary to assess whether a historical distribution (i.e., a distribution created from the observed returns) is normally distributed.

For example, a property of the normal probability distribution is that the distribution is symmetric around the expected value. However, a given probability distribution might be best characterized like those shown in Exhibits 1(b) and 1(c). Such distributions are referred to as skewed distributions. In addition to skewness, a historical distribution may have more outliers (i.e., observations in the “tails”) than the normal distribution predicts. Distributions with this characteristic are said to have fat tails. This is depicted in Exhibit 1(d). Notice that, if a distribution does indeed have fat tails but it is assumed to be normally distributed, then the probability of an actual value in a tail is assumed to be less than the true probability.²⁶³

In order to determine whether a historical distribution can be characterized as a normal distribution, the following two questions must be addressed:

Let’s look at the evidence on bond returns. For bonds, there is a lower limit on the loss. For Treasury securities, the limit depends on how high interest rates can rise. Since Treasury rates have never exceeded 15%, this places a lower bound on negative returns from holding a bond. There is a maximum return. Assuming that negative interest rates are not possible, the maximum price for a bond is the undiscounted value of the cash flows (i.e., the sum of the interest payments and maturity value). In turn, this determines the maximum return. On balance, government bond return distributions are negatively skewed. Studies by JP Morgan RiskMetrics suggest that this occurs for government bonds. Moreover, government bond returns exhibit fat tails.²⁶⁵

Another issue in using a normal distribution is the independence of observations. Casting this in terms of returns, it is important to know whether the return that might be realized today is affected by the return that was realized in a prior period. The term serial correlation is used to describe the correlation between returns in different periods. Studies by JP Morgan’s RiskMetrics suggest that there is a small positive serial correlation for government bond returns.²⁶⁶

Other measures of risk focus on only that portion of an investment’s return distribution that is below a specified level. These measures of risk are referred to as downside risk measures. For downside risk measures, the portfolio manager defines the target return, and returns less than the target return represent adverse consequences. In the case of the standard deviation, the target return is the expected value. However, in practice, this need not be the case. For example, in managing money against a benchmark index, the target return might be X basis points over the return of the benchmark index. Outcomes that are less than the return of the benchmark index plus X basis points represent downside risk. In managing money against liabilities, the target return might be the rate guaranteed on liabilities plus a spread of S basis points. Returns with an outcome less than the rate guaranteed on liabilities plus S basis points then represent downside risk.

The three more popular downside risk measures are:

If the standard deviation is used as the measure of risk, the variance of each bond and the covariance for each pair of bonds must be estimated in order to be able to compute the portfolio’s standard deviation. Let’s look at two major problems with this approach. After we discuss these problems, we will see how to resolve these issues.

The first problem with this approach is that the number of required estimated inputs increases dramatically as the number of bonds in the portfolio or the number of bonds considered for inclusion in the portfolio increases. For example, consider a manager who wants to construct a portfolio for which there are 5,000 bonds that are candidates for inclusion. (If this number sounds large, consider that the broad-based bond market indexes are comprised of far more than 5,000 bonds.) Then the number of variances and covariances that must be estimated is 12,502,500.²⁷²

The second problem is that, whether it is 55 variances and covariances for a 10-bond portfolio or 12,502,500 for a 5,000-bond portfolio, these values must all be estimated. Where does the portfolio manager obtain these values? The answer is that they must be estimated from historical data. While equity portfolio managers have the luxury of working with a long time series of returns on stocks, bond portfolio managers often do not have good sources of historical bond data. In addition, even with time series data on the return for a particular bond, a portfolio manager must question what these returns mean. The reason is that the characteristics of a bond change over time.

