While managers can alter the duration of their portfolios with cash market instruments (buying or selling Treasury securities), using interest rate futures has the following four advantages:

To appreciate the last advantage, suppose that in a certain interest rate environment a pension fund manager must structure a portfolio to have a duration of 15 to accomplish a particular investment objective. Bonds with such a long duration may not be available. By buying the appropriate number and kind of interest rate futures contracts, a pension fund manager can increase the portfolio’s duration to the target level of 15.

There are two commonly used measures for approximating the change in the dollar value of a bond or bond portfolio resulting from changes in interest rates: price value of a basis point (PVBP) and duration. PVBP is the dollar price change resulting from a one-basis-point change in yield. Duration is the approximate percentage change in price for a 100-basis-point change in rates. (Given the percentage price change, the dollar price change for a given change in interest rates can be computed.) There are two measures of duration: modified and effective. Effective duration is the appropriate measure for bonds with embedded options. In this chapter when we refer to duration, we mean effective duration. Moreover, since the manager is interested in dollar price exposure, the effective dollar duration should be used. For a one basis point change in rates, PVBP is equal to the effective dollar duration for a one-basis-point change in rates.

As emphasized in earlier chapters, a good valuation model is needed to estimate the effective dollar duration. The valuation model is used to determine the new values for the bonds in the portfolio if rates change. Consequently, the starting point in controlling interest rate risk is the development of a reliable valuation model. A reliable valuation model is also needed to value the derivative contracts that the manager wants to use to control interest rate exposure.

Suppose that a manager seeks a target duration for the portfolio based on either expectations of interest rates or client-specified exposure. Given the target duration, a target dollar duration for a small basis point change in interest rates can be computed. For a 50 basis point change in interest rates the target dollar duration can be found by multiplying the target duration by the dollar value of the portfolio and then dividing by 200. (We divide by 2 because we are dealing with one half of a 100 basis point change.) For example, suppose that the manager of a $500 million portfolio wants a target duration of 6. That is, the manager seeks a 3% change in the value of the portfolio for a 50 basis point change in rates (assuming a parallel shift in rates of all maturities). Multiplying the target duration of 6 by $500 million and dividing by 200 gives a target dollar duration of $15 million.

The manager must then determine the dollar duration of the current portfolio. The current dollar duration for a 50 basis point change in interest rates is calculated by multiplying the current duration by the dollar value of the portfolio and dividing by 200. For our $500 million portfolio, suppose that the current duration is 4. The current dollar duration is then $10 million (4 times $500 million divided by 200).

The target dollar duration is then compared to the current dollar duration; the difference is the dollar exposure that must be provided by a position in the futures contract. If the target dollar duration exceeds the current dollar duration, a futures position must increase the dollar exposure by the difference. To increase the dollar exposure, an appropriate number of futures contracts must be purchased. If the target dollar duration is less than the current dollar duration, an appropriate number of futures contracts must be sold. That is,

Once a futures position is taken, the portfolio’s dollar duration is equal to the sum of the current dollar duration without futures plus the dollar duration of the futures position. That is,

The objective is to control the portfolio’s interest rate risk by establishing a futures position such that the portfolio’s dollar duration is equal to the target dollar duration. Thus,

Our focus above is on duration. However, duration measures price sensitivity to changes in interest rates assuming a parallel shift in the yield curve. For bond portfolios, nonparallel shifts in the yield curve should be taken into account. The same is true for individual mortgage-backed securities because, as explained in the next chapter, these securities are sensitive to changes in the yield curve. In the next chapter, a framework will be presented that shows how to hedge both a change in the level of interest rates and a change in the shape of the yield curve.

The dollar duration of a futures position is then the number of futures contracts multiplied by the dollar duration per futures contract. That is,

How many futures contracts are needed to obtain the target dollar duration? Substituting equation (2) into equation (1), we get

Equation (4) gives the approximate number of futures contracts needed to adjust the portfolio’s dollar duration to the target dollar duration. A positive number means that futures contracts must be purchased; a negative number means that futures contracts must be sold. Notice that if the target dollar duration is greater than the current dollar duration without futures, the numerator is positive and futures contracts are purchased. If the target dollar duration is less than the current dollar duration without futures, the numerator is negative and futures contracts are sold.

