# Interest Rate Futures

### Interest Rate Futures [viewable here in Excel]

For the Eurodollar contract, the contract size is \$1 million with the yield being quoted on an add-on basis. The add-on yield is given:

add-on yield = (discount / price) x (360/DTM)

where DTM is the days until maturity. For example, assume the discount yield is quoted at 8.32 percent. To get the price associated with this discount yield, we apply the following formula:

Equation (5.2) price = \$1,000,000 = discount x \$1,000,000 x DTM / 360

Using Equation 5.2, we see that for this discount yield the price for a \$1 million face value three-month instrument is \$979,200. Therefore, the dollar discount is \$20,800 and 9-days remain until the bill will mature. With these values, we have:

add-on yield = (\$20,800/\$979,200) x (360/90) = 0.0850

\$1,000,000 x .0832 x 90/360 = \$20,800
=1000000-20,800 = \$979,200
= 979,200 x 2.12% = \$1,000,000

Add-on yields exceed corresponding discount yields. In our example, the discount yield is 8.32 percent, and the add-on yield equivalent is 8.5 percent. However, for both measures, a shift of one basis point (1/100 of 1 percent) is worth \$25 on a \$1 million contract. This amount is determined from the following equation:

\$1,000,000 x 0.0001 x 90/360 = \$25 x 832 = \$20,800

Also note that these yields and relationships vary with maturity, so the statements made here hold only for three-month maturities.

Eurodollar futures contract prices are quoted using the IMM index, which is a function of the three-month LIBOR rate:

IMM index = 100.00 - three-month LIBOR
IMM index = 100.00 - DY

where DY is the discount yield (for example 7.1 is 7.1 percent). As an example, a discount yield of 8.32 percent implies an IMM index value of 100 - 91.68 = 8.32

Futures invoice price = \$1,000,000 - .0832 x \$1,000,000 x 90 / 360 = \$979,200

of 832 x 25 = \$20,800 - \$1,000,000

### Hedging with Interest Rate Futures

In this section, we explore the concept of hedging with interest rate futures. We present a series of examples, progressing from simple cases to situations that are more complex. In essence, the hedger in interest rate futures attempts to take a futures position that will generate a gain to offset a potential loss in the cash market. This also implies that the hedger takes a futures position that will generate a loss to offset a potential gain in the cash market. Thus, the interest rate futures hedger is attempting to reduce risk, not make profits.

### A Long Hedge Example

A portfolio managers learns on December 15 that he will have \$970,000 to invest in 90-day T-bills six month from now. Current yields on T-bills stand at 12 percent and the yield curve is flat, so forward rates are all 12 percent as well. The manager finds the 12 percent rate attractive and decides to lock in by going long in a T-bill futures contract maturing on June 15, exactly when the funds come available for investment. As Table 5.13 shows, the manager anticipates the cash position on December 15 and buys one T-bill futures contract to hedge the risk that yields might fall before the funds are available for investment on June 15. With the current yield and, more importantly, the forward rate on T-bills of 12 percent, the portfolio manager expects to be able to buy \$1 million face value of t-bills because

Here, you're investing in the instrument. Going long means going long on "PRICE", not "Interest Rates", so if the instrument "PRICE" moves higher, "INTEREST RATES" move LOWER.

\$970,000 = \$1,000,000 - 0.1200 x \$1,000,000 x 90/360 = \$970,000.