For example, consider a 10-year Treasury note issued 8 years ago for which quarterly returns have been calculated. The first quarterly return is the return on a 10-year Treasury note. However, the second quarterly return is the return on a 9.75-year Treasury note. The third quarterly return is the return on a 9.5-year Treasury note, and so on. If the original 10-year Treasury note is in the current portfolio and the manager wants to estimate the standard deviation for that security, the historical standard deviation is not meaningful. Since it was purchased 8 years ago, this security is now a 2-year Treasury note and does not share the return volatility characteristics of the earlier maturities. Not only does the changing time to maturity affect the historical data and render it of limited use, but some securities’ characteristics change dramatically over time because of call/prepayment provisions. For example, we discussed mortgage-backed securities. CMO support bonds have average lives that change dramatically due to prepayments, which affects the historical return pattern.

Now, if we couple the problem of a large number of required estimates and the lack of relevant data, we see another major problem. Consider a 100-bond portfolio. There are 5,050 inputs to be estimated. Suppose that just 10% of the inputs are misestimated because of a lack of good historical or meaningful data. This means that there are 505 misestimated numbers which could have a material impact on the estimated portfolio standard deviation.

When a manager’s benchmark is a bond market index, performance relative to that benchmark index is important. A manager who realizes a −1% annual return when the benchmark index returns −3% has performed well. So, it is important that such a manager understand the risk of a portfolio relative to the risk of the benchmark index. The measure used for this purpose is tracking error.

Exhibit 2 shows the calculations of the tracking error for returns to a hypothetical bond portfolio observed for a 12-month period. The benchmark index is the Lehman Aggregate Bond Index. The tracking error is shown at the bottom of the exhibit. Details regarding the interim calculations necessary to compute the standard deviation are not explained here.²⁷³

A portfolio manager seeking to match the benchmark index will regularly have excess returns close to zero, and therefore a small tracking error will be observed. In theory, if a manager could perfectly match the performance of the portfolio, a tracking error of zero would be realized. An actively managed portfolio that takes positions substantially different from those of the benchmark index (i.e., has a different risk profile) would likely have large observed excess returns, both positive and negative, and thus would have a large observed tracking error.

We will use the portfolio shown in Exhibit 3 to explain tracking error as well as the other risk measures discussed throughout this chapter. There are 45 bonds in the portfolio and, throughout this chapter and the next, we will refer to this portfolio as the “45-bond portfolio.” The 45-bond portfolio was constructed on November 5, 2001 based on prices as of October 31,2001.²⁷⁴ The market value (which includes accrued interest) of the portfolio is $1,001,648,000. The benchmark index is the Lehman Aggregate Bond Index, a broad-based bond market index described in the previous chapter. The market value (which includes accrued interest) of this index is $6,999,201,792,000.

Lehman Brothers has historical information on excess returns for bonds. Using that information, Lehman Brothers has calculated that the tracking error for the 45-bond portfolio, relative to the Lehman Aggregate Bond Index, is 62 basis points per year. Assuming that the tracking error is normally distributed, the probability is about 68% that the portfolio return over the next year will be within ± 62 basis points of the benchmark index return.

In Section IX we will see how this 62 basis point tracking error is decomposed into the risks that cause this tracking error.

One problem with the use of actual tracking error is that it does not reflect the effect of the portfolio manager’s current decisions on the future active returns and hence the tracking error that may be realized in the future. If, for example, the manager significantly changes the portfolio’s effective duration or sector allocations today, then the actual tracking error calculated using data from prior periods would not accurately reflect the current portfolio risks. That is, the actual tracking error will have little predictive value and can be misleading regarding portfolio risks.

The portfolio manager needs a forward looking estimate of tracking error to accurately reflect future portfolio risk. In practice, this is accomplished by using the services of a commercial vendor that has a model, called a multi-factor risk model, that has identified and defined the risks associated with a benchmark index. Such a model is described in Section X. Statistical analysis of historical return data for the bonds in the benchmark index is used to obtain the risk factors and to quantify the risks. (This involves the use of variances and correlations/covariances.) Using the manager’s current portfolio holdings, the portfolio’s current exposure to the various risk factors can be calculated and compared to the benchmark’s exposures to the risk factors. Using the differential factor exposures and the risks of the factors, a forward looking tracking error for the portfolio can be computed. This tracking error is also referred to as ex ante tracking error or predicted tracking error. We will use the term predicted tracking error in this chapter and the next.