Given a valuation model, the general methodology for computing the dollar duration of a futures position for a given change in interest rates is straightforward. The procedure is used for computing the dollar duration of any cash market instrument—shock (change) interest rates up and down by the same number of basis points and compute the average dollar price change.

An adjustment is needed for the Treasury bond and note futures contracts. As explained, the pricing of the futures contract depends on the cheapest-to-deliver (CTD) issue.³⁴⁸The calculation of the dollar duration of a Treasury bond or note futures contract requires computing the effect of a change in interest rates on the price of the CTD issue, which in turn determines the change in the futures price. The dollar duration of a Treasury bond and note futures contract is determined as follows:

A short hedge (or sell hedge) is used to protect against a decline in the cash price of a bond. To execute a short hedge, futures contracts are sold. By establishing a short hedge, the manager has fixed the future cash price and transferred the price risk of ownership to the buyer of the futures contract. To understand why a short hedge might be executed, suppose that a pension fund manager knows that bonds must be liquidated in 40 days to make a $5 million payment to beneficiaries. If interest rates rise during the 40-day period, more bonds will have to be liquidated at a lower price than today to realize $5 million. To guard against this possibility, the manager can lock in a selling price by selling bonds in the futures market.³⁴⁹

A long hedge (or buy hedge) is undertaken to protect against an increase in the cash price of a bond. In a long hedge, the manager buys a futures contract to lock in a purchase price. A pension fund manager might use a long hedge when substantial cash contributions are expected and the manager is concerned that interest rates will fall. Also, a money manager who knows that bonds are maturing in the near future and expects that interest rates will fall before receipt can employ a long hedge to lock in a rate for the proceeds to be reinvested.

In bond portfolio management, typically the bond or portfolio to be hedged is not identical to the bond underlying the futures contract. This type of hedging is referred to as cross hedging.

The hedging process can be broken down into four steps:

Correlation is not, however, the only consideration if the hedging program is of significant size. If, for example, a manager wants to hedge $600 million of a cash position in a distant delivery month, liquidity becomes an important consideration. In such a case, it might be necessary for the manager to spread the hedge across two or more different contracts.

While our focus in this chapter is on hedging changes in the level of interest rates, when more than one type of change in the yield curve is to be hedged, then more than one hedging instrument must be used. For example, hedging a mortgage security is covered in the next chapter. As explained in the next chapter, because of the characteristic of mortgage securities, hedging both a change in the level and slope of the yield curve would be more effective. In such cases, two hedging instruments are used. One can also hedge the level, slope, and curvature changes of the yield curve. In that case, three hedging instruments are used.

To illustrate how a Treasury bond futures contract held to the delivery date locks in the futures rate, assume for the sake of simplicity that the manager knows which Treasury bond will be delivered and that delivery will take place on the last day of the delivery month. Suppose that for delivery on the September 1999 futures contract, the conversion factor for a deliverable Treasury issue—the of 2/15/15—is 1.283, implying that the investor who delivers this issue would receive from the buyer 1.283 times the futures settlement price plus accrued interest. An important principle to remember is that at delivery, the spot price and the futures price times the conversion factor must converge. Convergence refers to the fact that at delivery there can be no discrepancy between the spot price and futures price for a given security. If convergence does not take place, arbitrageurs would buy at the lower price and sell at the higher price and earn risk-free profits. Accordingly, a manager could lock in a September 1999 sale price for this issue by selling Treasury bond futures contracts equal to 1.283 times the par value of the bonds. For example, $100 million face value of this issue would be hedged by selling $128.3 million face value of bond futures (1,283 contracts).

The sale price that the manager locks in would be 1.283 times the futures price. This is the converted price. Thus, if the futures price is 113 when the hedge is set, the manager locks in a sale price of 144.979 (113 times 1.283) for September 1999 delivery, regardless of where rates are in September 1999. Exhibit 1 shows the cash flows for various final prices for this issue and illustrates how cash flows on the futures contracts offset gains or losses relative to the target price of 144.979.

