1. Computing the Total Return The total return takes into account all three potential sources of dollar return over the investor’s investment horizon. It is the rate of return that will grow the amount invested (i.e., price plus accrued interest) to the projected total dollar return at the investment horizon.²⁸⁹ The calculation of the total return requires that the investor specify:

• the investment horizon

• the reinvestment rate

• the price of the bond at the investment horizon.

A graphical depiction of the total return calculation is presented in Exhibit 4. More formally, the steps for computing total return are as follows:

Step 1: Compute the total coupon payments plus the reinvestment income based on an expected reinvestment rate. If coupon payments are semiannual, the reinvestment rate is one-half the annual interest rate that the investor assumes can be earned on the reinvestment of coupon interest payments.⁵

EXHIBIT 4 Graphical Depiction of Total Return Calculation

Step 2: Determine the projected price of the bond at the investment horizon, which is referred to as the horizon price. We explained how the price of a bond is computed given the term structure of default-free interest rates (i.e., the Treasury spot rate curve) and the term structure of credit spreads. Moreover, for bonds with embedded options, the price depends on the option-adjusted spread (OAS). So, to determine the horizon price it is necessary to use a forecasted Treasury spot rate curve, term structure of credit spreads, and OAS at the horizon date. Obviously, the predicted values reflect changes in interest rates and spreads from the current date to the investment horizon. We shall refer to these rates as the structure of rates at the horizon date. However, in the illustrations to follow, we will simplify by assuming a single yield at the horizon date. This yield reflects the Treasury rate plus a spread and is referred to as the horizon yield.

Step 3: Add the values computed in Steps 1 and 2. Reduce this value by any borrowing cost to obtain the total future dollars that will be received from the investment, given the predicted reinvestment rate and projected structure of rates at the horizon date (or horizon yield in our illustrations to follow).

Step 4: Compute the semiannual total return as follows:

where the full price is the price plus accrued interest and h is the number of periods in the investment horizon. For bonds with semiannual coupon payments, h is the number of semiannual periods.

Step 5: For semiannual-pay bonds, double the interest rate found in Step 4. The result is the total return expressed on a bond-equivalent basis. Alternatively, the total return can be computed on an effective rate basis, using the following formula:

(1 + semiannual total return)² − 1

Whether the total return is calculated on a bond-equivalent basis or an effective rate basis depends on the situation. If the total return is compared to a benchmark index whose rate of return is calculated on a bond-equivalent basis, then the total return should be calculated in the same way. However, if the bond is used to satisfy liabilities with rates calculated on an effective basis, then the total return should be calculated on an effective rate basis.

To illustrate the computation of total return, suppose that an investor with a 1-year investment horizon is considering the purchase of a 20-year 6% corporate bond. The issue is selling for $86.4365 at a yield of 7.3%, and will be purchased for cash (i.e., no funds will be borrowed). Assume that the yield curve is flat and the projected yield for a 20-year Treasury issue is 6.5%. This means that the yield spread over the projected Treasury issue is 80 basis points. The investor expects that:

1. he can reinvest the coupon payments at 6%

2. the Treasury yield curve will shift down by 25 basis points and remain flat at the end of 1 year, so that the yield for the 19-year Treasury issue is 6.25% (6.5% minus 25 basis points)

3. the yield spread to the 19-year Treasury issue is unchanged at 80 basis points, so the horizon yield is 7.05% (6.25% plus 80 basis points)

The calculations are shown below.

Step 1: Compute the total coupon payments plus reinvestment income using an annual reinvestment rate of 6%, or 3% every six months. The semiannual coupon payment is $3. The future value of an annuity formula can be used for this calculation or, because the investment horizon is only one year, the future value can be computed as follows:

First coupon payment reinvested for six months = $3 (1.03)	=	$3.09
Second coupon payment	=	$3.00
Total	=	$6.09

Step 2: The horizon price is determined as follows. The price of this bond when discounted at a flat 7.05% yield is $89.0992.

Step 3: Adding the amounts in Steps 1 and 2 gives the total future dollars of $95.1892.

Step 4: Compute the following (h is 2 in our illustration):

Step 5: The total return on a bond-equivalent basis and on an effective rate basis are:

2 × 4.94% = 9.88%	(BEY)
(1 + 0.0494)² − 1 = 10.13%	(effective rate basis)

a. OAS-Total Return A valuation model can be used to incorporate the option-adjusted spread (OAS) into the calculation of the horizon price. The manager must specify the changes he expects in the OAS at the investment horizon. The horizon price can be “backed out” of a valuation model. This technique can be extended to the total return framework by making assumptions about the required variables at the horizon date.

Assumptions about the OAS at the investment horizon reflect the expectations of the portfolio manager. It is common to assume that the OAS at the horizon date will be the same as the OAS at the time of purchase. A total return calculated under this assumption is referred to as a constant-OAS total return. Alternatively, managers take positions that reflect their views on how the OAS will change. The total return framework can be used to assess the sensitivity of the performance of a bond with an embedded option to changes in the OAS.

b. Total Return for a Mortgage-Backed and Asset-Backed Security In calculating the total return for mortgage-backed and asset-backed securities, the total future dollars depend on (1) the projected principal repayment (scheduled repayment plus projected prepayments) and (2) the interest earned on reinvestment of projected interest payments and principal payments. To compute the total future dollars, a prepayment rate over the investment horizon must be projected.

The monthly total return for a mortgage-backed security or an asset-backed security with monthly payments is computed as follows:

The monthly total return can be annualized on a bond-equivalent yield basis as follows:

bond-equivalent annual return = 2[(1 + monthly total return)⁶ − 1]

Recall from our discussion that we calculate the bond-equivalent yield for a monthly pay security by first computing the effective 6-month yield and then doubling the effective 6-month yield. This is the calculation indicated by the bond-equivalent annual return formula above.