There is no guarantee that the predicted tracking error at the start of, say, a year would exactly match the actual tracking error calculated at the end of the year. There are two reasons for this. The first is that, as the year progresses and changes are made to the portfolio, the predicted tracking error changes to reflect the new exposures. The second is that the accuracy of the predicted tracking error depends on stability in the variances and correlations/covariances used in the analysis. These problems notwithstanding, the average of predicted tracking errors obtained at different times during the year will be reasonably close to the actual tracking errors observed at the end of the year.

Both calculations of tracking error have their use. The predicted tracking error is useful in risk control and portfolio construction. The manager can immediately see the likely effect on tracking error of any intended change in the portfolio. Thus, the portfolio manager can do a what-if analysis for various portfolio strategies and eliminate those that would result in tracking errors beyond his tolerance for risk. The actual tracking error can be useful for assessing performance as discussed in the next chapter.

When actual tracking error is reported, the following supplementary information is often provided: average return difference (i.e., average of the active returns), the maximum active return difference (i.e., the highest of the active returns), and the minimum active return difference (i.e., the lowest of the active returns). It should be noted that some market participants refer to the active return for a given observation period as tracking error.²⁷⁵

Three different duration measures are used to quantify a bond’s or a portfolio’s exposure to a parallel shift in the yield curve—modified duration, Macaulay duration, and effective duration. As explained, modified duration assumes that, when interest rates change, the cash flows do not change. This is a limitation when measuring the exposure of bonds with embedded options (e.g., callable bonds, mortgage-backed securities, and some asset-backed securities) to changes in interest rates. Macaulay duration is related to modified duration and suffers from the same failure to consider changes in cash flows when interest rates change. In contrast, effective duration takes these changes in cash flows into account and thus, is the appropriate measure for bonds with embedded options. Effective duration is also referred to as option-adjusted duration.

The calculation of a portfolio’s duration begins with the calculation of the duration for each of the individual bonds comprising the portfolio. This calculation requires the use of a valuation model. Duration is found by shocking (i.e., changing) interest rates and computing the new value of the bond will be. Consequently, the duration measure is only as good as the valuation model. We explained how the effective duration of a bond with an embedded option is computed using the binomial model and also explained mortgage-backed and asset-backed securities.

For the 45-bond portfolio, the duration of each issue is shown in the column labeled “adjusted duration.” Lehman Brothers uses this term instead of effective duration or option-adjusted duration. It is important to understand that the measure used by Lehman Brothers takes into consideration the impact of changes in interest rates on the cash flows of a security. Here we see a clear example of the differences in the terminology used by major bond market participants.

To illustrate this calculation, consider the following 3-bond portfolio in which all three bonds are option free and pay a semiannual coupon:

In this illustration, it is assumed that the next coupon payment for each bond is six months from now. The market value of the portfolio is $9,609,961. The market price per $100 par value, the yield, and the duration for each bond are given below:

An alternative procedure for calculating the duration of a portfolio is to first calculate the dollar price change for a given number of basis points for each security and then add the price changes. Next, divide the total of the price changes by the initial market value of the portfolio to produce a percentage price change that can be adjusted to obtain the portfolio’s duration.

For example, consider the 3-bond portfolio shown above. Suppose we calculate the dollar price change for each bond in the portfolio, based on its duration, for a 50 basis point change in yield. We then have:

If the above formula is applied to the 45-bond portfolio, it can be shown that the portfolio’s duration is 4.59. Alternatively, portfolio duration can be computed from the durations of the sectors. For example, below we have the duration and the percentage of the portfolio for each sector:

This bond market index consists of various sectors which we described in detail in Chapter 16. A breakdown of the index by detailed sector, as of October 31, 2001, and the effective duration (“adjusted duration” in Lehman Brother’s terminology) are given below:

Duration is calculated by multiplying each sector weight by the duration for the sector and then summing these products. The third column above shows the product of each sector weight times the duration. The sum of the last column is 4.1. So the duration of the index is 4.1.