Table 5.13 A long hedge with T-Bill futures

 Date Cash Market Futures Market December 15 A portfolio manger learns he will receive \$970,000 in six months to invest in T-bills Market yield: 12% The manager buys one T-bill futures contract to mature in six months Expected face value of bills to purchase: \$1 million Futures price: \$970,000 June 15 Manager receives \$970,000 to invest. The manager sells one T-bill futures with the contract maturing immediately. Market yield: 10% \$1 million face value of T-bills now costs \$975,000. Futures yield: 10% Loss = -\$5,000 Profit = \$5,000 Net wealth change = 0

The hedge is initiated and time passes. On June 15, the 90-day T-bill yield has fallen to 10 percent, confirming the portfolio manager's fears. Consequently, the \$1 million face value of 90-day T-bills is worth:

\$975,000 = \$1,000,000 - 0.1000 x \$1,000,000 x 90/360 = \$975,000

Just before the futures contract matures, the manager sells one June T-bill futures contract, making a profit of \$5,000. But in the spot market (where now the portfolio manager buys what he thought would be \$970,000 in T-bills), the cost of the \$1 million face value of 90-day T-bills has risen from \$970,000 t0 \$975,000, generating a cash market loss of \$5,000. However, the futures profit exactly offsets the cash market loss for a zero change in wealth. With the receipt of the \$970,000 that was to be invested, plus the \$5,000 futures profit, the original plan may be executed, and the portfolio manager purchases \$1 million face value in 90-day T-bills.

By design, this example is extremely artificial in order to illustrate the long hedge. Notice that the yield curve is flat at the outset, and only its level changes. Figure 5.2 portrays the kind of yield curve shift that was assumed. This idealized yield curve shift is unlikely to occur. Moreover, the assumption of a flat yield curve plays a crucial role in accounting for the simplicity of this example. If the yield curve is flat, spot and forward rates are identical. When one "locks in" some rate via futures trading, it is necessarily a forward rate that is locked in, as the next example shows. We also assume that the portfolio manager received exactly the right amount of funds at exactly the right time to purchase \$1 million of T-bills. These unrealistic assumptions are gradually relaxed in the following examples (A Short Hedge).

Figure 5.2 The idealized yield curve shift of the long hedge ### A Short Hedge Example

Interest rate futures can be used to hedge a variety of risks posed by fluctuating interest rates. For example, hedges constructed with interest rate futures are useful for banks facing an interest-sensitivity mismatch between their assets and liabilities. Suppose that in March a bank customer demands a \$1 million fixed-rate loan for nine months. The problem the bank faces is estimating its cost of funds over the life of the loan. If the bank could issue a nine-month \$1 million dollar fixed-rate certificate of deposit (CD), it would have a precise match between the interest sensitivity of its asset (the loan) and the interest sensitivity of its liabilities (the CD). However, suppose the bank can only lock in its funding for six months at 3.00 percent. To fund the loan for the entire nine months, the bank will need to issue a \$1 million three-month CD quoted in March is 3.5 percent. This yield provides the bank with the market's assessment of the three-month CD rates that will prevail in September. This assessment is helpful to the bank in determining an expected cost of funds over the life of the loan, but still leaves the bank vulnerable to rates rising above the expected rate.

Here, you are investing in the instrument. Going short means going short on "PRICE", so if the instrument "PRICE" moves LOWER, "INTEREST RATES" move HIGHER.

To hedge this risk, the bank can establish a short position in SEP Eurodollar futures in March. In constructing the hedge, the bank will use a single futures contract. If rates rise unexpectedly, the Eurodollar futures price will fall and the bank's short position will become more valuable. Suppose, in fact, that the three-month rate rises to 4.5 percent in September. We know that the Eurodollar futures price yield must converge to the prevailing three-month rate at contract expiration. This means that the bank's Eurodollar futures position has gained \$2,500. This amount precisely offsets the bank's increase in its cost of funds over and above the 3.5 percent rate it expected to prevail at the time the bank loan was priced and the hedge was constructed. Table 5.14 displays the dash flows associated with the construction of this hedge. Of course, by hedging, the bank has given up the opportunity to make extra money if its cost of funds had fallen unexpectedly. However, by hedging, and locking in its cost of funds, the bank is able to price the nine-month fixed rate loan with certainty and lock in an acceptable profit.

