CHAPTER
SIXTY-ONE
CONTROLLING INTEREST-RATE RISK WITH FUTURES AND OPTIONS
Professor of Finance
EDHEC Business School
SHRIKANT RAMAMURTHY
Consultant
MARK PITTS, PH.D.
In Chapter 59 the features and characteristics of interest-rate futures and options were explained. In this chapter, our focus is on how these derivative instruments can be used to control the interest-rate risk of a portfolio.
CONTROLLING INTEREST-RATE RISK WITH FUTURES
The price of an interest-rate futures contract moves in the opposite direction from the change in interest rates: when rates rise, the futures price will fall; when rates fall, the futures price will rise. By buying a futures contract, a portfolio’s exposure to a rate increase is increased. That is, the portfolio’s duration increases. By selling a futures contract, a portfolio’s exposure to a rate increase is decreased. Equivalently, this means that the portfolio’s duration is reduced. Consequently, buying and selling futures can be used to alter the duration of a portfolio.
While managers can alter the duration of their portfolios with cash-market instruments (buying or selling Treasury securities), using interest-rate futures instead of trading long-term Treasuries themselves has the following three advantages:
This chapter is adapted from an article by the authors appearing in Professional Perspectives on Fixed Income Portfolio Management, Vol. 1 (New Hope, PA: Frank J. Fabozzi Associates, 2000).
Advantage 1: Transaction costs for trading futures are lower than trading in the cash market.
Advantage 2: Margin requirements are lower for futures than for Treasury securities; using futures thus permits greater leverage.
Advantage 3: It is easier to sell short in the futures market than in the Treasury market.
Futures also can be used in constructing a portfolio with a longer duration than is available with cash-market securities. For example, 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.
General Principles of Interest-Rate Risk Control
The general principle in controlling interest-rate risk with futures is to combine the dollar exposure of the current portfolio and the dollar exposure of a futures position so that the total dollar exposure is equal to the target dollar exposure. This means that the manager must be able to accurately measure the dollar exposure of both the current portfolio and the futures contract employed to alter the exposure.
There are two commonly used measures for approximating the change in the dollar value of a bond or bond portfolio to changes in interest rates: price value of a basis point (PVBP) and duration. PVBP is the dollar price change resulting from a 1 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 that should be used 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, it is the effective dollar duration that should be used. For a 1 basis point change in rates, PVBP is equal to the effective dollar duration for a 1 basis point change in rates.
To estimate the effective dollar duration, it is necessary to have a good valuation model. It is the valuation model that is used to determine what the new values for the bonds in the portfolio will be if rates change. The difference between the current values of the bonds in the portfolio and the new values estimated by the valuation model when rates are changed is the dollar price exposure. Consequently, the starting point in controlling interest-rate risk is the development of a reliable valuation model. A reliable valuation model also is 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 obtained. For a 50 basis point change in interest rates, for example, the target dollar duration can be found by multiplying the dollar value of the portfolio by the target duration and then dividing by 200. For example, suppose that the manager of a $500 million portfolio wants a target duration of 6. This means that 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 then must determine the dollar duration of the current portfolio. The current dollar duration for a 50 basis point change in interest rates is found by multiplying the current duration by the dollar value of the portfolio and dividing by 200. Thus, 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 with the current dollar duration. The difference between the two dollar durations 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,
If target dollar duration – current dollar duration > 0, buy futures
If target dollar duration – current dollar duration < 0, sell futures
Once a futures position is taken, the portfolio’s dollar duration is equal to the current dollar duration without futures plus the dollar duration of the futures position. That is,
Portfolio’s dollar return = current dollar duration without futures + dollar duration of futures position
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
Portfolio’s dollar duration = target dollar duration
Or equivalently,
Over time, the portfolio’s dollar duration will move away from the target dollar duration. The manager can alter the futures position to adjust the portfolio’s dollar duration to the target dollar duration.
Determining the Number of Contracts
Each futures contract calls for delivery of a specified amount of the underlying instrument. When interest rates change, the value of the underlying instrument changes, and therefore, the value of the futures contract changes. How much the futures dollar value will change when interest rates change must be estimated. This amount is called the dollar duration per futures contract. For example, suppose that the futures price of an interest-rate futures contract is 70 and that the underlying interest-rate instrument has a par value of $100,000. Thus the futures delivery price is $70,000 (0.70 times $100,000). Suppose that a change in interest rates of 50 basis points results in the futures price changing by about $0.03 per contract. Then the dollar duration per futures contract is $2,100 (0.03 times $70,000).
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 Eq. (61–2) into Eq. (61–1), we get
Solving for the number of futures contracts, we have
Equation (61–4) gives the approximate number of futures contracts that are necessary to adjust the portfolio’s dollar duration to the target dollar duration. A positive number means that the futures contract must be purchased; a negative number means that the futures contract must be sold. Notice that if the target dollar duration is greater than the current dollar duration without futures, the numerator is positive, and therefore, futures contracts are purchased. If the target dollar duration is less than the current dollar duration without futures, the numerator is negative, and therefore, futures contracts are sold.
Dollar Duration for a Futures Position
Now we turn to how to measure the dollar duration of a bond futures position. Keep in mind what the goal is: it is to measure the sensitivity of a bond futures position to changes in rates.
The general methodology for computing the dollar duration of a futures position for a given change in interest rates is straightforward given a valuation model. The procedure is the same as 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 determine the average dollar price change.