So, for example, if the monthly total return for a monthly pay mortgage-backed security or asset-backed security is 0.7%, the bond-equivalent annual return is

2[(1 + 0.007)⁶ − 1] = 0.0855 = 8.55%

Or, the effective annual return can be computed as follows:

effective annual return = (1 + monthly total return)¹² − 1

The effective annual return is computed by compounding the monthly return. For the previous example, the effective annual return is:

(1 + 0.007)¹² − 1 = 0.0873 = 8.73%

c. Scenario Analysis The computation of a total return is based on assumptions regarding interest rates expected at the investment horizon, spreads at the investment horizon, and reinvestment rates during the investment period. A manager does not rely on one set of assumptions when making an investment decision. Instead, a manager computes total return under different sets of assumptions. A set of assumptions is referred to as a scenario. Evaluating returns for a strategy under several scenarios is called scenario analysis. Regulators also require certain institutions to perform scenario analysis based on assumptions specified by regulations.²⁹⁰

Exhibits 5 and 6 provide illustrations of scenario analysis. The bond used in the illustrations is the 6% 20-year corporate bond selling for $86.4365 at a yield of 7.3%. In both exhibits, the assumptions in the scenario analysis are that the yield curve is flat and that shifts in the yield curve are parallel shifts. In Exhibit 5, we assume that only the Treasury yield curve shifts. In Exhibit 6, we assume that the yield spread changes and that the change varies with shifts in the Treasury yield curve.

EXHIBIT 5 Scenario Analysis Assuming only the Treasury Yield Curve Changes (1-Year Investment Horizon)

EXHIBIT 6 Scenario Analysis Assuming Shift in Treasury Yield Curve and Change in Yield Spread (1-Year Investment Horizon)

2. Controlling for Interest Rate Risk in Trades When assessing strategies, it is critical to compare positions with the same dollar duration, unless the objective of a trade is to alter the duration. To understand why this is so, consider two bonds, X and Y. Suppose that the price of bond X is 80 and the duration is 5, while bond Y has a price of 90 and a duration of 4. A 100 basis point change in the yield of bond X would change its price by about 5%, or a change of about $4 per $80 of market value. Thus, the dollar duration for a 100 basis point change in yield is $4 per $80 of market value. Similarly, for bond Y, the dollar duration for a 100 basis point change in yield per $90 of market value is $3.60. So, if bonds X and Y are being considered as alternative investments, the amount of each bond in the strategy should be such that both have the same dollar duration.

To illustrate, suppose that a portfolio manager owns $10 million par value of bond X which has a market value of $8 million. The dollar duration of bond X per 100 basis point change in yield is $400,000. Suppose further that the portfolio manager is considering exchanging bond X, which it owns, for bond Y. If the portfolio manager wants to have the same interest rate exposure (i.e., dollar duration) for bond Y that she currently has for bond X, she will buy a market value amount of bond Y that has the same dollar duration. If the portfolio manager purchased $10 million par value of bond Y, and therefore $9 million of market value, the dollar price change per 100 basis point change in yield would be only $360,000. If, instead, the portfolio manager purchased $10 million market value of bond Y, the dollar duration per 100 basis point change in yield would be $400,000. Since bond Y is trading at 90, $11.11 million par value of bond Y must be purchased to keep the dollar duration of bond Y equal to that of bond X.

Mathematically, the market value of bond Y that is required in order to have the same dollar duration (per 100 basis point change in rates) as bond X is:

The par value of bond Y that must be purchased to obtain the same dollar duration as bond X is:

Using our previous illustration to demonstrate how to use these formulas, we know that:

This means that $10 million market value of bond Y has the same dollar duration as the $8 million market value position in bond X. The amount of par value of bond Y that must be purchased, given its price of 90, or 0.90 per $1 of par value, is:

Failure to adjust a trade to account for an expected change in yield spread, keeping the dollar duration constant, leads to the result that the outcome of the trade will be affected by the expected change in the yield spread and the change in the yield level. Thus, a manager would be taking a conscious yield spread view and possibly an undesired view on the level of interest rates.

Also note that equating the dollar durations of two positions only means that they will be equal for small changes in rates because of the convexity of a bond.

3. Assessing the Portfolio There is no shortage of trading strategies suggested by bond dealer firms or in the popular press. Investment management firms have developed proprietary trading strategies. All of these strategies are based on a set of expectations about the bond market over the investment horizon. Furthermore, these strategies all involve risk. A trading strategy might involve borrowing in the repo market and/or shorting bonds. Some managers and dealers incorrectly use the term “arbitrage” to refer to these trading strategies they tout to customers. In fact, such strategies incur risk, even though the proponent of the strategy might perceive the risk to be small.

The potential performance of any trading strategy can be quantified using total return analysis. More specifically, scenario analysis is used to determine the total return under different assumptions about what might occur over the investment horizon. Scenario analysis identifies the range of possible outcomes and therefore provides the manager with a feel for the risk associated with a trade.

EXHIBIT 7 Three Hypothetical Treasury Securities Information on three Treasury securities:

In this section, we use a basic illustration illustrate how a trade can be evaluated. We begin with three Treasury securities—A, B, and C. Information about each of these three securities is provided in Exhibit 7. Security A is a short-term Treasury, security B is a long-term Treasury, and security C is an intermediate-term Treasury. Each security is selling at par, and it is assumed that the next coupon payment is six months from now. The duration and convexity for each security are calculated in the exhibit. Since all three securities are trading at par value, the respective durations and convexities are equal to the dollar durations and dollar convexities per $100 of par value.

Suppose that two Treasury portfolios are constructed. The first portfolio consists entirely of security C, the 10-year issue, and shall be referred to as the bullet portfolio, because the principal is returned at one time—the maturity date of the 10-year issue. The second portfolio invested 51.86% in security A and 48.14% in security B. This portfolio shall be referred to as the barbell portfolio, because the maturity dates are both shorter than and longer than that of the bullet portfolio.

As shown in Exhibit 7, the duration of the bullet portfolio is 6.94821. The duration of the barbell portfolio is the market value weighted average of the durations of the securities in the portfolio, and is computed below:

0.5186 (4.21122) + 0.4814 (9.89681) = 6.94826

The duration of the barbell is equal to the duration of the bullet. In fact, the barbell portfolio was designed to produce this result.