Table 5.14 Hedging a bank's cost of funds using interest rate futures

 Date Cash Market Futures Market March Bank makes nine-month fixed rate loan financed by a six-month CD at 3.0% and rolled over for three months at an expected rate of 3.5%. Establish a short position in SEP Eurodollar futures at 96.5 reflecting a 3.5% futures yield. September The three-month LIBOR is now at 4.5%; the bank's cost of funds of 3.5%; the additional cost equals \$2,500 - i.e., 90/360 x 0.01 x \$1,million. Offset one SEP Eurodollar futures contract at 95.5 reflecting a 4.5% futures yield; this produces a profit of \$2,500 - i.e., 100 basis points x \$25 per basis point x one contract. Total additional cost of funds: \$2,500 Futures profit: \$2,500 Net interest expense after hedge: 0

### The Cross-Hedge [viewable here in Excel]

The financial vice-president of a large manufacturing firm has decided to issue \$1 billion worth of 90-day commercial paper in three months. The outstanding 90-day commercial paper of the firm yields 17 percent, or 2 percent above the current 90-day T-bill rate of 15 percent. Fearing that rate might rise, the vice-president decides to hedge against the risk of increasing yields by entering the interest rate futures market.

He decides to hedge the firm's commercial paper in the T-bill futures market, since rates on commercial paper and T-bills tend to be highly correlated. Since one type of instrument is hedged with another, this hedge becomes a "cross-hedge." In general, a cross-hedge occurs when the hedged and hedging instrument differ with respect to (1) risk level, (2) coupon, (3) maturity, or (4) the time span covered by the instrument being hedged and the instrument deliverable against the futures contract. This means that the vast majority of all hedges in the interest rate futures markets are cross-hedges. The hedge contemplated by the vice-president is a cross-hedge, because the commercial paper and the T-bill differ in risk. Assuming that the commercial paper is to be issued in 90 days (and that the T-bill futures contract matures at the same time) ensures that the commercial paper and the T-bill delivered on the futures contract cover the same time span. Therefore, the vice-president decides to sell 1,000 T-bill futures contracts to mature in three months. Table 5.15 shows the transaction. The futures price is \$960,000, implying a futures yield of 16 percent. Notice that this differs by 1 percent from the current 90-day T-bill yield of 15 percent. Time passes, and in three months the futures yield has not changed, remaining at 16 percent. However, since the futures contract is about to mature, the spot and futures rates and now equal. Consequently, the trade incurs no gain or loss in the futures contract.

Table 5.15 A cross-hedge between T-bill futures and commercial paper

 Date Cash Market Futures Market t = 0 The financial vice-president plans to sell 90-day commercial paper in three months in the amount of \$1 billion, at an expected yield of 17%, which should net the firm \$957.5 million The vice-president sells 1,000 T-bill futures contracts to mature in three months with a futures yield of 16%, a futures price per contract of \$960, 000, and a total futures price of \$960 million. t = 3 months The spot commercial paper rate is now 18%, the usual 2% above the sport T-bill rate; consequently, the sale of the \$1 billion of commercial paper nets \$955 million, not the expected \$957.5 million. Opportunity loss = ? The T-bill futures contract is about to mature, so the T-bill futures rate = spot rate = 16%; the futures price is still \$960,000 per contract, so there is no gain or loss Net wealth change = ? Gain/loss = 0.

In the cash market, the 90-day commercial paper spot rate at the end of the hedging period has become 18 percent, not the 17 percent that was the original 90-day spot rate at the initiation date of the hedge. Since the vice-president thought he was "locking-in" the 17 percent spot rate, he expected to receive \$957.5 million for the commercial paper issue. But the commercial paper rate at the time of issue is 18 percent, so the firm receives only \$955 million. This appears to be a loss in the cash market of \$2.5 million. However, this is only appearance. The vice-president may have thought he was locking in the prevailing spot rate of 17 percent at the time the hedge was initiated, but such a belief was unwarranted. By hedging the issuance of the commercial paper, the vice-president should have expected to lock in the three-month forward rate for the 90-day commercial paper.