An adjustment is needed for the Treasury bond and note futures contracts. The pricing of the futures contract depends on the cheapest-to-deliver (CTD) issue.1 Calculation of the dollar duration of a Treasury bond or note futures contract requires determining the impact of a change in interest rates will have on the price of a futures contract, which, in turn, affects how the futures price will change. The dollar duration of a Treasury bond and note futures contract is determined as follows:
There is a conversion factor for each issue that is acceptable for delivery for the futures contract. The conversion factor makes deliverable equitable to both the buyer and seller of the futures contract. For each deliverable issue, the product of the futures price and the conversion factor is the adjusted futures price for the issue. This adjusted price is called the converted price. Relating this to the preceding equation, the second ratio is approximately equal to the conversion factor of the cheapest-to-deliver issue. Thus we can write
Dollar duration of futures contract
= dollar duration of the CTD issue × conversion factor for the CTD issue
Why did we focus on dollar duration rather than duration? Recall that duration is the approximate percentage change in price. But what is the price of this leveraged instrument? The investor does not put up the full price of the position in order to acquire the position. Only the initial margin need be made in cash or a cash equivalent. Consequently, what is the base investment made by the investor? Rather than debate what should be used as the base investment in order to compute duration, let’s simply ask why we are interested in calculating the exposure to changes in rates. As we have emphasized, it is to determine how a futures position will alter the exposure of a portfolio to changes in rates. Once we know how a futures position changes the dollar duration of a portfolio, we can determine for a portfolio its dollar duration. Given the funds invested by the investor in the portfolio, the portfolio’s duration can be computed.
Hedging with Interest-Rate Futures
Hedging with futures calls for taking a futures position as a temporary substitute for transactions to be made in the cash market at a later date. If cash and futures prices move together, any loss realized by the hedger from one position (whether cash or futures) will be offset by a profit on the other position. Hedging is a special case of controlling interest-rate risk. In a hedge, the manager seeks a target duration or target dollar duration of zero.
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 sell bonds in the futures market to lock in a selling price.
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 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:
Step 1. Determining the appropriate hedging instrument
Step 2. Determining the target for the hedge
Step 3. Determining the position to be taken in the hedging instrument
Step 4. Monitoring and evaluating the hedge
We discuss each step below.
Determining the Appropriate Hedging Instrument
A primary factor in determining which futures contract will provide the best hedge is the degree of correlation between the rate on the futures contract and the interest rate that creates the underlying risk that the manager seeks to eliminate. For example, a long-term corporate bond portfolio can be better hedged with Treasury bond futures than with Treasury bill futures because long-term corporate bond rates are more highly correlated with Treasury bond futures than Treasury bill futures. Using the right delivery month is also important. A manager trying to lock in a rate or price for September will use September futures contracts because September futures contracts will give the highest degree of correlation.
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.
Determining the Target for the Hedge
Having determined the right contract and the right delivery months, the manager then should determine what is expected from the hedge—that is, what rate will, on average, be locked in by the hedge. This is the target rate or target price. If this target rate is too high (if hedging a future sale) or too low (if hedging a future purchase), hedging may not be the right strategy for dealing with the unwanted risk. Determining what is expected (calculating the target rate or price for a hedge) is not always simple. We’ll see how a manager should approach this problem for both simple and complex hedges.
Risk and Expected Return in a Hedge. When a manager enters into a hedge, the objective is to “lock in” a rate for the sale or purchase of a security. However, there is much disagreement about what rate or price a manager should expect to lock in when futures are used to hedge. Here are the two views:
View 1. The manager can, on average, lock in the current spot rate for the security (i.e., current rate in the cash market).
View 2. The manager can, on average, lock in the rate at which the futures contracts are bought or sold.
The truth usually lies somewhere in between these two views. However, as the following cases illustrate, each view is entirely correct in certain situations.
The Target for Hedges Held to Delivery. Hedges that are held until the futures delivery date provide an example of a hedge that locks in the futures rate (i.e., the second view). The complication in the case of using Treasury bond futures and Treasury note futures to hedge the value of intermediate- and long-term bonds is that because of the delivery options the manager does not know for sure when delivery will take place or which bond will be delivered. This is because of the delivery options granted to the short.2
To illustrate how a Treasury bond futures 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 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 61–1 shows the cash-flows for a number of 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.
EXHIBIT 61–1
Treasury Issue Hedge Held to Delivery
Let’s look at all the columns in Exhibit 61–1 and explain the computations for one of the scenarios—that is, for one actual sale price for the 11/4% Treasury bond. 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. Specifically, the adjustment is as follows. We know that
Converted price = Treasury bond’s price × conversion factor
and by convergence
Final futures price = converted price
so that
Final futures price = Treasury bond’s actual sale price × conversion factor
Thus, to compute the final futures price in column (2) of Exhibit 61–1 given the Treasury bond’s actual sale price in column (1), the following is computed:
Since the conversion factor is 1.283 for the 11/4% Treasury issue, for the first actual sale price of 140, the final futures price is
Column (3) shows the market value of the Treasury bonds. This is found by multiplying 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,
Market value of Treasury bonds = (actual sale price/100) × $100,000,000
For the actual sale price of 140, the value in column (3) is
Market value of Treasury bonds = (140/100) × $100,000,000
= $140,000,000
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,
Value of futures position
= (final futures price/100) × $100, 0000 × number of futures contracts
In our illustration, the number of futures contracts is 1,283. For the actual sale price of the bond of 140, the final futures price is 109.1193. Thus, the value shown in column (4) is
Value of futures position = (109.1193/100) × $100,000 × 113
= $140,000,062
The value shown in column (4) is $140,000,000 because the final futures price of 109.1193 was rounded. Using more decimal places, the value would be $140,000,000.