Duration is a first approximation of the change in market value resulting from a change in interest rates. As explained, the convexity measure provides an improvement to the duration estimate. The convexity measures of the bullet and barbell portfolios are not equal. We discussed the issues associated with computing the convexity measure previously so we won’t repeat the discussion here. The important concept to understand regarding the convexity measure in Exhibit 7 is the relative size of the convexity measures for the two portfolios. The convexity of the bullet portfolio is 31.09724. The convexity of the barbell is a market weighted average of the convexities of the securities in the portfolio. That is:

0.5186 (10.67912) + 0.4814 (73.63737) = 40.98722

Thus, the convexity of the bullet portfolio is less than that of the barbell portfolio. Below is a summary of the duration and convexity measures of the two portfolios:

	Treasury	portfolio
Parameter	Bullet	Barbell
Dollar duration	6.94821	6.94826
Dollar convexity	31.09724	40.98722

Suppose that a manager is considering the following trade: buy one portfolio and sell the other. In this case, selling a portfolio means that the manager sells short, or shorts, the security or securities in the portfolio. When an investor shorts a bond, the coupon interest paid to holders of the bond must be paid by the investor to the owner of the bond. Also assume that the manager is basing the trade on a 6-month investment horizon.

Since both portfolios have the same duration and the same dollar amount invested in each portfolio, then the portfolios have the same dollar duration, but different convexity. Suppose the manager believes that there will be significant interest rate volatility over the next six months. It is precisely under such circumstances that a portfolio with higher convexity experiences better performance. Consequently, suppose that the manager decides to buy the barbell portfolio because it has higher convexity, and short the bullet portfolio.

EXHIBIT 8 Performance of Trading Strategy Over a 6-Month Horizon Assuming a Parallel Yield Curve Shift: Scenario Analysis

Let’s assess this trade for the manager. First, note that the statement that a “large” change is expected is vague. Can the manager be more specific about how much rates must change in order to benefit from the better convexity of the barbell portfolio relative to the bullet portfolio? Also, is there an implicit assumption about what happens to the shape of the yield curve at the investment horizon? This can be quantified by using total return analysis and scenario analysis.

The last column of Exhibit 8 shows the total return over a six-month period for this trading strategy, assuming that the yield curve shifts in a parallel fashion. Since the manager owns the barbell portfolio and is short the bullet portfolio, then the difference between the total returns in columns (5) and (6) is the total return for this trading strategy. For example, if the yield curve shifts down by 150 basis points, the barbell portfolio would earn a 6-month total return of 29.26%. If the manager owned the bullet portfolio, then the total return would be 28.99% for the 150 basis point decline in yield. However, the manager is short the bullet portfolio so he must pay 28.99%. Thus, the barbell portfolio earned 29.26% but the manager had to pay 28.99%, resulting in a 27 basis point 6-month total return for this trading strategy.

The total return in the various scenarios shown in the last column allows the manager to quantify the size of the yield curve shift required in order to realize a positive return. The yield curve must change in a parallel fashion by more than 100 basis points (the precise number of basis points is not shown in Exhibit 8) in order to benefit from the better convexity of the barbell portfolio. Thus, the manager now knows more than simply the fact that he is taking a view on a “large” rate change, but, more specifically, a view that rates will change by more than 100 basis points.

Exhibit 9 shows the return for this strategy if the yield curve does not shift in a parallel fashion. In computing the total return for the trading strategy in Exhibit 9, we assume a steepening of the yield curve. There are an infinite number of ways that the yield curve can steepen. The scenario analysis in Exhibit 9 assumes that, for the change in yield for security C shown in the first column, the yield of A will change by the same amount less 30 basis points, whereas the yield of B will change by the same amount plus 30 basis points. The last column shows that the trade results in a loss for all scenarios except a 300 basis point shift.

Thus, we see the power of total return analysis and scenario analysis to sharpen our skills in assessing a trading strategy.

EXHIBIT 9 Performance of Trading Strategy Over a 6-Month Horizon Assuming a Steepening of the Yield Curve: Scenario Analysis

B. Assessing Strategies Relative to a Benchmark Index

One way to assess the strategic bets a manager makes is to examine how the portfolio is expected to perform relative to the benchmark index under various scenarios. That is, scenario analysis can be helpful and, again, performance is measured in terms of total return.

To illustrate how this is done, let’s consider once again the 45-bond portfolio described in the previous chapter to demonstrate the risk profile of a portfolio and benchmark index and the risk factors. (See Exhibit 3 in Chapter 17 for the portfolio composition.) The analysis was performed as of November 5, 2001, based on prices of October 31, 2001. The benchmark is the Lehman Aggregate Bond Index.

The duration and convexity for the 45-bond portfolio and for the benchmark index are:

	Duration	Convexity
Portfolio	4.59	0.13
Benchmark	4.10	− 0.26

The value of the 45-bond portfolio has greater sensitivity to a parallel shift in interest rates than does the benchmark index. The 45-bond portfolio exhibits positive convexity while the benchmark index exhibits negative convexity. Consequently, the manager should expect that for a large decline in interest rates, the 45-bond portfolio will outperform the benchmark index by an amount greater than the duration would indicate.

Now let’s do some simple scenario analysis to assess the performance of both the 45-bond portfolio and the benchmark index. The scenario analysis assumes a 1-year investment horizon and a parallel shift in interest rates. The results are shown in Exhibit 10, which reports the 1-year total return of both the portfolio and the benchmark index based on parallel shifts in the yield curve of 0, ±50, ±100, ±150, ±200, ±250, and ±300 basis points. Notice that the total return profile is as expected, given the duration and the convexity. Because the duration of the portfolio is greater than the duration of the benchmark index, the portfolio outperforms the benchmark when interest rates decline and underperforms when interest rates rise. Moreover, the positive convexity of the portfolio and the negative convexity of the benchmark imply that the better performance of the portfolio is further enhanced for a large decline in interest rates.