Let’s look at each of the columns in Exhibit 1 and explain the computations for one of the scenarios—that is, for one actual sale price for the of 2/15/15 Treasury issue. Consider the first actual sale price of 140. By convergence, at the delivery date the final futures price shown in Column (2) must equal the Treasury bond’s actual sale price adjusted by the conversion factor. Thus, the final futures price in Column (2) of Exhibit 1 is computed as follows:

Column (3) shows the market value of the Treasury bonds. This is found by dividing the actual sale price in Column (1) by 100 to obtain the actual sale price per $1 of par value and then multiplying by the $100 million par value. That is,

Column (4) shows the value of the futures position at the delivery date. This value is computed by first dividing the futures price shown in Column (2) by 100 to obtain the futures price per $1 of par value. Then this value is multiplied by the par value per contract of $100,000 and further multiplied by the number of futures contracts. That is,

Now let’s look at the gain or loss from the futures position. This value is shown in Column (5). Recall that the futures contract was shorted. The futures price at which the contracts were sold was 113. So, if the final futures price exceeds 113, this means that there is a loss on the futures position—that is, the futures contract is purchased at a price greater than that at which it was sold. In contrast, if the futures price is less than 113, this means that there is a gain on the futures position—that is, the futures contract is purchased at a price less than that at which it was sold. The gain or loss is determined by the following formula:

Finally, Column (6) shows the effective sale price for the Treasury bond. This value is computed as follows: which is the sum of the numbers in each of the rows of Columns (3) and (5). For the actual sale price of $140 million, the gain is $4,979,000. Therefore the effective sale price for the Treasury bond is

When we admit the possibility that bonds other than the issue used in our illustration can be delivered, and that it might be advantageous to do so, the situation becomes somewhat more involved. In this more realistic case, the manager may decide not to deliver this issue, but if she does decide to deliver it, the manager is still assured of receiving an effective sale price of approximately 144.979. If the manager does not deliver this issue, it would be because another issue can be delivered more cheaply, and thus the manager does better than the targeted price.

In summary, if a manager establishes a futures hedge that is held until delivery, the manager can be assured of receiving an effective price dictated by the futures rate (not the spot rate) on the day the hedge is set.

c. The Target for Hedges with Short Holding Periods When a manager must lift (remove) a hedge prior to the delivery date, the effective rate that is obtained is more likely to approximate the current spot rate than the futures rate and this likelihood increases the shorter the term of the hedge. The critical difference between this hedge and the hedge held to the delivery date is that convergence will generally not take place by the termination date of the hedge.

To illustrate why a manager should expect the hedge to lock in the spot rate rather than the futures rate for very short-lived hedges, let’s return to the simplified example used earlier to illustrate a hedge maintained to the delivery date. It is assumed that this issue is the only deliverable instrument for the Treasury bond futures contract. Suppose that the hedge is set three months before the delivery date and the manager plans to lift the hedge one day later. It is much more likely that the spot price of the bond will move parallel to the converted futures price (that is, the futures price times the conversion factor), than that the spot price and the converted futures price will converge by the time the hedge is lifted.

A 1-day hedge is, admittedly, an extreme example. Other than underwriters, dealers, and traders who reallocate assets very frequently, few money managers are interested in such a short horizon. The very short-term hedge does, however, illustrate a very important point: when hedging, a manager should not expect to lock in the futures rate (or price) just by hedging with futures contracts. The futures rate is locked in only if the hedge is held until delivery, at which point convergence must take place. If the hedge is held for only one day, the manager should expect to lock in the 1-day forward rate,³⁵¹ which will very nearly equal the spot rate. Generally hedges are held for more than one day, but not necessarily to delivery.