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. Thus, 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 for 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 for which it was sold. The gain or loss is determined by the following formula:
(113/100 – final futures price/100) × $100,000 × number of futures contracts
In our illustration, for a final futures price of 109.1193 and 1,283 futures contracts, we have
(113/100 – 109.1193/100) × $100,000 × 1,283 = $4,978,938.1
The value shown in column (5) is $4,979,000 because that is the more precise value using more decimal places for the final futures price than shown in Exhibit 61–1. The value is positive which means that there is a gain in the futures position. Note that for all the final futures prices above 113 in Exhibit 61–1, there is a negative value, which means that there is a loss on the futures position.
Finally, column (6) shows the effective sale price for the Treasury bond. This value is found as follows:
Effective sale price for Treasury bond
= actual sale price of Treasury bond + gain or loss on futures position
For the actual sale price of $140 million, the gain is $4,979,000. Therefore, the effective sale price for the Treasury bond is
$140,000,000 + $4,979,000 = $144,979,000
Note that this is the target price for the Treasury bond. In fact, it can be seen from column (6) of Exhibit 61–1 that the effective sale price for all the actual sale prices for the Treasury bond is the target price. However, the target price is determined by the futures price, so the target price may be higher or lower than the cash (spot) market price when the hedge is set.
When we admit the possibility that bonds other than the deliverable issue used in our illustration can be delivered and that it might be advantageous to deliver other issues, 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.
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 much more likely to approximate the current spot rate than the futures rate the shorter the term of the hedge. The critical difference between this hedge and the hedge held to the delivery date is that convergence generally will 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 to the delivery date. It is assumed that this issue is the only deliverable Treasury bond 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 after one day. It is much more likely that the spot price of the bond will move parallel to the converted futures price (i.e., 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 one-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 because he is 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 one-day forward rate,3 which will very nearly equal the spot rate. Generally, hedges are held for more than one day but not necessarily to delivery.
How the Basis Affects the Target Rate for a Hedge. The proper target for a hedge that is to be lifted prior to the delivery date depends on the basis. The basis is simply the difference between the spot (cash) price of a security and its futures price; that is:
Basis = spot price – futures price
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
Price basis = spot price – futures delivery price
For hedging purposes, it is also useful frequently to define the basis in terms of interest rates rather than prices. The rate basis is defined as
Rate basis = spot rate – futures rate
where spot rate refers to the current rate on the instrument to be hedged and the futures rate is the interest rate corresponding to the futures delivery price of the deliverable instrument.
The rate basis is helpful in explaining why the two views of hedges explained 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. A hedge lifted on the delivery date is expected to have, and by convergence will have, a zero rate basis when the hedge is lifted. Thus the target rate for the hedge should be the rate on the futures contract plus the expected rate basis of zero or, in other words, just the futures rate. When a hedge is lifted prior to the delivery date, one would not expect the basis to change very much in one day, so the target rate basis equals the futures rate plus the current difference between the spot rate and futures rate, that is, the current spot rate.
The manager can set the target rate for any hedge equal to the futures rate plus the target rate basis. That is,
Target rate for hedge = futures rate + target rate basis
If projecting the basis in terms of price rather than rate is more manageable (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 just the projected price basis for the day the hedge is to be lifted. For a deliverable security, the target for the hedge then becomes
Target price for hedge = futures delivery price+ target price basis
The idea of a target price or rate basis explains why a hedge held until the delivery date locks in a price with certainty, 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 is certain to be zero. For the short-lived hedge, the basis probably will approximate the current basis when the hedge is lifted, but its actual value is not known. 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. Consequently, when hedges do not produce the desired results, it is customary to place the blame on basis risk. However, basis risk is the real culprit only if the target for the hedge is defined properly. 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 defined properly if one is to assess the risk and expected return in a hedge correctly.
Determining the Number of Futures Contracts
The final step that must be determined before the hedge is set is the number of futures contracts needed for the hedge. This is called the hedge ratio. Usually the hedge ratio is expressed in terms of relative par amounts. Accordingly, a hedge ratio of 1.20 means that for every $1 million par value of securities to be hedged, one needs $1.2 million par value of futures contracts to offset the risk. In our discussion, the values are defined so that the hedge ratio is the number of futures contracts.
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. 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 manger also might 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 is 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. Practical considerations at times may 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 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 futures position. Equation (61–4) indicated the number of futures contracts 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 Eq. (61–4) 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 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 volatility is the point at which the hedge will be lifted. Dollar duration at any other point 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 Eq. (61–5) is the dollar duration on the date the hedge is expected to be delivered. The yield that is to be used on this date in order to determine the dollar duration is the forward rate.
Let’s look at how we apply Eq. (61–5) 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 Eq. (61–5) as follows:
As noted earlier, the conversion ratio for the CTD issue is a good approximation of the second ratio. Thus Eq. (61–6) can be rewritten as
An Illustration. An example for a single bond shows why dollar duration weighting leads to the correct number of contracts to use to hedge. The hedge illustrated is a cross-hedge. Suppose that on 6/24/99 a manager owned $10 million par value of a 6.25% Fannie Mae (FNMA) option-free bond maturing on 5/15/29 selling at 88.39 to yield 7.20%. The manager wants to sell September 1999 Treasury bond futures to hedge a future sale of the FNMA bond. At the time, the price of the September Treasury bond futures contract was at 113. The CTD issue was the 11.25% of 2/15/15 issue that was trading at the time at 146.19 to yield 6.50%. The conversion factor for the CTD issue was 1.283. To simplify, assume that the yield spread between the FNMA bond and the CTD issue remains at 0.70% (i.e., 70 basis points) and that the anticipated sale date is the last business day in September 1999.