EXHIBIT 10 Scenario Analysis to Compare Performance (Total Return %) of Portfolio to Benchmark Index Assuming a Parallel Shift in Interest Rates

The scenario analysis framework in Exhibit 10 is useful for giving the portfolio manager a feel for the relationship between the performance of the portfolio relative to the benchmark index. However, this analysis considers only one risk: interest rate risk. In addition, the analysis of interest rate risk is limited to a parallel shift in the yield curve so that a nonparallel shift is not considered. Of course, a portfolio manager can “play” with different yield curve shifts to assess the impact on the portfolio and benchmark index. In fact, some portfolio managers seek to identify “realistic yield curve shifts” and analyze a portfolio based on these shifts. The simple scenario analysis of Exhibit 10 also ignores changes in spreads for spread products. Again, one can attempt to assess the performance assuming different changes in spreads. However, a more insightful methodology for quantifying the risk exposure of a portfolio relative to a benchmark index was discussed in the previous chapter—a multi-factor risk model.

V. USING MULTI-FACTOR RISK MODELS IN PORTFOLIO CONSTRUCTION

In the previous chapter we explained how multi-factor risk models can be used to quantify the risk exposure of a portfolio. But a multi-factor risk model is also an invaluable tool for constructing a portfolio, where the term “constructing a portfolio” means either the initial construction or the rebalancing of a portfolio.

In constructing a portfolio using the multi-factor risk model, an optimizer is used. The purpose of an optimizer is to obtain the optimal value for an objective function subject to constraints. Optimizers use mathematical programming, but a discussion of the types of mathematical programming and the algorithms used to solve mathematical programming problems is beyond the scope of this chapter. It is important to understand that, if a portfolio manager specifies an objective, the optimizer or mathematical programming model will compute the optimal solution. We will give two illustrations to see how this is done. In our illustrations, we will use the 45-bond portfolio and continue to use the Lehman U.S. Aggregate Bond Index as the benchmark index.

A. Rebalancing to Construct a More Passive Portfolio

In the previous chapter, we saw that the predicted tracking error for the 45-bond portfolio is 62 basis points per year. Suppose that the portfolio manager wants to reduce the predicted tracking error significantly. That is, the manager wants to rebalance the portfolio in such a manner that its risk exposure is much closer to that of the benchmark index. Depending on the extent to which the manager reduces the predicted tracking error, the manager shifts from actively managing a portfolio to an approach that more closely resembles enhanced indexing. The goal is to rebalance the portfolio in a cost efficient manner. Here is where the optimizer comes in.

An optimizer can be used to identify trades that reduce tracking error in a cost efficient way. Specifically, the objective specified for the optimizer is to minimize tracking error, the constraint is to keep the turnover of the portfolio to a minimum. The optimizer begins with a universe of market prices for acceptable securities. The acceptable securities are those permitted by the investment guidelines. The optimizer is designed to select a trade that identifies a 1-for-1 bond swap with the greatest reduction in tracking error per unit of each bond purchased.

When the optimizer was applied to the 45-bond portfolio, the following bond swap was identified:

The trades are stated here in terms of par value, but the cash values for each side of the trade are approximately equal. If this bond swap is undertaken, predicted tracking error for the portfolio would be reduced from 62 basis points to 43 basis points.

The next eight bond swaps identified by the optimizer, along with the resulting predicted tracking error after each trade, are shown below. The trades’ impact on the predicted tracking error is cumulative. While the cost of each transaction is not shown, the portfolio manager has the opportunity to compare the reduction in predicted tracking error with the cost of the trade.

After these nine bond swaps, the portfolio’s predicted tracking error would be reduced from 62 basis points to 22 basis points. The total transaction cost of these swaps would have been $1 million.

The same optimization approach described above can be used to rebalance an indexed portfolio as the index changes.

B. Incorporating Active Strategies

We have shown how a multi-factor risk model and an optimizer can be used to rebalance a portfolio in order to reduce predicted tracking error. An optimizer can also be used to structure a portfolio so as to incorporate a market view (on sectors or on the term structure) or a view on individual issues or issuers. To see how, let’s suppose that a manager follows a top down approach to the selection of sectors. Consequently, the manager seeks exposure to sector risk, while at the same time, seeking to minimize exposure to other risk factors such as term structure risk and non-sector non-term structure systematic risk. How can this be done?

An optimizer can be used to rebalance the current portfolio in a manner that maintains the current predicted tracking error for sector risk but reduces the predicted tracking error for the other systematic risks. The optimizer is set up so that there is a substantial penalty for reducing exposure to sector risk.

For example, consider the 45-bond portfolio. As discussed in the previous chapter, the predicted tracking error for the portfolio is 62 basis points and the predicted tracking error due to sector risk is 22.7 basis points. The optimizer was run to keep the predicted tracking error due to sector risk as close as possible to 22.7 basis points while reducing the predicted tracking error due to the other risk factors. The following nine bond swaps were identified by the optimizer:

After these bond swaps, the predicted tracking error for the portfolio declines from 62 basis points to 53 basis points. An analysis of the predicted tracking error risks for the original 45-bond portfolio and for the new portfolio is shown in Exhibit 11. Note first that the predicted tracking error due to sector risk was effectively unchanged (an increase of a mere 0.1 basis points). The major risk eliminated is the term structure risk. The predicted tracking error attributable to this source is reduced by 11 basis points. So, while the tracking error due to sector risk is unchanged, the portfolio’s predicted tracking error is reduced from 62 basis points to 53 basis, effectively attributable to a reduction in term structure risk.

VI. PERFORMANCE EVALUATION

Thus far in this chapter we have analyzed various portfolio strategies. In this section we turn our attention to the measurement and evaluation of the performance of a bond portfolio manager. Performance measurement requires the calculation of realized return over a specified time interval, called the evaluation period.

EXHIBIT 11 Predicted Tracking Error Before and After Nine Bond Swaps Seeking to Hold Tracking Error Due to Sector Risk Constant

Performance evaluation is concerned with two issues. The first is whether the manager added value by outperforming the designated benchmark index. The second is how the manager achieved the realized return. The decomposition of performance results used in order to understand how the results were achieved is called performance attribution analysis.

A bond performance and attribution analysis should satisfy three basic requirements.²⁹¹ First, the process should be accurate. For example, as explained below, there are several ways to measure portfolio return. The measure that is used should recognize the timing of each cash flow, resulting in a much more accurate measure of the actual portfolio performance.