Notice that, if the benchmark index for the 45-bond portfolio is the Lehman Aggregate Bond Index, then the portfolio is more sensitive to a parallel shift in interest rates than the benchmark index (4.59 versus 4.10).

To illustrate this, look at Exhibit 4, which shows the sector breakdown of both the 45-bond portfolio and the benchmark index. The contribution to portfolio duration and the contribution to the benchmark index duration are shown.

A portfolio manager who wants to determine the contribution of a sector to portfolio duration relative to the contribution of the same sector in a broad-based market index can compute the difference between the two contributions. The difference in the percentage distribution by sector is not as meaningful as is the difference in the contribution to duration.

The convexity of the 45-bond portfolio is +0.13, and the convexity of the Lehman Aggregate Bond Index is −0.26. How does one interpret these two values? An extensive discussion explained why convexity value is difficult to interpret since it can be scaled in different ways. It is important to observe that the portfolio exhibits positive convexity while the benchmark exhibits negative convexity. This means that the manager should expect that, for a large decline in interest rates, the portfolio will outperform the benchmark index. We will confirm this when we describe scenario analysis in the next chapter.

One way to get a feel for the risk exposure resulting from yield curve shifts is to analyze the distribution of the present values of the cash flows for the portfolio and the benchmark index.²⁷⁶ Exhibit 5 shows the distribution of the present values of the cash flows for the 45-bond portfolio and for the Lehman Aggregate Bond Index. The exhibit demonstrates that the portfolio has greater cash flows (in present value terms) with shorter and longer maturities relative to the benchmark index and they are more barbelled.

Another way to see this exposure to changes in the yield curve is to compute the key rate durations of the portfolio and the benchmark. Key rate duration, which was described earlier, is the sensitivity of a portfolio’s value to the change in a particular key spot rate. The specific maturities on the spot rate curve for which key rate durations are measured vary from vendor to vendor. Exhibit 6 reports six key rate durations as computed by Lehman Brothers.²⁷⁷ Consider the 5-year key rate duration for the portfolio, 0.504. This measure is interpreted as follows: For a 100 basis point change in the 5-year rate holding all other rates constant, the portfolio’s value will change by approximately 0.5%.

The key rate durations reported in Exhibit 6 support the observation from the analysis of the present values of the cash flows: There is more exposure to changes in the short and long end of the maturity sectors and less exposure to the maturity sectors in between. (Note that the key rate durations shown are based on modified durations, not effective durations. At the time of the analysis, Lehman Brothers’ model did not produce key rate durations based on effective duration.)

The nominal spread is the traditional spread measure. That is, it is the difference between the yield on a spread product and the yield on a comparable maturity Treasury issue. Thus, when spread is defined as the nominal spread, spread duration indicates the approximate percentage change in price for a 100 basis point change in the nominal spread, holding the Treasury yield constant. It is important to note that, for any spread product, spread duration is the same as duration if the nominal spread is used. For example, suppose that the duration of a corporate bond is 5. This means that, for a 100 basis point change in interest rates, the value of the corporate bond changes by approximately 5%. It does not matter whether the change in rates is due to a change in the level of rates (i.e., a change in the Treasury rate) or a change in the nominal spread.

The zero-volatility spread, or static spread, is the spread that, when added to the Treasury spot rate curve, makes the present value of the cash flows (when discounted at the spot rates plus the spread) equal to the price of the bond plus accrued interest. It is a measure of the spread over the Treasury spot rate curve. When spread is defined in this way, spread duration is the approximate percentage change in price for a 100 basis point change in the zero-volatility spread, holding the Treasury spot rate curve constant.

The option-adjusted spread (OAS) is another spread measure²⁷⁸ that can be interpreted as the approximate percentage change in price of a spread product for a 100 basis point change in the OAS, holding Treasury rates constant. So, for example, if a corporate bond has a spread duration of 3, this means that, if the OAS changes by 20 basis points, then the price of this corporate bond will change by approximately 0.6% (0.03 × 0.002 × 100).