The target price for hedging the CTD issue would be 144.979 (from 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 duration for a 50 basis point change in rates for the CTD issue and FNMA bond per $100 of par value is $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 $10 million par value of the FNMA bond is then $545,300 ($10 million/100 times $5.453). Per $100,000 par value for the CTD issue, the dollar duration per futures contract is $6,255 ($100,000/100 times $6.255).
Thus we know
Current dollar duration without futures
= dollar duration of the FNMA bond
= $545,300
Dollar duration of the CTD issue = $6,255
Conversion factor for CTD issue = 1.283
Substituting these values into Eq. (61–7), we obtain
Consequently, to hedge the FNMA bond position, 112 Treasury bond futures contracts must be shorted.
Exhibit 61–2 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. Given the assumed sale price for the FNMA bond, the corresponding yield can be determined. Column (3) shows the yield for the CTD issue. This yield is computed based on the assumption regarding the yield spread of 70 basis points between the FNMA bond and the CTD issue. Thus, 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).
EXHIBIT 61–2
Hedging a Nondeliverable Bond to a Delivery Date with Futures
Now we must move from the price of the CTD issue to the futures price. As explained in the description of the columns in Exhibit 61–1, by dividing the price for the CTD issue shown in column (4) by the conversion factor of the CTD issue (1.283), the futures price is obtained. This price is shown in column (5).
The value of the futures position is found in the same way as in Exhibit 61–1. 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,
Value of futures position
= (futures price/100) × $100,0000 × number of futures contracts
Since the number of futures contracts sold is 112,
Value of futures position = (final futures price/100) × $100,0000 × 112
The values shown in column (6) use the preceding formula. Using the first assumed actual sale price for the FNMA of $8 million as an example, the corresponding futures price in column (5) is 105.85581. Therefore, the value of the futures position is
Value of futures position = (105.85581/100) × $100,000 × 112
= $11,855,850
Now let’s calculate the gain or loss on the futures position shown in column (7). This is done in the same manner as explained for Exhibit 61–1. 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:
(113/100 – final futures price/100) × $100,000 × number of futures contracts
For example, for the first scenario in Exhibit 61–2, the futures price is 105.85581, and 112 futures contract were sold. Therefore,
(113/100 – 105.85581/100) × $100,000 × 112 = $800,150
There is a gain from the futures position because the futures price is less than 113. Note that for all the final futures prices above 113 in Exhibit 61–2, there is a negative value, which means that there is a loss on the futures position. For all futures prices below 113, there is a loss.
Finally, column (8) shows the effective sale price for the FNMA bond. This value is found as follows:
Effective sale price for FNMA bond
= actual sale price of FNMA bond + gain or loss on futures position
For the actual sale price of $8 million, the gain is $800,150. Therefore, the effective sale price for the FNMA bond is
$8,000,000 + $800,150 = $8,800,150
Looking at column (8) of Exhibit 61–2, we see that if the simplifying assumptions hold, a futures hedge using the recommended number of futures contracts (112) very nearly locks in the target price for $10 million par value of the FNMA bonds.
Refining for Changing Yield Spread. Another refinement in the hedging strategy is usually necessary for hedging nondeliverable securities. This refinement concerns the assumption about the relative yield spread between the CTD issue and the bond to be hedged. In the prior discussion, we assumed that the yield spread was constant over time. Yield spreads, however, are not constant over time. They vary with the maturity of the instruments in question and the level of rates, as well as with many unpredictable and nonsystematic factors.
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
The regression procedure provides an estimate of b, which is the expected relative yield change in the two bonds. This parameter b is called the yield beta. Our example that used constant spreads implicitly assumes that the yield beta b equals 1.0 and a equals 0.70 (because 0.70 is the assumed spread).
For the two issues in question, that is, the FNMA bond and the CTD issue, suppose that the estimated yield beta was 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.
The formula for the number of futures contracts is revised as follows to incorporate the impact of the yield beta:
where the yield beta is derived from the yield of the bond to be hedged regressed on the yield of the CTD issue (Eq. 61–8).
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 that of Exhibit 61–2 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 61–2), 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.
Monitoring and Evaluating the Hedge
After a target is determined and a hedge is set, there are two remaining tasks. The hedge must be monitored during its life and evaluated after it is over. Most futures hedges require very little active monitoring during their life. In fact, overactive management poses more of a threat to most hedges than does inactive management. The reason for this is that the manager usually will not receive enough new information during the life of the hedge to justify a change in the hedging strategy. For example, it is not advisable to readjust the hedge ratio every day in response to a new data point and a possible corresponding change in the estimated value of the yield beta.
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 normally can 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:
1. The dollar duration for the hedged instrument was incorrect.
2. The projected value of the basis at the date the hedge is removed can be in error.
3. The parameters estimated from the regression (a and b) can be inaccurate.
Recall from the calculation of duration in Chapter 7 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 problem just listed 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 the more difficult problem. Unfortunately, there are no satisfactory simple models like regression 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 soluble only by complex numerical methods. Modeling the basis is undoubtedly one of the most important and difficult problems that managers seeking to hedge face.
HEDGING WITH OPTIONS
Hedging strategies using options involve taking a position in an option and a position in the underlying bond in such a way that changes in the value of one position will offset any unfavorable price (interest rate) movement in the other position. We begin with the basic hedging strategies using options. Then we illustrate these basic strategies using futures options to hedge the FNMA bond for which a futures hedge was used earlier in this chapter. Using futures options in our illustration of hedging the bond is a worthwhile exercise because it shows how complicated hedging with futures options is and the key parameters involved in the process. We also compare the outcome of hedging with futures and hedging with futures options.