Second, the process should be informative. It should measure the managerial skills that go into bond portfolio management. To be informative, the process must effectively address the key management skills, and measure these skills in terms of realized performance. For example, the process should identify the degree to which performance is attributed to changes in the level of interest rates, changes in the shape of the yield curve, changes in spreads, and individual security selection. The effect of transaction costs on performance should also be recognized.

Third, the process should be simple. The output of the process should be understood by manager and client, or others concerned with the performance of the portfolio.

While the process must provide accurate information, this does not mean that analysis needs to be accurate to the n-th decimal place. Rather, it should provide the client with an accurate picture of the major areas, as well as order of magnitude, of outperformance or underperformance. A client engages a manager, at least in part because of claims made by the manager regarding the strategies that will be pursued. Information that indicates whether a manager’s performance is consistent with these claims is critical in the decision to retain or discharge the manager.

A. Performance Measurement

The starting point for evaluating a manager’s performance is the measurement of return. Essentially the measurement process begins with the calculation of total return for the sub-periods of the evaluation period. For example, if the evaluation period is one year, the sub-period might be a month, so that there would be 12 sub-period returns. Three methodologies that can be used to calculate the average sub-period return in order to compute the return for the evaluation period: (1) the arithmetic average rate of return, (2) the time-weighted rate of return (also called the geometric rate of return), and (3) the dollar-weighted rate of return. These methodologies and their limitations are discussed elsewhere.⁸

Because rate of return is influenced by the client’s decisions with respect to the periodic withdrawal and contribution of funds, a return measure should be reflective of the manager’s performance without being affected by client decisions to add or take away funds. The proper methodology to resolve this problem is the time-weighted rate of return. The CFA Institute Performance Presentation Standards require the use of time-weighted rates of return that adjust for cash flows (i.e., contributions and withdrawals). Time-weighted rates of return that adjust for daily-weighted cash flows must be used for periods beginning January 1, 2005 and later.

B. Single-Index Performance Evaluation Measures

In the 1960s, several single-index measures were used to evaluate the relative performance of portfolio managers. The three measures are referred to as the Treynor measure, the Sharpe measure, and the Jensen measure. All three measures assume that there is a linear relationship between the portfolio’s return and the return on the benchmark index.²⁹² These measures did not indicate how or why a portfolio manager outperformed or underperformed a benchmark index.

In early studies of portfolio manager performance, these measures were used primarily to evaluate the performance of equity mutual fund managers. However, these measures are less useful today because of the subsequent development of the performance attribution models discussed next.

C. Performance Attribution Analysis

Single-index performance evaluation measures do not explain how a manager achieved a particular observed return. The reason a client must have the answer to this question is that a manager may tell a client that he plans to pursue an active strategy. The client would then expect that any superior return achieved by the manager results from this active strategy. But how can the client be certain?

For example, suppose a manager who solicits funds from a client claims that he can achieve superior performance by selecting underpriced bonds. Suppose also that this manager generates a superior return compared to the client-designated bond index. The client should not necessarily be satisfied with this performance until the return realized by the manager is decomposed into the components that generated the return. A client may find that the superior performance is due to the manager’s timing of the market (i.e., modifying portfolio duration in anticipation of interest rate movements) rather than due to selecting underpriced bonds. In such an instance, the manager may have outperformed the benchmark index, but not by following the strategy that the manager stated that he intended to pursue.

Performance attribution analysis seeks to identify and quantify the active management decisions that contributed to the performance of a portfolio. The performance of a portfolio can be decomposed in terms of the risk factors discussed in Chapter 17.

Several commercial vendors have developed models for performance attribution analysis. The analysis is performed relative to a benchmark index. We will illustrate performance attribution models using the system developed by Global Advanced Technology (G.A.T.).²⁹³ This model decomposes a portfolio’s total return into the following factors: (1) static return, (2) interest sensitive return, (3) spread change return, and (4) trading return. The difference between the total return and the sum of the four factors is called the residual (error). Each of the return factors can be further decomposed as described below.

The static return is the portion of a portfolio’s total return that is attributable to “rolling down the yield curve.” That is, the static return is the portion of the return that would be earned assuming a static (meaning zero volatility) world, in which the yield curve evolves to its implied forward curve. The static return is further decomposed into two components: (1) risk-free return and (2) accrual of OAS return. The risk-free return is based on the assumption that the portfolio consists of only Treasury strips. The risk-free return is then calculated based on the rolling down of the yield curve. The accrual of OAS return is also calculated from rolling down the yield curve, but, unlike the risk-free return, it is based on investing in spread products.

The interest sensitive return is that portion of a portfolio’s return attributable to changes in the level, slope, and shape of the yield curve. In turn, this return is decomposed into two components: (1) effective duration return and (2) convexity return. Key rate durations can be used to measure sensitivity to changes in the shape of the yield curve. The effective duration return is the sum of the returns attributable to the key rate durations. The convexity return is the return due to change in the portfolio’s duration over the evaluation period.

The spread change return is the portion of a portfolio’s return that is due to changes in both (1) bond sector spreads and (2) individual security richness/cheapness. The portion of the spread return attributable to changes in the sector’s OAS is called the delta OAS return and the spread return due to a widening or tightening of a specific issue’s spread is called the delta rich/cheap return.

The portion of a portfolio’s total return that is attributable to changes in the composition of the portfolio is called the trading return. The trading return identifies the manager’s value added from changes in the composition of the portfolio, as opposed to a simple buy-and-hold strategy.

In the illustration, a performance attribution analysis is performed on a portfolio of corporate securities. We will refer to this portfolio as Portfolio A. The evaluation period is September 1996, a month when the yield curve shifted downward. The shift was almost a parallel shift. The effective duration of Portfolio A was 7.09. The effective duration of the benchmark index, the Merrill Lynch Corporate Index, was 5.76. Thus, Portfolio A had a higher duration than did the benchmark index.

Exhibit 12 presents the results of the performance attribution analysis for the portfolio and for the individual sectors. The second column provides information about the manager’s sector views (i.e., underweighting or overweighting). The third column shows how the allocation paid off for each sector.