How do you know whether a spread duration for a fixed-rate bond is a spread based on the nominal spread, zero-volatility spread, or the OAS? You do not know. You must ask the broker/dealer or vendor of the analytical system.

The spread duration for the 45-bond portfolio is 3.22 compared to 2.69 for the benchmark index. That is, there is greater spread risk for the 45-bond portfolio than for the benchmark index. The greater spread duration is the result of the underweight in the Treasury sector combined with the longer adjusted duration in spread sectors, specifically Industrials. Notice that the calculation of spread duration is simply the sum of the contribution to duration for each sector.

Optionality risk can be quantified using measures commonly used in the option pricing. In option pricing theory, the delta of an option is an estimate of the sensitivity of the value of the option to changes in the price of the underlying instrument.²⁷⁹ For bonds, delta can be computed for each issue that has an embedded option and then these deltas can be aggregated to obtain an estimate of the delta for a portfolio or a benchmark index.

Exhibit 9 shows the delta of the 45-bond portfolio and the benchmark index, as well as the difference between the two. The percentage of the portfolio represents all holdings other than the mortgage sector. (Thus, the value of 65.30% for the portfolio is 100% minus the mortgage holdings of 34.7%.) Notice that the holdings are partitioned in terms of the embedded option and how the securities trade. Bullet bonds do not have embedded options and therefore the delta for these bonds is zero. A security with an embedded call option is classified as trading either to its call date or to its maturity date, depending on its market price. In this context, “trading to” indicates that the price of the security is such that the market is pricing the security either as if it will be called or as if it will not be called. The same is true for securities with a put option—they are classified as either trading to the put date or trading to maturity.

The three major risks associated with investing in the MBS sector are:

Exhibit 10 shows the spread risk for the different subsectors of the MBS market for both the 45-bond portfolio and the benchmark index. The risk is measured in terms of contribution to adjusted duration. Notice that the portfolio is substantially underweighted in unseasoned MBS and overweighted in moderately seasoned and seasoned MBS.

For example, suppose that, for some MBS at 500 PSA, the price is 110.08. A 1% increase in the PSA prepayment rate means that PSA increases from 500 PSA to 505 PSA. Suppose that, at 505 PSA, the price is recomputed using a valuation model and found to be 110.00. Therefore, the price decline is −0.08, in terms of basis points, −8, so that the prepayment sensitivity is −8.

Some MBS products increase in value when prepayments increase and some MBS products decrease in value when prepayments increase. Examples of the former are passthrough securities trading at a discount (i.e., passthrough securities whose coupon rate is less than the prevailing mortgage rate) and principal-only mortgage strips. These securities have positive prepayment sensitivity. Examples of MBS products that decrease in value when prepayments increase are passthrough securities trading at a premium (i.e., passthrough securities whose coupon rate is greater than the prevailing mortgage rate) and interest-only mortgage strips. These securities have negative prepayment sensitivity.

Exhibit 11 shows the risk exposure in terms of prepayment sensitivity for both the 45-bond portfolio and the benchmark. The risk exposure is provided for the same coupon subsectors used for the MBS sector risk in Exhibit 10. In addition, the subsectors are further partitioned in terms of conventional issues (i.e., Fannie Mae and Freddie Mac passthroughs), 30-year Ginnie Mae issues, 15-year MBS, and MBS backed by balloon mortgages.²⁸² A summary of the portfolio’s exposure relative to the benchmark is provided in the last three lines of the exhibit. The portfolio has a greater percentage invested in the GNMA 30-year and Balloon sectors than the benchmark and less invested in the 15-year MBS sector. However, the portfolio manager is not interested in the percentage difference for the amount invested in each sector, but rather the contribution to prepayment sensitivity. As seen in the exhibit, the prepayment sensitivity difference is primarily due to the greater exposure to the GNMA 30-year sector (−0.26).