Basic Hedging Strategies
There are three popular hedging strategies: (1) a protective put-buying strategy, (2) a covered call-writing strategy, and (3) a collar strategy. We discuss each strategy below.
Protective Put-Buying Strategy
Consider a manager who has a bond and wants to hedge against rising interest rates. The most obvious options hedging strategy is to buy put options on bonds. This hedging strategy is referred to as a protective put-buying strategy. The puts are usually out-of-the-money puts and may be either puts on cash bonds or puts on interest-rate futures. If interest rates rise, the puts will increase in value (holding other factors constant), offsetting some or all of the loss on the bonds in the portfolio.
This strategy is a simple combination of a long put option with a long position in a cash bond. Such a position has limited downside risk, but large upside potential. However, if rates fall, the price appreciation on the bonds in the portfolio will be diminished by the amount paid to purchase the puts. Exhibit 61–3 compares the protective put-buying strategy to an unhedged position.
EXHIBIT 61–3
Protective Put-Buying Strategy
The protective put-buying strategy is very often compared with purchasing insurance. Like insurance, the premium paid for the protection is nonrefundable and is paid before the coverage begins. The degree to which a portfolio is protected depends on the strike price of the options; thus the strike price is often compared with the deductible on an insurance policy. The lower the deductible (i.e., the higher the strike price for the put), the greater is the level of protection, and the more the protection costs. Conversely, the higher the deductible (the lower the strike price on the put), the more the portfolio can lose in value, but the cost of the insurance is lower. Exhibit 61–4 compares an unhedged position with several protective put positions, each with a different strike price, or level of protection. As the exhibit shows, no one strategy dominates any other strategy, in the sense of performing better at all possible rate levels. Consequently, it is impossible to say that one strike price is necessarily the “best” strike price or even that buying protective puts is necessarily better than doing nothing at all.
EXHIBIT 61–4
Protective Put-Buying Strategy with Different Strike Prices
Covered Call-Writing Strategy
Another options hedging strategy used by many portfolio managers is to sell calls against the bond portfolio. This hedging strategy is called a covered call-writing strategy. The calls that are sold are usually out-of-the-money calls and can be either calls on cash bonds or calls on interest-rate futures. Covered call writing is just an outright long position combined with a short call position. Obviously, this strategy entails much more downside risk than buying a put to protect the value of the portfolio. In fact, many portfolio managers do not consider covered call writing a hedge.
Regardless of how it is classified, it is important to recognize that while covered call writing has substantial downside risk, it has less downside risk than an unhedged long position alone. On the downside, the difference between the long position alone and the covered call-writing strategy is the premium received for the calls that are sold. This premium acts as a cushion for downward movements in prices, reducing losses when rates rise. The cost of obtaining this cushion is that the manager gives up some of the potential on the upside. When rates decline, the call options become greater liabilities for the covered call writer. These incremental liabilities decrease the gains the manager would otherwise have realized on the portfolio in a declining rate environment. Thus the covered call writer gives up some (or all) of the upside potential of the portfolio in return for a cushion on the downside. The more upside potential that is forfeited (i.e., the lower the strike price on the calls), the more cushion there is on the downside. Exhibit 61–5 illustrates this point by comparing an unhedged position to several covered call-writing strategies, each with a different strike price. Like the protective put-buying strategy, there is no “right” strike price for the covered call writer.
EXHIBIT 61–5
Covered Call-Writing Strategy with Different Strike Prices
Collar Strategy
There are other hedging strategies employing options that are used frequently by managers. For example, many managers combine the protective put-buying strategy and the covered call-writing strategy. By combining a long position in an out-of-the-money put and a short position in an out-of-the-money call, the manager creates a long position in a collar. Consequently, this hedging strategy is called a collar strategy. The manager who uses the collar eliminates part of the portfolio’s downside risk by giving up part of its upside potential. A long position hedged with a collar is shown in Exhibit 61–6.
EXHIBIT 61–6
Long Position Hedged with a Collar
The collar in some ways resembles the protective put, in some ways resembles covered call writing, and in some ways resembles an unhedged position. The collar is like the protective put-buying strategy in that it limits the possible losses on the portfolio if interest rates go up. Like the covered call-writing strategy, the portfolio’s upside potential is limited. Like an unhedged position, within the range defined by the strike prices, the value of the portfolio varies with interest rates.
Selecting the “Best” Strategy
Comparing the two basic strategies for hedging with options, one cannot say that the protective put-buying strategy or the covered call-writing strategy is necessarily the better or more correct options hedge. The best strategy (and the best strike price) depends on the manager’s view of the market. Purchasing a put and paying the required premium are appropriate if the manager is fundamentally bearish. If, instead, the manager is neutral to mildly bearish, it is better to receive the premium on the covered call-writing strategy. If the manager prefers to take no view on the market at all, and as little risk as possible, then the futures hedge is the most appropriate. If the manager is fundamentally bullish, then no hedge at all is probably the best strategy.
Steps in Options Hedging
Like hedging with futures, there are steps that managers should consider before setting their hedges. These steps include
Step 1. Determine the option contract that is the best hedging vehicle. The best option contract to use depends on several factors. These include option price, liquidity, and correlation with the bond(s) to be hedged. In price-inefficient markets, the option price is important because not all options will be priced in the same manner or with the same volatility assumption. Consequently, some options may be overpriced and some underpriced. Obviously, with other factors equal, it is better to use the underpriced options when buying and the overpriced options when selling.