Because the yield curve shifted downward in an almost parallel fashion, Portfolio A would be expected to outperform the benchmark index because of the portfolio’s higher duration. This is captured in the interest sensitive return, the fifth column of Exhibit 12. This indicates that, holding all other factors constant, the outperformance would have been 37.9 basis points. Portfolio A did not outperform by that much because the spread change return which was −18.5 basis points. The static return and the trading return were minimal.

Exhibit 13 provides a summary of the analysis. The return on Portfolio A was 2.187% and the return on the benchmark index was 1.954%, so that Portfolio A outperformed the index by 23 bps in September 1996.

VII. LEVERAGING STRATEGIES

A manager may be permitted to use leverage as part of a trade or trading strategy. Leverage means that funds are borrowed to purchase some of the securities involved in the strategy.

EXHIBIT 12 Performance Attribution Example

Source: G.A.T. Integrative Bond System

So, we begin this section with a discussion of the advantages and disadvantages of leverage, and then we discuss the use of repurchase agreements as a source of borrowed funds. Scenario analysis, discussed in Section IV, can then be used to assess a trade or a portfolio strategy that uses leverage.

A. The Principle of Leverage

Leveraging is the investment approach of borrowing funds with the expectation of earning a return in excess of the cost of funds. The attractive feature of leveraging is that it magnifies the return realized from investment in a security for a given change in the price of that security. That’s the good news. The bad news is that leveraging also magnifies losses.

EXHIBIT 13 Summary of Performance Attribution Analysis

To illustrate, consider an investor who plans to purchase a 30-year U.S. Treasury bond in anticipation of a decline in interest rates six months from now. The investor has $1 million to invest, which is referred to as the investor’s equity. Assuming that the coupon rate for the Treasury bond is 8%, the next coupon payment is six months from now, and the bond can be purchased at par value, then the investor can purchase $1 million par value of the Treasury bond with the equity available.

Exhibit 14 shows the return that will be realized for various assumed yields six months from now for the Treasury bond. The dollar return consists of the coupon payment six months from now plus the change in the value of the Treasury bond. There is no reinvestment income. At the end of six months, the 30-year Treasury bond is a 29.5-year Treasury bond. The percent return is found by first dividing the dollar return by the $1 million of investor’s equity and then multiplying by 2 to annualize so the return is computed on a bond-equivalent basis. Notice that the range for the annualized rate of return is from −29.8% to +63.0%.

In our illustration, the investor does not borrow any funds, so that the strategy is referred to as an unleveraged strategy. Now suppose that the investor borrows $1 million in order to purchase an additional $1 million of par value of the Treasury bond. Assume further that the loan agreement specifies that:

1. the maturity of the loan is six months

2. the annual interest rate for the loan is 9%, and

EXHIBIT 14 Annual Return from a $1 Million Investment in a 30-Year 8% Coupon Treasury Bond Held for Six Months

EXHIBIT 15 Annual Return from a $2 Million Investment in a 30-Year 8% Coupon Treasury Bond Held for Six Months Using $1 Million of Borrowed Funds

3. $1 million par value of the 30-year 8% coupon Treasury bond is used as collateral for the loan

Therefore, the loan is a collateralized loan.

The amount invested is then $2 million, which comes from the investor’s equity of $1 million and $1 million of borrowed funds. In this strategy, the investor uses leverage. Since the investor has the use of $2 million in proceeds and has equity of $1 million, this is called “2-to-1 leverage.” (This means $2 invested for $1 in investor’s equity.)

Exhibit 15 shows the annual rate of return for this leveraged strategy for the same yields shown in Exhibit 14. The return is measured relative to the investor’s equity of $1 million, not the $2 million. The dollar return shown in the exhibit adjusts for the cost of borrowing.

Because of the use of borrowed funds, the range for the annualized percent return is wider (−68.5% to +117.0%) than in the unleveraged case (−29.8% to 63.0%). This example clearly shows that leverage is a two-edged sword—it magnifies returns both up and down. Notice that, if the market yield does not change at the end of six months, then the unleveraged strategy generates an 8% annual return. That is, for the $1 million invested, the coupon interest is $40,000 for six months. Since there is no change in the market value of the security, this produces a 4% semiannual return, or 8% on a simple annual basis (i.e., a bond-equivalent basis). In contrast, consider what happens if $2 million is invested in the 2-for-1 leveraging strategy. Since $2 million is invested, the coupon interest is $80,000 for six months. But the interest cost of the $1 million loan for six months is $45,000 ($1 million × 9%/2). Thus, the dollar return after the financing cost is $35,000 ($80, 000 − $45, 000). Hence, the return on the investor’s $1 million equity is 3.5% for six months ($35,000/$1 million) and 7% annualized. Without leverage, the investor earns 8% if interest rates do not change but only 7% in the same scenario in the 2-for-1 leveraging strategy.

Suppose that, instead of borrowing $1 million, the investor is able to borrow $11 million for six months at an annual interest rate of 9%. The investor can now purchase $12 million of Treasury bonds, using $1 million of investor’s equity and $11 million of borrowed funds. The lender requires that the $11 million of Treasury bonds be used as collateral for this loan.

EXHIBIT 16 Annual Return from a $12 Million Investment in a 30-Year 8% Coupon Treasury Bond Held for Six Months Using $11 Million of Borrowed Funds

Since there is $12 million invested and $1 million of investor’s equity, this strategy is said to have “12-to-1 leverage.”

Exhibit 16 shows the annual return assuming the same yields used in Exhibits 14 and 15. Notice the considerably wider range of annual returns for the 12-to-1 leverage strategy compared to the 2-to-1 leverage strategy or the unleveraged strategy. When the yield remains at 8%, the 12-to-1 strategy results in an annual return of −3%. This result occurs because the coupon interest earned on the $12 million invested for six months is $480,000 ($12 million × 8%/2) but the interest expense is $495,000 ($11 million borrowed × 9%/2). The dollar return to the investor for the 6-month period is then −$15,000 or −1.5% (−$15,000/$1 million). Doubling the −1.5% semiannual return gives the −3% annual return.