In the bond market, a problem arises when trying to make practical use of the concept of the basis. The quoted futures price does not equal the price that one receives at delivery. For the Treasury bond and note futures contracts, the actual futures price equals the quoted futures price times the appropriate conversion factor. Consequently, to be useful, the basis in the bond market should be defined using actual futures delivery prices rather than quoted futures prices. Thus, the price basis for bonds should be redefined as:

For hedging purposes it is also frequently useful to define the basis in terms of interest rates rather than prices. The rate basis is defined as:

The rate basis is helpful in explaining why the two views of hedging described earlier are expected to lock in such different rates. To see this, we first define the target rate basis. This is defined as the expected rate basis on the day the hedge is lifted. That is,

Consider first a hedge lifted on the delivery date. On the delivery date, the spot rate and the futures rate will be the same by convergence. Thus, the target rate basis if the hedge is expected to be removed on the delivery date is zero. Substituting zero for the target rate basis in the equation above we have the following:

Now consider if a hedge is lifted prior to the delivery date. Let’s consider the case where the hedge is removed the next day. One would not expect the basis to change very much in one day. Assume that the basis does not change. Then,

If projecting the basis in terms of price rather than rate is easier (as is often the case for intermediate- and long-term futures), it is easier to work with the target price basis instead of the target rate basis. The target price basis is the projected price basis for the day the hedge is to be lifted. For a deliverable security, the target for the hedge then becomes

The idea of a target price or rate basis explains why a hedge held until the delivery date locks in a price, and other hedges do not. The examples have shown that this is true. For the hedge held to delivery, there is no uncertainty surrounding the target basis; by convergence, the basis on the day the hedge is lifted will be zero. For the short-lived hedge, the basis will probably approximate the current basis when the hedge is lifted, but its actual value is not known in advance. For hedges longer than one day but ending prior to the futures delivery date, there can be considerable basis risk because the basis on the day the hedge is lifted can end up being anywhere within a wide range. Thus, the uncertainty surrounding the outcome of a hedge is directly related to the uncertainty surrounding the basis on the day the hedge is lifted (i.e., the uncertainty surrounding the target basis).

The uncertainty about the value of the basis at the time the hedge is removed is called basis risk. For a given investment horizon, hedging substitutes basis risk for price risk. Thus, one trades the uncertainty of the price of the hedged security for the uncertainty of the basis. A manager would be willing to substitute basis risk for price risk if the manager expects that basis risk is less than price risk. Consequently, when hedges do not produce the desired results, it is common to place the blame on basis risk. However, basis risk is the real culprit only if the target for the hedge is properly defined. Basis risk should refer only to the unexpected or unpredictable part of the relationship between cash and futures prices. The fact that this relationship changes over time does not in itself imply that there is basis risk.

Basis risk, properly defined, refers only to the uncertainty associated with the target rate basis or target price basis. Accordingly, it is imperative that the target basis be properly defined if one is to correctly assess the risk and expected return in a hedge.

Earlier, we defined a cross hedge in the futures market as a hedge in which the security to be hedged is not deliverable on the futures contract used in the hedge. (A bond that does not meet the specific criteria for delivery to satisfy a particular futures contract is referred to as a nondeliverable bond.) For example, a manager who wants to hedge the sale price of long-term corporate bonds might hedge with the Treasury bond futures contract, but since non-Treasury bonds cannot be delivered in satisfaction of the contract, the hedge would be considered a cross hedge. A manager might also want to hedge a rate that is of the same quality as the rate specified in one of the contracts, but that has a different maturity. For example, it might be necessary to cross hedge a Treasury bond, note, or bill with a maturity that does not qualify for delivery on any futures contract. Thus, when the security to be hedged differs from the futures contract specification in terms of either quality or maturity, one is led to the cross hedge.

Conceptually, cross hedging is somewhat more complicated than hedging deliverable securities, because it involves two relationships. First, there is the relationship between the cheapest-to-deliver (CTD) issue and the futures contract. Second, there is the relationship between the security to be hedged and the CTD issue. Practical considerations may at times lead a manager to shortcut this two-step relationship and focus directly on the relationship between the security to be hedged and the futures contract, thus ignoring the CTD issue altogether. However, in so doing, a manager runs the risk of miscalculating the target rate and the risk in the hedge. Furthermore, if the hedge does not perform as expected, the shortcut makes it difficult to tell why the hedge did not work out as expected.