Whenever there is a possibility that the option position may be closed out prior to expiration, liquidity is also an important consideration. If the particular option is illiquid, closing out a position may be prohibitively expensive, and the manager loses the flexibility of closing out positions early or rolling into other positions that may become more attractive. Correlation with the underlying bond(s) to be hedged is another factor in selecting the right contract. The higher the correlation, the more precisely the final profit and loss can be defined as a function of the final level of rates. Poor correlation leads to more uncertainty.
While most of the uncertainty in an options hedge usually comes from the uncertainty of interest rates themselves, the degree of correlation between the bonds to be hedged and the instruments underlying the options contracts add to that risk. The lower the correlation, the greater the risk.
Step 2. Find the appropriate strike price. For a cross-hedge, the manager will want to convert the strike price on the options that are actually bought or sold into an equivalent strike price for the actual bonds being hedged.
Step 3. Determine the number of contracts. The hedge ratio is the number of options to buy or sell.
Steps 2 and 3, determining the strike price and the number of contracts, can best be explained with examples using futures options.
Protective Put-Buying Strategy Using Futures Options
As explained earlier, managers who want to hedge their bond positions against a possible increase in interest rates will find that buying puts on futures is one of the easiest ways to purchase protection against rising rates. To illustrate a protective put-buying strategy, we can use the same FNMA bond that we used to demonstrate how to hedge with Treasury bond futures.4 In that example, a manager held $10 million par value of a 6.25% FNMA bond maturing 5/15/29 and used September 1999 Treasury bond futures to lock in a sale price for those bonds on the futures delivery date. Now we want to show how the manager could use futures options instead of futures to protect against rising rates. 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.
Selecting the Strike Price
The manager must determine the minimum price that he wants to establish for the FNMA bonds. In our illustration, we will assume that the minimum price before the cost of the put options purchased is 84.453. This is equivalent to saying that the manager wants to establish a strike price for a put option on the hedged bonds of 84.453. But the manager is not buying a put option on the FNMA bond. He is buying a put option on a Treasury bond futures contract. Therefore, the manager must determine the strike price for a put option on a Treasury bond futures contract that is equivalent to a strike price of 84.453 for the FNMA bond.
This can be done with the help of Exhibit 61–7. We begin at the top left-hand box of the exhibit. Since the minimum price is 84.453 for the FNMA bond, this means that the manager is attempting to establish a maximum yield of 7.573%. This is found from the relationship between price and yield: given a price of 84.453 for the FNMA bond, this equates to a yield of 7.573%. (This gets us to the lower left-hand box in Exhibit 61–7.) From the assumption that the spread between the FNMA bond and the cheapest-to-deliver issue is a constant 70 basis points, setting a maximum yield of 7.573% for the FNMA bond is equivalent to setting a maximum yield of 6.873% for the cheapest-to-deliver issue. (Now we are at the lower box in the middle column of Exhibit 61–7.) Given the yield of 6.873% for the CTD issue, the minimum price before the cost of the puts purchased can be determined (the top box in the middle column of the exhibit). A 6.873% yield for the CTD issue gives us a price of 141.136. (This is determined from the characteristics of the CTD issue.) The corresponding futures price is found by dividing the price of the CTD issue by the conversion factor. This gets us to the box in the right-hand column of Exhibit 61–7. Since the conversion factor is 1.283, the futures price is about 110 (141.136 divided by 1.283). This means that a strike price of 110 for a put option on a Treasury bond futures contract is roughly equivalent to a put option on our FNMA bond with a strike price of 84.453.
EXHIBIT 61–7
Calculating Equivalent Strike Prices and Yields for Hedging with Futures Options
The foregoing steps are always necessary to obtain the appropriate strike price on a put futures option. The process is not complicated. It simply involves (1) the relationship between price and yield, (2) the assumed relationship between the yield spread between the bonds to be hedged and the cheapest-to-deliver issue, and (3) the conversion factor for the cheapest-to-deliver issue. As with hedging employing futures illustrated earlier in this chapter, the success of the hedging strategy will depend on (1) whether the cheapest-to-deliver issue changes and (2) the yield spread between the bonds to be hedged and the cheapest-to-deliver issue.
Calculating the Number of Options Contracts
The hedge ratio is determined using the following equation similar to Eq. (61–7) because we will assume a constant yield spread between the bond to be hedged and the cheapest-to-deliver issue:
The dollar durations are as follows per 50 basis point change in rates:
Current dollar duration without options = $512,320
Dollar duration of the CTD issue = $6,021
Notice that the dollar durations are different from those used in calculating the number of futures contracts for the futures hedge. This is so because the dollar durations are calculated at prices corresponding to the strike price of the futures option (110) rather than the futures price (113). The number of futures options contracts is then
Thus, to hedge the FNMA bond position with put options on Treasury bond futures, 109 put options must be purchased.
Outcome of the Hedge
To create a table for the protective put hedge, we can use some of the numbers from Exhibit 61–2. Exhibit 61–8 shows the scenario analysis for the protective put-buying strategy. The first five columns are the same as in Exhibit 61–2. For the put option hedge, column (6) shows the value of the put option position at the expiration date. The value of the put option position at the expiration date will be equal to zero if the futures price is greater than or equal to the strike price of 110.