Exhibit 17 shows the range of outcomes for different degrees of leverage. The greater the leverage, the wider the range of potential outcomes.

B. Borrowing Funds via Repurchase Agreements

EXHIBIT 17 Annual Return for Various Degrees of Leverage

A repurchase agreement is the sale of a security with a commitment by the seller to buy back the same security from the purchaser at a specified price at a designated future date. The price at which the seller must subsequently repurchase the security is called the repurchase price and the date by which the security must be repurchased is called the repurchase date. Basically, a repurchase agreement is a collateralized loan, where the collateral is the security that is sold and subsequently repurchased.²⁹⁴ The agreement is best described with an illustration.

Suppose a government securities dealer has purchased $10 million of a particular Treasury security. The dealer can finance the position with its own funds or by borrowing from a bank. Typically, however, the dealer uses the repurchase agreement or “repo” market to obtain financing. In the repo market, the dealer uses the $10 million Treasury security as collateral for the loan. The term of the loan and the interest rate the dealer agrees to pay are specified. The interest rate is called the repo rate. When the term of the loan is one day, it is called an overnight repo (or RP); a loan for more than one day is called a term repo (or term RP). The difference between the purchase (repurchase) price and the sale price is the dollar interest cost of the loan.

Suppose now that a customer of the dealer firm has $10 million. The dealer firm agrees to deliver (“sell”) $10 million of the Treasury security to the customer in exchange for $10 million and simultaneously agree to buy back (i.e., “repurchase”) the same Treasury security the next day for $10 million plus interest.

The dollar amount of the interest is determined by the repo rate, the number of days that the funds are borrowed (i.e., the term of the loan), and the amount borrowed. The formula for the dollar interest is:

dollar interest = amount borrowed × repo rate × repo term/360

Notice that the dollar interest is computed on an actual/360-day basis.

In our example, if the repo rate is 5%, then we know:

amount borrowed	=	$10, 000, 000
repo rate	=	0.05
repo term	=	1 day

Therefore the dollar interest is:

dollar interest = $10, 000, 000 × 0.05 × 1/360 = $1, 388.89

So, the dealer sells the Treasury security to the customer for $10 million and agree to repurchase it the next day for $10,001,388.89 ($10, 000, 000 + $1, 388.89).

The advantage for the dealer in using the repo market to borrow on a short-term basis is that the repo rate is lower than the cost of bank financing. (The reason for this is explained below.) From the customer’s perspective, the repo market offers an attractive yield on a short-term secured transaction that is highly liquid.

While the example illustrates the use of the repo market to finance a dealer’s long position, dealers can also use the repo market to cover a short position. For example, suppose a government dealer sold short $10 million of Treasury securities two weeks ago and must now cover the position—that is, deliver the securities. The dealer can do a reverse repo (agree to buy the securities and then sell them back). Of course, the dealer eventually would have to buy the Treasury security in the market in order to cover the short position. In this case, the dealer actually makes a collateralized loan to the customer. The customer (or other dealer) then uses the funds obtained from the collateralized loan to create leverage.

1. Industry Jargon A good deal of industry jargon is used to describe repo transactions. In order to understand the terminology, remember that one party lends money and accepts a security as collateral for the loan; the other party borrows money and provides collateral in the form of a security.

We discuss the following terminology below:

1. “repo” versus “reverse repo”

2. “reversing out a security” and “reversing in a security”

3. “selling collateral” and “buying collateral”

First, if a party loans a security and receives cash, then, from this party’s perspective, the transaction is called a “repo.” If a party loans cash and receives a security as collateral, this is a “reverse repo” transaction from this party’s perspective.

When someone lends a security (i.e., uses a security as collateral) in order to receive cash (i.e., borrow money), that party is said to be “reversing out” a security. A party that lends money with a security as collateral is said to be “reversing in” a security.

Finally, the expressions “selling collateral” and “buying collateral” are used to describe a party financing a security with a repo on the one hand, and lending on the basis of collateral, on the other.

To summarize, the following expressions are used for the transaction:

Borrower of funds	Lender of funds
repo	reverse repo
reversing out a security	reversing in a security
selling collateral	buying collateral

Rather than using industry jargon, investment guidelines should be clear as to what a manager is permitted to do. For example, a client may have no objections to its portfolio manager using a repo as a short-term investment—that is, the portfolio manager may lend funds on a short-term basis. The investment guidelines will set forth how the loan arrangement should be structured to protect against credit risk. We’ll discuss this below. However, if a client does not want a money manager to use the repo agreement as a vehicle for borrowing funds (thereby, creating leverage), it should state so.

2. Margin and Marking to Market Despite the fact that high-quality collateral underlies a repo transaction, both parties to the transaction are exposed to credit risk. Why does credit risk exist in a repo transaction? Consider our initial example of the dealer who uses $10 million of government securities as collateral for a loan. If the dealer cannot repurchase the government securities, the customer keeps the collateral; if interest rates on government securities increase subsequent to the repo transaction, however, the market value of the government securities declines, and the customer then owns securities with a market value less than the amount of the loan. If the market value of the security rises instead, the dealer will be concerned with the return of the collateral, which then has a market value greater than the amount of the loan.

Repos should be carefully structured to reduce credit risk exposure. The amount lent should be less than the market value of the security used as collateral, thereby providing the lender with cushion should the market value of the security decline. The amount by which the market value of the security used as collateral exceeds the value of the loan is called repo margin or simply margin. Margin is also referred to as the “haircut.” Repo margin is generally between 1% and 3%, but can be 10% or more for borrowers of lower credit worthiness and/or when less liquid or more price sensitive securities are used as collateral.

For example, consider the dealer who needs to borrow $10 million to finance the purchase of a Treasury security. Suppose that the repo margin is 2%. Then for a Treasury security with a market value of $10 million, only 98% of that amount, $9.8 million, will be lent. That is, the dealer will agree to deliver (sell) $10 million of the Treasury security to the customer for $9.8 million and agree to repurchase the $10 million of the Treasury security the next day for $9.8 million plus the dollar interest. The dollar interest for this overnight repo, assuming a repo rate of 5%, is:

dollar interest = $9, 800, 000 × 0.05 × 1/360 = $1, 361.11

Note that the dollar interest is based on $9.8 million (the amount actually lent by the customer), not $10 million as in our earlier example.