The key to minimizing risk in a cross hedge is to choose the right number of futures contracts. This depends on the relative dollar duration of the bond to be hedged and the dollar duration of the futures position. Equation (4) indicated the number of futures contracts required to achieve a particular target dollar duration. The objective in hedging is to make the target dollar duration equal to zero. Substituting zero for target dollar duration in equation (4) we obtain:

To calculate the dollar duration of a bond, the manager must know the precise point in time that the dollar duration is to be calculated (because price volatility generally declines as a bond matures) as well as the price or yield at which to calculate dollar duration (because higher yields generally reduce dollar duration for a given yield change). The relevant point in the life of the bond for calculating price volatility is the point at which the hedge will be lifted. Dollar duration at any other point in time is essentially irrelevant because the goal is to lock in a price or rate only on that particular day. Similarly, the relevant yield at which to calculate dollar duration initially is the target yield. Consequently, the numerator of equation (5) is the dollar duration on the date the hedge is expected to be lifted. The yield that can be used on this date in order to determine the dollar duration is the forward rate.

Let’s look at how we apply equation (5) when using the Treasury bond futures contract to hedge. The number of futures contracts will be affected by the dollar duration of the CTD issue. We can modify equation (5) as follows:

The target price for hedging the CTD issue would be 144.979 (113 × 1.283), and the target yield would be 6.56% (the yield at a price of 144.979). Since the yield on the FNMA bond is assumed to stay at 0.70% above the yield on the CTD issue, the target yield for the FNMA bond would be 7.26%. The corresponding price for the FNMA bond for this target yield is 87.76. At these target levels, the dollar durations for a 50 basis point change in rates for the CTD issue and the FNMA bond per $100 of par value are $6.255 and $5.453, respectively. As indicated earlier, all these calculations are made using a settlement date equal to the anticipated sale date, in this case the end of September 1999. The dollar duration for a 50 basis point change in rates for $10 million par value of the FNMA bond is then $545,300 ([$10 million/100] × $5.453). Per $100,000 par value for the CTD issue, the dollar duration per futures contract is $6,255 ([$100,000/100] × $6.255).

Thus, we know current dollar duration without futures = dollar duration of the FNMA bond = $545, 300

Substituting these values into equation (7) we obtain

Exhibit 2 uses scenario analysis to show the outcome of the hedge based on different prices for the FNMA bond at the delivery date of the futures contract. Let’s go through each of the columns. Column (1) shows the assumed sale price for the FNMA bond and Column (2) shows the corresponding yield based on the actual sale price in Column (1). This yield is found from the price/yield relationship. Column (3) shows the yield for the CTD issue, computed on the assumed 70 basis point yield spread between the FNMA bond and CTD issue. So, by subtracting 70 basis points from the yield for the FNMA bond in Column (2), the yield on the CTD issue (the 11.25% of 2/15/15) is obtained. Given the yield for the CTD issue in Column (3), the price per $100 of par value of the CTD issue can be computed. This CTD price is shown in Column (4).

Now we move from the price of the CTD issue to the futures price. As explained in the description of the columns in Exhibit 1, the futures price is computed by dividing the price for the CTD issue shown in Column (4) by the conversion factor of the CTD issue (1.283). This price is shown in Column (5).

The value of the futures position is found in the same way as in Exhibit 1. First the futures price per $1 of par value is computed by dividing the futures price by 100. Then this value is multiplied by $100,000 (the par value for the contract) and the number of futures contracts. That is,

Now let’s calculate the gain or loss on the futures position shown in Column (7). Note that the negative values in Column (7) for all futures prices above 113 mean there is a loss on the futures position. Since the futures price at which the contracts are sold at the inception of the hedge is 113, the gain or loss on the futures position is found as follows:

Finally, Column (8) shows the effective sale price for the FNMA bond. This value is found as follows:

Regression analysis allows the manager to capture the relationship between yield levels and yield spreads and use it to advantage. For hedging purposes, the variables are the yield on the bond to be hedged and the yield on the CTD issue. The regression equation takes the form:

For the two issues in question, the FNMA bond and the CTD issue, suppose the estimated yield beta is 1.05. Thus, yields on the FNMA issue are expected to move 5% more than yields on the Treasury issue. To calculate the number of futures contracts correctly, this fact must be taken into account; thus, the number of futures contracts derived in our earlier example is multiplied by the factor 1.05. Consequently, instead of shorting 112 Treasury bond futures contracts to hedge $10 million of the FNMA bond, the investor would short 118 (rounded up) contracts.

