In the spectrum of strategies defined by Volpert, this strategy is called an “enhanced strategy,” although some investors refer to this as simply an indexing strategy. Two commonly used techniques to construct a portfolio that replicates an index are:

Volpert identifies two types of active strategies. In the more conservative of these strategies, the manager creates larger mismatches, in terms of risk factors, relative to the benchmark index. This includes minor duration mismatches. Typically, there is a limit to the degree of duration mismatch. For example, the manager might be constrained to within ±1 of the duration of the index.²⁸⁷ So, if the duration of the index is 4, the manager may construct a portfolio with duration between 3 and 5. To take advantage of anticipated reshaping of the yield curve, the manager might create a portfolio such that there are significant differences in cash flow distribution between the benchmark index and the portfolio. As another example, if the manager believes that, within the corporate sector, A rated issues will outperform AA rated issues, the manager might overweight the A rated issues and underweight AA rated issues.

The manager can choose to invest a small percentage of the portfolio in one or more sectors that are not in the bond market index. For example, the portfolio might have a small allocation to nonagency mortgage-backed securities. If the index includes only investment-grade products, the manager can allocate some portion of the portfolio, if permitted by investment guidelines, to non-investment-grade products.

Most active managers fall into this category of active management rather than the unrestricted active management category described below.

We will discuss various strategies used in active bond portfolio management in the next section.

Strategic strategies, sometimes referred to as top down value added strategies, include the following:

Tactical strategies, sometimes referred to as relative value strategies, are short-term trading strategies. They include:

To explain strategic strategies, we will use the portfolio recommended by Lehman Brothers to its clients in a August 27, 2001 publication (Global Relative Value). The recommended portfolio, shown in Exhibit 1, is for a manager whose benchmark index is the Lehman U.S. Aggregate Bond Index. Note that, in this exhibit, the credit sector is basically the corporate sector, as discussed in Chapter 16.

A portfolio’s duration may be altered in the cash market by swapping (or exchanging) bonds in the portfolio for new bonds that will achieve the target portfolio duration. Or, a manager can sell bonds and stay in cash (i.e., invest in money market instruments) if he believes that interest rates are going to increase. Alternatively, a more efficient means for altering the duration of a bond portfolio is to use interest rate futures contracts. As we explain in Chapter 22, buying futures increases a portfolio’s duration, while selling futures decreases the duration.

The key to this active strategy is, of course, an ability to forecast the direction of interest rates. The academic literature does not support the view that interest rates can be forecasted with accuracy in order to realize positive active returns on a consistent basis. It is doubtful that betting on future interest rates will provide consistently superior returns.

While a manager might not pursue an active strategy based strictly on future interest rate movements, certain circumstances can lead a manager to make an interest rate bet to cover inferior performance relative to a benchmark index. For example, suppose a manager represents himself to a client as pursuing one of the active strategies discussed later in this chapter. Suppose further that the manager is evaluated over a 1-year investment horizon, and that, three months before the end of the investment horizon, the manager’s performance is significantly below the client-specified benchmark index. If the manager believes the account will be lost due to the underperformance, the manager has an incentive to bet on interest rate movements. If the manager is correct, the account will be saved, while an incorrect bet will result in further underperforming the index. In this case, the account will probably be lost regardless of the extent of the underperformance. A client can prevent this type of gaming by imposing constraints on the extent to which the portfolio’s duration can diverge from that of the index. Also, in the performance evaluation stage of the investment management process, described later in this chapter, decomposing the portfolio’s return into the risk factors that generate the return highlights the extent to which a portfolio’s return is attributable to changes in the level of interest rates.

Exhibit 1 shows the option-adjusted duration for the Lehman Brothers Aggregate Index and for the recommended portfolio. In this case, Lehman Brothers uses the term “option-adjusted duration” rather than effective duration, which in other reports, the term “adjusted duration” is used. Exhibit 1 shows that the recommended duration for a U.S. aggregate portfolio is 4.63, compared to 4.56 for the index. That is, the recommended portfolio duration is 102% of the index duration and therefore the portfolio has slightly greater exposure to changes in interest rates.

Top down yield curve strategies involve positioning a portfolio to capitalize on expected changes in the shape of the Treasury yield curve. There are three yield curve strategies: (1) a bullet strategy, (2) a barbell strategy, and (3) a ladder strategy. In a bullet strategy, the portfolio is constructed so that the maturities of the bonds in the portfolio are highly concentrated at one point on the yield curve. In a barbell strategy, the maturities of the bonds in the portfolio are concentrated at two extreme maturities. In practice, when managers refer to a barbell strategy, they are referring to a comparison with a bullet strategy. For example, a bullet strategy might be to create a portfolio with maturities concentrated around 10 years, while a corresponding barbell strategy might be a portfolio with 5-year and 20-year maturities. In a ladder strategy, the portfolio has approximately equal amounts of each maturity. So, for example, a portfolio might have equal amounts of bonds with one year to maturity, two years to maturity, etc.

Each of these strategies results in different performance when the yield curve shifts. The actual performance depends on both the type and magnitude of the shift. Thus, no general statements can be made about the optimal yield curve strategy. This conclusion will be demonstrated in Section IV.