EXHIBIT 61–8
Hedging a Nondeliverable Bond to a Delivery Date with Puts on Futures
If the futures price is below 110, then the options expire in the money, and the value of the put option position is
Value of put option position
= (110/100 – futures price/100) × $100,000 × number of put options
For example, for the first scenario in Exhibit 61–8 of $8 million for the actual sale price of the FNMA bond, the corresponding futures price is 105.85581. The number of put options purchased is 109. Therefore
(110/100 – 105.85581/100) × $100,000 × 109 = $45,717
The effective sale price for the FNMA bonds is then equal to
Effective sale price = actual sale price + value of put option position – option cost
Let’s look at the option cost. Suppose that the price of the put option with a strike price of 110 is $500 per contract. With a total of 109 options, the cost of the protection is $54,500 (109 × $500, not including financing costs and commissions). This cost is shown in column (7) and is equivalent to 0.545 per $100 par value hedged.
The effective sale price for the FNMA bonds for each scenario is shown in the last column of Exhibit 61–8. This effective sale price is never less than 83.908. This equals the price of the FNMA bonds equivalent to the futures strike price of 110 (i.e., 84.453) minus the cost of the puts (i.e., 0.545 per $100 par value hedged). This minimum effective price is something that can be calculated before the hedge is ever initiated. (As prices decline, the effective sale price actually exceeds the target minimum sale price of 83.908 by a small amount. This is due only to rounding and the fact that the hedge ratio is left unaltered, although the relative dollar durations that go into the hedge ratio calculation change as yields change.) As prices increase, however, the effective sale price of the hedged bonds increases as well; unlike the futures hedge shown in Exhibit 61–2, the options hedge protects the investor if rates rise but allows the investor to profit if rates fall.
Covered Call-Writing Strategy with Futures Options
Unlike the protective put-buying strategy, covered call writing is not entered into with the sole purpose of protecting a portfolio against rising rates. The covered call writer, believing that the market will not trade much higher or much lower than its present level, sells out-of-the-money calls against an existing bond portfolio. The sale of the calls brings in premium income that provides partial protection in case rates increase. The premium received does not, of course, provide the kind of protection that a long put position provides, but it does provide some additional income that can be used to offset declining prices. If, instead, rates fall, portfolio appreciation is limited because the short call position constitutes a liability for the seller, and this liability increases as rates decline. Consequently, there is limited upside price potential for the covered call writer. Of course, this is not so bad if prices are essentially going nowhere; the added income from the sale of call options is obtained without sacrificing any gains.
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. With futures selling around 113 on the hedge initiation date, a sale of a 117 call option on futures might be appropriate. As before, it is assumed that the hedged bond will remain at a 70 basis point spread over the CTD issue. We also assume for simplicity that the price of the 117 calls is $500 per contract. The number of options contracts sold will be the same, namely, 109 contracts for $10 million face value of underlying bonds. Thus, the proceeds received from the sale of the 109 call options is $54,500 (109 contracts × $500) or 0.545 per $100 par value hedged.
Exhibit 61–9 shows the outcomes of the covered call-writing strategy given these assumptions. The first five columns of the exhibit are the same as for Exhibit 61–8. In column (6), the liability resulting from the call option position is shown. The liability is zero if the futures price for the scenario 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:
(Futures price/100 – 117/100) × $100,000 × number of put options
For example, consider the scenario in Exhibit 61–9, where the actual sale price of the FNMA bond is $9.5 million. The corresponding futures price is 119.31255. The number of call options sold is 109. Therefore
(119.31255/100 – 117/100) × $100,000 × 109 = $252,068
That is,
Effective sale price = actual sale price
+ proceeds from sale of the call options
- liability of call position
Since the proceeds from sale of the call options is $54,500, then
Effective sale price = actual sale price + $54,000 – liability of call position
The last column of Exhibit 61–9 shows the effective sale price for each scenario.
EXHIBIT 61–9
Writing Calls on Futures against a Nondeliverable Bond
Just as the minimum effective sale price could be calculated beforehand for the protective put-buying strategy, the maximum effective sale price can be calculated beforehand for the covered call-writing strategy. The maximum effective sale price will be the price of the hedged bond corresponding to the strike price of the option sold plus the premium received. In this case, the strike price on the futures call option is 117. A futures price of 117 corresponds to a price of 150.111 (from 117 times the conversion factor of 1.283) and a corresponding yield of 6.159% for the cheapest-to-deliver issue. The equivalent yield for the hedged bond is 70 basis points higher, or 6.859%, for a corresponding price of 92.313. Adding on the premium received of 0.545 per $100 par value hedged, the final maximum effective sale price will be about 92.858. As Exhibit 61–10 shows, if the hedged bond does trade at 70 basis points over the CTD issue as assumed, the maximum effective sale price for the hedged bond is, in fact, slightly over 92.858. The discrepancies shown in the exhibit are due to rounding and the fact that the position is not adjusted even though the relative dollar durations change as yields change.
EXHIBIT 61–10
Comparison of Alternative Strategies
Comparing Alternative Strategies
In this chapter we reviewed three basic strategies for hedging a bond position: (1) hedging with futures, (2) hedging with out-of-the-money puts, and (3) covered call writing with out-of-the-money calls. Similar but opposite strategies exist for managers who are concerned that rates will decrease. As might be expected, there is no “best” strategy. Each strategy has its advantages and its disadvantages, and we never get something for nothing. To get anything of value, something else of value must be forfeited.
To make a choice among strategies, it helps to lay the alternatives side by side. Using the futures example and the futures options examples, Exhibit 61–10 shows the final values of the portfolio for the various hedging alternatives. It is easy to see from Exhibit 61–10 that if one alternative is superior to another alternative at one level of rates, it will be inferior at some other level of rates.
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.
Hedging with Options on Cash Instruments
Hedging a position with options on cash bonds is relatively straightforward. Most strategies, including the purchase of protective puts, covered call writing, and buying collars, are essentially the same whether futures options or options on physicals are used. There are some mechanical differences in the way the two types of option contracts are traded, and there may be substantial differences in their liquidity. Nonetheless, the basic economics of the strategies are virtually identical.