To incorporate the impact of a yield beta, the formula for the number of futures contracts is revised as follows:

The effect of a change in the CTD issue and the yield spread can be assessed before the hedge is implemented. An exhibit similar to Exhibit 2 can be constructed under a wide range of assumptions. For example, at different yield levels at the date the hedge is to be lifted (the second column in Exhibit 2), a different yield spread may be appropriate and a different acceptable issue will be the CTD issue. The manager can determine what this will do to the outcome of the hedge.

There are, however, exceptions to this general rule. As rates change, dollar duration changes. Consequently, the hedge ratio may change slightly. In other cases, there may be sound economic reasons to believe that the yield beta has changed. While there are exceptions, the best approach is usually to let a hedge run its course using the original hedge ratio with only slight adjustments.

A hedge can normally be evaluated only after it has been lifted. Evaluation involves, first, an assessment of how closely the hedge locked in the target rate—that is, how much error there was in the hedge. To provide a meaningful interpretation of the error, the manager should calculate how far from the target the sale (or purchase) would have been, had there been no hedge at all. One good reason for evaluating a completed hedge is to ascertain the sources of error in the hedge in the hope that the manager will gain insights that can be used to advantage in subsequent hedges. A manager will find that there are three major sources of hedging errors:

Let’s discuss the first two sources of hedging error. The third source is self-explanatory. Recall from the calculation of duration that interest rates are changed up and down by a small number of basis points and the security is revalued. The two recalculated values are used in the numerator of the duration formula. The first source of error listed above recognizes that the instrument to be hedged may be a complex instrument (i.e., one with embedded options) and that the valuation model does not do a good job of valuing the security when interest rates change.

The second major source of errors in a hedge—an inaccurate projected value of the basis—is a more difficult problem. Unfortunately, there are no satisfactory easy models like regression analysis that can be applied to the basis. Simple models of the basis violate certain equilibrium relationships for bonds that should not be violated. On the other hand, theoretically rigorous models are very unintuitive and usually solvable only by complex numerical methods. Modeling the basis is undoubtedly one of the most important and difficult problems that managers face when seeking to hedge.

Suppose a bank has a portfolio consisting of 4-year commercial loans with a fixed interest rate. The principal value of the portfolio is $100 million, and the interest rate on every loan in the portfolio is 11%. The loans are interest-only loans; interest is paid semiannually, and the principal is paid at the end of four years. That is, assuming no default on the loans, the cash flow from the loan portfolio is $5.5 million every six months for the next four years and $100 million at the end of four years. To fund its loan portfolio, assume that the bank can borrow at 6-month LIBOR for the next four years.

The risk that the bank faces is that 6-month LIBOR will be 11% or greater. To understand why, remember that the bank is earning 11% annually on its commercial loan portfolio. If 6-month LIBOR is 11% when the borrowing rate for the bank’s loan resets, there will be no spread income for that 6-month period. Worse, if 6-month LIBOR rises above 11%, there will be a loss for that 6-month period; that is, the cost of funds will exceed the interest rate earned on the loan portfolio. The bank’s objective is to lock in a spread over the cost of its funds.

The other party in the interest rate swap illustration is a life insurance company that has committed itself to pay an 8% rate for the next four years on a $100 million guaranteed investment contract (GIC) it has issued. Suppose that the life insurance company has the opportunity to invest $100 million in what it considers an attractive 4-year floating-rate instrument in a private placement transaction. The interest rate on this instrument is 6-month LIBOR plus 120 basis points. The coupon rate is set every six months.