Another practice used to limit credit risk is to mark the collateral to market on a regular basis. Marking a position to market means recording the value of a position at its market value. When the market value changes by a certain percentage, the repo position is adjusted accordingly. The decline in market value below a specified amount results in a margin deficit. In such cases, the borrower of funds typically has the option to resolve the margin deficit either by providing additional cash or by transferring additional acceptable securities to the lender. When the market value rises above the amount required, excess margin will result. When this occurs, the lender of funds has the option to give cash to the borrower equal to the amount of the excess margin or to transfer purchased securities to the borrower.

3. Delivery and Credit Risk One concern of the parties to a repo is delivery of the collateral to the lender. The most obvious procedure is for the borrower to deliver the collateral to the lender or to the lender’s clearing agent. In such instances, the collateral is said to be “delivered out.” At the end of the repo term, the lender returns the collateral to the borrower in exchange for the principal and interest payment. This procedure may be expensive though, particularly for short-term repos, because of costs associated with delivering the collateral. The cost of delivery would be factored into the repo rate. The lender’s risk in terms of taking possession of the collateral derives from the fact that the borrower might sell the security, or go under so that the lender has nothing to liquidate, or use the same security as collateral for a repo with another party.

As an alternative to delivering out the collateral, the lender might agree to allow the borrower to hold the security in a segregated customer account. Of course, in this arrangement, the lender still faces the risk that the borrower might use the collateral fraudulently by offering it as collateral for another repo transaction. If the borrower holds the collateral, then the transaction is called a hold-in-custody repo (HIC repo). Despite the credit risk associated with a HIC repo, this arrangement is sometimes used in transactions when the collateral is difficult to deliver or when the transaction amount is small and the lender is comfortable with the reputation of the borrower.

Another arrangement is for the borrower to deliver the collateral to the lender’s custodial account at the borrower’s clearing bank. The custodian then has possession of the collateral on behalf of the lender. This practice reduces the cost of delivery because it requires only a transfer within the borrower’s clearing bank. If, for example, a dealer enters into an overnight repo with Customer A, the collateral is transferred back to the dealer the next day. The dealer can then enter into a repo with Customer B for, say, five days without having to redeliver the collateral. The clearing bank simply establishes a custodian account for Customer B and holds the collateral in that account. This specialized type of arrangement is called a tri-party repo. Tri-party repos account for about half of all repo arrangements.

The third party bank is responsible for marking the collateral to market and reporting these values each day to the two parties. Also, if the borrower wishes to substitute collateral (i.e., change the specific Treasury securities collateralizing the loan), the third-party agent verifies that the collateral satisfies the requirements set forth in the repo agreement.

4. Determinants of the Repo Rate The repo rate varies from transaction to transaction depending on a variety of factors, including:

• quality of collateral

• term of the repo

• delivery requirement

• availability of collateral

• prevailing federal funds rate

The higher the credit quality and liquidity of the collateral, the lower the repo rate. The repo rate varies with the term of the repo agreement. This is basically the very short-term end of the yield curve. The maturity of the security used as collateral for the repo does not affect the repo rate. If delivery of the collateral to the lender is required, the repo rate is lower.

The more difficult it is to obtain the collateral, the lower the repo rate. To understand why this is so, remember that the borrower (or equivalently the seller of the collateral) has a security that lenders of cash want. Such collateral is referred to as hot collateral or special collateral (or just as “on special”). Collateral that does not have this characteristic is referred to as general collateral. The party that needs the hot collateral will be willing to lend at a lower rate in order to obtain the collateral.

While these factors determine the repo rate on a particular transaction, the federal funds rate determines the general level of repo rates in the United States. Banks borrow funds from each other via the federal funds market. The interest rate charged on such borrowing is called the federal funds rate (or “fed funds”). The repo rate is generally lower than the federal funds rate because a repo is a form of collateralized borrowing, while a federal funds transaction is unsecured borrowing.

C. Computing Duration for a Leveraged Portfolio

When a manager borrows to leverage a portfolio, the following procedure should be used to compute the portfolio’s duration. The portfolio is then composed of the assets and the liabilities. The change in the portfolio value that includes borrowed funds is equal to:

Dividing the total change in the portfolio value by the initial value of the portfolio and adjusting for the basis point change in rates equals the duration of the portfolio. If the liabilities are short term, then the duration of the liability is low.

To illustrate these calculations, consider the following three-bond portfolio, with duration and change in value for a 50-basis point change as shown:

Note that this is the portfolio used in Chapter 17 to illustrate the duration calculation for an unlevered portfolio. Suppose that $2 million was borrowed to buy the securities in the portfolio, so that the amount of the client’s funds invested (i.e., the equity) is $7,609,961. The client is interested in how the rate changes affect the equity investment. Suppose further that the funds are borrowed using a 3-month repurchase agreement so that the duration of the liabilities is close to zero and the dollar change in the value of the liabilities for a 50 basis point change in rates is close to zero. Then the change in the value of the portfolio for a 50 basis point change in rates is computed as follows:

Thus, the percentage change in the portfolio’s value for a 50 basis point change in rates is 4.08% ($310,663 divided by $7,609,961), so the portfolio’s duration is 8.16. The higher duration that results from the $2 million short-term reverse repo borrowing (8.16 versus 6.47) is due to the leveraging of the portfolio.

I. INTRODUCTION

II. DEGREES OF ACTIVE MANAGEMENT

A. Pure Bond Index Strategy

B. Enhanced Indexing/Matching Risk Factors

C. Enhanced Indexing/Minor Risk Factor Mismatches

D. Active Management/Larger Risk Factor Mismatches

E. Unrestricted Active Management

III. STRATEGIES

A. Interest Rate Expectations Strategies

B. Yield Curve Strategies

C. Inter- and Intra-Sector Allocation Strategies

D. Individual Security Selection Strategies

IV. SCENARIO ANALYSIS FOR ASSESSING POTENTIAL PERFORMANCE

A. Assessing Trades