Using options on physicals frequently relieves the manager of much of the basis risk associated with a futures options hedge. For example, a manager of Treasury bonds or notes usually can buy or sell options on the exact security held in the portfolio. Using options on futures, rather than options on Treasury bonds, is sure to introduce additional elements of uncertainty.
Given the illustration presented earlier, and given that the economics of options on physicals and options on futures are essentially identical, additional illustrations for options on physicals are unnecessary. The only important difference is the hedge ratio calculation and the calculation of the equivalent strike price. To derive the hedge ratio, we always resort to an expression of relative dollar durations. Thus, for options on physicals assuming a constant spread, the hedge ratio 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 very similar. Given the strike price of the option, the strike yield is easily 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.
KEY POINTS
• Buying an interest-rate futures contract increases a portfolio’s duration; selling an interest-rate futures contract decreases a portfolio’s duration.
• The advantages of adjusting a portfolio’s duration using futures rather than cash-market instruments are that transaction costs are lower, margin requirements are lower, and short selling in the futures market is easier.
• The general principle in controlling interest-rate risk with futures is to combine the dollar exposure of the current portfolio and that of a futures position so that it is equal to the target dollar exposure. The number of futures contracts needed to achieve the target dollar duration depends on the current dollar duration of the portfolio without futures and the dollar duration per futures contract.
• Hedging with futures calls for taking a futures position as a temporary substitute for transactions to be made in the cash market at a later date, with the expectation that any loss realized by the manager from one position (whether cash or futures) will be offset by a profit on the other position.
• Hedging is a special case of controlling interest-rate risk in which the target duration or target dollar duration is zero.
• Cross-hedging occurs when the bond to be hedged is not identical to the bond underlying the futures contract. A short (or sell hedge) is used to protect against a decline in the cash price of a bond; a long or buy hedge is employed to protect against an increase in the cash price of a bond.
• The steps in hedging include (1) determining the appropriate hedging instrument, (2) determining the target for the hedge, (3) determining the position to be taken in the hedging instrument, and (4) monitoring and evaluating the hedge.
• The key factor to determine which futures contract will provide the best hedge is the degree of correlation between the rate on the futures contract and the interest rate that creates the underlying risk that the manager seeks to eliminate.
• The manager should determine the target rate or target price, which is what is expected from the hedge. The hedge ratio is the number of futures contracts needed for the hedge.
• The basis is the difference between the spot price (or rate) and the futures price (or rate).
• In general, when hedging to the delivery date of the futures contract, a manager locks in the futures rate or price. Hedging with Treasury bond futures and Treasury note futures is complicated by the delivery options embedded in these contracts. When a hedge is lifted prior to the delivery date, the effective rate (or price) that is obtained is much more likely to approximate the current spot rate than the futures rate the shorter the term of the hedge.
• The proper target for a hedge that is to be lifted prior to the delivery date depends on the basis. Basis risk refers only to the uncertainty associated with the target rate basis or target price basis.
• Hedging substitutes basis risk for price risk.
• Hedging non-Treasury securities with Treasury bond futures requires that the hedge ratio consider two relationships: (1) the cash price of the non-Treasury security and the cheapest-to-deliver issue, and (2) the price of the cheapest-to-deliver issue and the futures price.
• In computing the hedge ratio for nondeliverable securities, the yield beta should be considered; regression analysis is used to estimate the yield beta and captures the relationship between yield levels and yield spreads.
• After a target is determined and a hedge is set, the hedge must be monitored during its life and evaluated after it is over. It is important to ascertain the sources of error in a hedge in order to gain insights that can be used to advantage in subsequent hedges.
• Three popular hedge strategies using options are the protective put-buying strategy, the covered call-writing strategy, and the collar strategy.
• A manager can use a protective put-buying strategy to hedge against rising interest rates. A protective put-buying strategy is a simple combination of a long put option with a long position in a cash bond.
• A covered call-writing strategy involves selling call options against the bond portfolio. A covered call-writing strategy entails much more downside risk than buying a put to protect the value of the portfolio, and many managers do not consider covered call writing a hedge.
• It is not possible to say that the protective put-buying strategy or the covered call-writing strategy is necessarily the better or more correct options hedge. The best strategy (and the best strike prices) depends on the manager’s view of the market.
• A collar strategy is a combination of a protective put-buying strategy and a covered call-writing strategy. A manager who implements a collar strategy eliminates part of the portfolio’s downside risk by giving up part of its upside potential.
• The steps in options hedging include determining the option contract that is the best hedging vehicle, finding the appropriate strike price, and determining the number of options contracts. At the outset of options hedging, a minimum effective sale price can be calculated for a protective put-buying strategy and a maximum effective sale price can be computed for a covered call-writing strategy.
• The best options contract to use depends on the option price, liquidity, and correlation with the bond(s) to be hedged.
• For a cross-hedge, the manager will want to convert the strike price for the options that are actually bought or sold into an equivalent strike price for the actual bonds being hedged. When using Treasury bond futures options, the hedge ratio is based on the relative dollar duration of the current portfolio, the cheapest-to-deliver issue, and the futures contract at the option expiration date, as well as the conversion factor for the cheapest-to-deliver issue.
• While there are some mechanical differences in the way options on physicals and options on futures are traded and there may be substantial differences in their liquidity, the basic economics of the hedging strategies are virtually identical for both contracts. Using options on physicals frequently relieves the manager of much of the basis risk associated with an options hedge.