The risk that the life insurance company faces in this instance is that 6-month LIBOR will fall so that the company will not earn enough to realize a spread over the 8% rate that it has guaranteed to the GIC policyholders. If 6-month LIBOR falls to 6.8% or less at a coupon reset date, no spread income will be generated. To understand why, suppose that 6-month LIBOR is 6.8% at the date the floating-rate instrument resets its coupon. Then the coupon rate for the next six months will be 8% (6.8% plus 120 basis points). Because the life insurance company has agreed to pay 8% on the GIC policy, there will be no spread income. Should 6-month LIBOR fall below 6.8%, there will be a loss for that 6-month period.

We can summarize the asset/liability problems of the bank and the life insurance company as follows.

Bank:

Life insurance company:

Now suppose the market has available a 4-year interest rate swap with a notional amount of $100 million, with the following swap terms available to the bank:

Now let’s look at the positions of the bank and the life insurance company after the swap. Exhibit 3 summarizes the position of each institution before and after the swap. Consider first the bank. For every 6-month period for the life of the swap, the interest rate spread will be as follows:

Now let’s look at the effect of the interest rate swap on the life insurance company:

The interest rate swap has allowed each party to accomplish its asset/liability objective of locking in a spread.³⁵² It permits the two financial institutions to alter the cash flow characteristics of its assets: from fixed to floating in the case of the bank, and from floating to fixed in the case of the life insurance company.

To see how to calculate the dollar duration, let’s work with the second economic interpretation of a swap—a package of cash flows from buying and selling cash market instruments. From the perspective of the fixed-rate receiver, the position can be viewed as follows:

This means that the dollar duration of an interest rate swap from the perspective of a fixed-rate receiver is the difference between the dollar durations of the two bond positions that comprise the swap. That is,

Most of the swap’s interest rate sensitivity results from the dollar duration of the fixed-rate bond since the dollar duration of the floating-rate bond will be small. The dollar duration of a floating-rate bond is smaller the closer the swap is to its reset date. If the dollar duration of the floating-rate bond is close to zero then:

Thus, adding an interest rate swap to a portfolio in which the manager pays a floating-rate and receives a fixed-rate increases the dollar duration of the portfolio by roughly the dollar duration of the underlying fixed-rate bond. This is because it effectively involves buying a fixed-rate bond on a leveraged basis.

We can use the cash market instrument economic interpretation to compute the dollar duration of a swap for the fixed-rate payer. The dollar duration is:

The dollar duration of a portfolio that includes a swap is:

Let’s look at our bank/life insurance illustration in terms of duration mismatch. The bank has a larger duration for its assets (the fixed-rate loans) than the duration for its liabilities (the short-term funds it borrows). Effectively, the position of the bank is as follows:

For the life insurance company, the duration of the liabilities is long while the duration of the floating-rate assets is short. That is,

While most of the uncertainty in an options hedge usually comes from the uncertainty of interest rates, the correlation between the prices of the bonds to be hedged and the instruments underlying the options contracts determines the extent to which the hedging objective is accomplished. Low correlation leads to greater uncertainty.

On 6/24/99 the FNMA bond was selling for 88.39 to yield 7.20% and the CTD issue’s yield was 6.50%. For simplicity, it is assumed that the yield spread between the FNMA bond and the CTD issue remains at 70 basis points.

To see how covered call writing with futures options works for the bond used in the protective put example, we construct a table much as we did before. We assume in this illustration that a sale of a call option with a strike price of 117 is selected by the manager.³⁵⁴ As in the futures hedging and protective buying strategies it is assumed that the hedged bond will remain at a 70 basis point spread over the CTD issue. We also assume that the price of each call option is $500.

Working backwards from box 5 in Exhibit 8, a strike price for the futures option of 117 would be equivalent to the following at the expiration date of the call option:

Given the above information, the current dollar duration for the FNMA bond and the dollar duration of the CTD issue can be computed. The information above represents values as of the expiration date of the call option. It can be shown that the dollar durations based on a 50 basis point change in rates are:

While the target price for the FNMA bond is 92.3104, the maximum effective sale price is determined by adjusting the target price by the proceeds received from the sale of the call options. The maximum effective sale price is 92.8904 (= 92.3104 + 0.5800).