When a yield curve strategy is applied by a manager whose benchmark index is a broad-based bond market index, a mismatch of maturities relative to the benchmark index exists in one or more of the bond sectors. Some managers use duration buckets rather than maturity, as a measure of the mismatch. For example, in Exhibit 1, the index and the recommended portfolio are grouped into five duration ranges: 0-2, 2-4, 4-7, 7-9, and 9+. Notice that the recommended portfolio underweights the 0-2 and 9+ duration buckets. That is, the short and long end of the yield curve are underweighted relative to the benchmark index.

Basically, this allocation strategy is aimed at benefiting from spread products—products that expose investors to credit risk and prepayment risk for mortgages (and some asset-backed securities). The recommended portfolio has less exposure to the Treasury sector and more exposure to the spread product sectors. Notice that all of the spread product sectors are overweighted.

As explained in the previous chapter, spread duration measures the exposure of a portfolio to changes in spreads. Exhibit 1 provides information about the exposure to spread risk. First is the difference in spread duration between the benchmark index (3.28) and the recommended portfolio (3.73). This differential is to be expected, given the overweighting of the spread sectors in the recommended portfolio relative to the benchmark index. The spread duration quantifies the extent of this overweighting. The second is the difference between the contribution to spread duration for each sector in the index and the corresponding sector in the recommended portfolio.

In an intra-sector allocation strategy, the manager’s allocation of funds within a sector differs from that of the index. Exhibit 1 shows the allocation within each sector as a percentage of the total market. Exhibit 2 displays Lehman Brothers’ intra-sector allocation recommendation for the credit sector as of August 24, 2001 in terms of contribution to spread duration for each credit quality and by sector. The credit sector is equivalent to the corporate bond sector as previously noted, and is further divided into the following subsectors: industrial, financial, utility, and non-corporate.²⁸⁸

Exhibit 3 shows the recommended allocation for the MBS sector as of the same date. This recommendation is presented in terms of market value and spread duration, by program and price. The MBS sector includes the mortgage passthrough sector and the commercial mortgage-backed securities sector. Let us first discuss the mortgage passthrough sector.

Information is classified by issue—Government National Mortgage Association (Ginnie Mae or GNMA), the Federal National Mortgage Association (Fannie Mae), and the Federal Home Loan Mortgage Corporation (Freddie Mac)—original maturity (30-year and 15-year passthroughs)—and by price. Why is there this detailed information? The differences in allocation, by price, are included to emphasize the differences in prepayment risk exposure between the benchmark index and the recommended portfolio. For example, in the 30-year programs, we observe an underweighting of premium products (i.e., passthroughs trading above par value). The allocation in the mortgage passthrough sector suggests a concern that prepayments will accelerate, causing premium products (i.e., high-coupon mortgages) to underperform the low-coupon and par-coupon mortgages. This is a result of the negative convexity of the premium products.

At this stage, a manager makes an intra-sector allocation decision to the specific issues. The manager might believe that some securities within a subsector are mispriced and will therefore outperform other issues within the same sector over the time horizon. Managers pursue several different active strategies in order to identify mispriced securities. The most common strategy identifies an issue as undervalued because either (1) its yield is higher than that of comparably rated issues or (2) its yield is expected to decline (and price therefore rise) because the manager’s credit analysis leads the manager to believe that an issue will be upgraded by the rating agency before the issue is put on credit watch/rating watch for an upgrade or a positive rating outlook is assigned by a rating agency.

Once a portfolio is constructed, a manager might undertake a swap of one bond for another that is similar in terms of coupon, maturity, and credit quality, but offers a higher yield. This is a substitution swap, and it depends on capital market imperfections. Such situations sometimes exist due to temporary market imbalances and the fragmented nature of the non-Treasury bond market. The risk the manager faces in making a substitution swap is that the bond purchased is not identical to the bond for which it is exchanged. Moreover, bonds typically have similar, but not identical, maturity and coupon, which might result in differences in convexity.

An active strategy used in the mortgage-backed securities market is to identify mispriced individual issues of passthroughs, CMO classes, or stripped MBS, given an assumed prepayment rate. In the case of CMOs and stripped MBS, this strategy involves investing in a non-index product—that is, a product not included in the benchmark index.

It is critical when evaluating any potential swaps to assess the impact of the swap on the risk exposure of the portfolio. We will see how a bond swap can be assessed when we discuss multifactor risk models in Section V. Also, we will see how a manager can use a multi-factor risk model to incorporate a security selection strategy in constructing a portfolio.

In this section, we will explain how scenario analysis can be used to assess potential performance.

For example, suppose that an investor purchases a security for $90 and expects a return over a 1-year investment horizon from the three sources equal to $6. Then the expected total rate of return is 6.7%(= $6/$90).

When a trade involves borrowing, the interest cost of the borrowed funds is deducted from the dollar return from these three sources. (In Section VII we will discuss how funds can be borrowed in the bond market via a repurchase agreement.) The dollar return, adjusted for the financial cost, is then related to the dollar amount invested. For example, suppose the investor purchased the $90 security by borrowing $80 and investing $10 of his own funds (i.e., the investor’s equity). Suppose also that the cost of the borrowed funds is 5%, or $4. Then the dollar return, after adjusting for the financing cost, is $2 ($6 − $4). The total rate of return is 20% ($2 return divided by the investor’s equity of $10).

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:

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.

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.

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.

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 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.

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.

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.

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.

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.

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.

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:

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.

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.

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:

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

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.

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.