Exposure to convexity risk in the MBS sector is computed from the difference between the convexity of a portfolio and the convexity of the benchmark index. (The appropriate convexity measure is effective convexity explained previously.) Convexity is −0.55 for the MBS sector of the 45-bond portfolio and −0.67 for the MBS sector of the benchmark index. So, the 45-bond portfolio has less negative convexity than the benchmark index. Exhibit 12 decomposes the convexity risk by subsectors within the MBS sector and shows the contribution to convexity by subsector.

In the fixed income area, several vendors and dealer firms have developed multi-factor risk models. In our illustration, we will use the Lehman Brothers multi-factor risk model.²⁸⁴ While the model covers U.S. dollar-denominated securities in most Lehman Brothers domestic fixed-rate bond indices (Aggregate, High Yield, Eurobond), we will use only the model for the Lehman Aggregate Bond Index. The historical data needed to estimate the model are updated monthly by Lehman Brothers. The portfolio we will analyze is the 45-bond portfolio that has been used throughout this chapter.

Much like other commercially available or dealer proprietary multi-factor risk models, the Lehman Brothers model focuses on predicted tracking error. Specifically, tracking error is defined as one standard deviation of the difference between the 45-bond portfolio and benchmark annualized returns. For the 45-bond portfolio, the predicted tracking error is 62 basis points per year.

If you add these two risks, you find a predicted tracking error risk greater than 62 basis points. What is wrong? There is nothing wrong. Remember that predicted tracking errors are standard deviations. We know that it is not correct to add standard deviations when computing the risk of a portfolio. To obtain the predicted tracking error for the portfolio, the sum of the squares of the two predicted tracking errors equals the square of the portfolio’s predicted tracking error. The square root of the sum of the squares of the two predicted tracking errors equals the portfolio’s predicted tracking error. That is, for the 45-bond portfolio:

The addition of variances assumes a zero correlation between the risk factors (i.e., the risk factors are statistically independent). If this is not the case, the correlation between the risk factors must be taken into account in estimating the predicted tracking error.

For the 45-bond portfolio, the Lehman model indicates that the predicted tracking error due to the non-term structure risk is 25.5 basis points. We now know that predicted tracking error due to systematic risk is 54.8 basis points, consisting of:

Again, the predicted tracking error due to systematic risk is not equal to the sum of these two components of predicted tracking error. If the term structure factor and the non-term structure factors are statistically independent, then the total predicted tracking error from all the systematic factors is 55.9 basis points (= [(49.8)²+ (25.5)²]^0.5). Why is there a difference between the 55.9 basis points just computed and the 54.8 basis points for the systematic risk? The reason is that the correlation (not reported here) between the two systematic risks is not zero. When the correlations are taken into account the predicted tracking error due to systematic risk is 54.8 basis points.

The Lehman model accounts for the correlation between risk factors in determining the predicted tracking errors due to each of the non-term structure risk factors. The results for the 45-bond portfolio are shown below:

The last column of the exhibit reports the percentage of the 45-bond portfolio’s market value invested in each issue. Since there are only 45 issues in the portfolio, each issue makes up a non-trivial fraction of the portfolio. Specifically, look at the exposure to two corporate issuers, Coca-Cola Enterprises and Daimler-Benz North America. The former represents about 10% of the portfolio and the latter almost 4%. If either firm is downgraded, this would cause large losses in the 45-bond portfolio, but it would not have a significant effect on the benchmark which includes 7,000 issues. Consequently, a large exposure to a specific corporate issuer represents a material mismatch between the exposure of the portfolio and the exposure of a benchmark index that must be taken into account in assessing a portfolio’s risk relative to a benchmark index.

As an example of non-systematic risk, suppose that a portfolio included Enron bonds in December 2001. For that month, Enron bonds lost more than three quarters of their value. The weighting of the bonds in the Merrill Lynch High Grade Index was 0.2%. If these bonds represent 1.6% of the portfolio, this would have resulted in a 100 basis point underperformance relative to the index at a time when sectors such as Consumer Cyclicals, Media, Telecom, and Technology outperformed the index by more than 200 basis points.²⁸⁵