Exhibit 10 shows the outcomes of the covered call writing strategy. The first five columns of the exhibit are the same as Exhibit 9. In Column (6), the liability resulting from the call option position is shown. The liability is zero if the futures price is less than the strike price of 117. If the futures price for the scenario is greater than 117, the liability is calculated as follows:

As Exhibit 10 shows, if the hedged bond trades at 70 basis points over the CTD issue as assumed, the maximum effective sale price for the hedged bond is, in fact, slightly over 92.8904. The discrepancies shown in the exhibit are due to rounding.

To make a choice among strategies, it helps to compare the alternatives side by side. Using the futures example from Section II B and the futures options examples from Section IV C, Exhibit 11 shows the final values of the portfolio for the various hedging alternatives. It is easy to see that no one strategy dominates the others in every scenario.

Consequently, we cannot conclude that one strategy is the best strategy. The manager who makes the strategy decision makes a choice among probability distributions, not usually among specific outcomes. Except for the perfect hedge, there is always some range of possible final values of the portfolio. Of course, exactly what that range is, and the probabilities associated with each possible outcome, is a matter of opinion.

Using options on physicals frequently eliminates much of the basis risk associated with a futures options hedge. For example, a manager of Treasury bonds or notes can usually buy or sell options on the exact security held in the portfolio. Using options on futures, rather than options on Treasury bonds, introduces additional elements of uncertainty.

Given the illustration presented above, and given that the risk-return profile of options on physicals and options on futures are essentially identical, additional illustrations for options on physicals are unnecessary. The only important differences are the calculations of the hedge ratio and the equivalent strike price. To derive the hedge ratio, we always resort to an expression of relative dollar durations. Thus, assuming a constant spread, the hedge ratio for options on physicals is:

If a relationship is estimated between the yield on the bonds to be hedged and the instrument underlying the option, the appropriate hedge ratio is:

Unlike futures options, there is only one deliverable, so there is no conversion factor. When cross hedging with options on physicals, the procedure for finding the equivalent strike price on the bonds to be hedged is similar. Given the strike price of the option, the strike yield is determined using the price/yield relationship for the instrument underlying the option. Then given the projected relationship between the yield on the instrument underlying the option and the yield on the bonds to be hedged, an equivalent strike yield is derived for the bonds to be hedged. Finally, using the yield-to-price formula for the bonds to be hedged, the equivalent strike price for the bonds to be hedged can be found.

To see how interest rate caps and floors can be used for asset/liability management, consider the problems faced by the commercial bank and the life insurance company discussed in Section III A. Recall that the bank’s objective is to lock in a spread over its cost of funds. Yet because the bank borrows short term, its cost of funds is uncertain. The bank may be able to purchase a cap, however, so that the cap rate plus the cost of purchasing the cap is less than the rate it is earning on its fixed-rate commercial loans. If short-term rates decline, the bank does not benefit from the cap, but its cost of funds declines. The cap therefore allows the bank to impose a ceiling on its cost of funds while retaining the opportunity to benefit from a decline in rates.

The bank can reduce the cost of purchasing the cap by selling a floor. In this case, the bank agrees to pay the buyer of the floor if the reference rate falls below the strike rate. The bank receives a fee for selling the floor, but it has sold off its opportunity to benefit from a decline in rates below the strike rate. By buying a cap and selling a floor, the bank has created a predetermined range for its cost of funds (i.e., a collar).

Recall the problem of the life insurance company that guarantees an 8% rate on a GIC for four years and is considering the purchase of a floating-rate instrument in a private placement transaction. The risk the company faces is that interest rates will fall so that it will not earn enough to realize the 8% guaranteed rate plus a spread. The life insurance company may be able to purchase a floor to set a lower bound on its investment return, yet retain the opportunity to benefit should rates increase. To reduce the cost of purchasing the floor, the life insurance company can sell an interest rate cap. By doing so, however, it forfeits the opportunity to benefit from an increase in the reference rate above the strike rate of the cap.