Figure 5.3 clarifies these relationships by presenting yield curves for T-bills and commercial paper based on bank discount yields. The yield curves are consistent with the data of the preceding discussion. At the outset of the hedge, the 90-day spot T-bill rate is 0.15, and the commercial paper rate equals 0.17. The 180-day spot rates are 0.152 and .171174 for T-bills and commercial paper, respectively. The shape of the yield curves gives sufficient information to calculate the forward, and hence the futures, rates for the time span covering the period from day 90 to day 180.

 \$962,500 15% =1000000-0.15*1000000*90/360 \$960,000 16% =1000000-0.16*1000000*90/360 \$957,500 17% =1000000-0.17*1000000*90/360 \$955,000 18% =1000000-0.18*1000000*90/360 \$962,000 15.2% =1000000-0.152*1000000*90/360 \$957,207 17.1174% =1000000-0.171174*1000000*90/360

Figure 5.3 Hypothetical yield curve for T-bills and commercial paper For the spot 90-day T-bill with a bank discount rate of 0.15, the price of \$1 million face value T-bill is \$962,500. The growth in this investment over the 90-day horizon is 1.038961 = \$1,000,000/\$962,500. For the 180-day T-bill with a bank discount rate of 15.20 percent, the price is \$924,000, and the growth would be 1.082251. The futures yield to cover from day 90 to day 180 has been specified to be 16.00 percent, a price of \$960,000 and an implied growth of 1.041667 = \$1,000,000/\$960,000. The values are mutually consistent because 1.082251 = 1.038961 x 1.041667.

Similarly, for the 90-day commercial paper with a rate of 0.17, the price of a \$1 million face value instrument would be \$957,500 with a growth of 1.044386 = \$1,000,000/\$957,500. For the 180-day commercial paper, the discount rate is 0.171174. This implies a price of \$914,413 and a growth rate of 1.093598 = \$1,000,000/\$914,413. These commercial paper rates of 0.17 for the 90-day paper and 0.17114 for 180-day paper imply a forward bank discount rate of 0.18 for commercial paper for the period from 90 to 180. For 90-day paper at a bank discount rate of 0.18, the price of a \$1 million instrument is \$955,000, which will grow by a factor of 1.047120 = \$1,000,000/\$955,000 over its life. These values are mutually consistent because 1.093598 = 1.044386 x 1.047120.

 Growth Rate T-Bill Received (Short Covered) Rate Forward Implied Forward Price 1.0389610 \$962,500 15% (Today's T-Bill spot rate) 1.0416667 \$960,000 16% (T-Bill 3-mo forward rate) 1.082251082 \$924,000 1.04439 \$957,500 17% (CP - 90 days) 1.087902524 \$919,200 1.04712 \$955,000 18% (CP - 180 days) 1.093598349 \$914,413 1.08225 \$924,000 15.2% (T-Bill-180 spot rate) 1.09360 \$914,413 17.1174% (CP 180-day spot rate) 924,000 Discount the 3-mo and 6-mo spot and forward rate to the 180-day T-bill spot rate 914,413 Discount the 3-mo and 6-mo spot and forward rate to the 180-day CP spot rate

These forward rates, evaluated at time = 0, are the expected future bank discount rates to prevail on three-month T-bills and commercial paper beginning in three months. Consequently, the implied yield on the commercial paper of this example is 0.18, not the 0.17 that the vice-president attempted to lock in.

Now, it is possible to understand exactly why the vice-president was unable to lock in 17 percent, even though it was the spot rate prevailing at the time the hedge was initiated. The reason is simple this: For the time period over which the commercial paper was to be issued (from three to six months into the future), the market believed that the 90-day commercial paper rate would be 18 percent in three months. The futures price and yield reflected this belief. Although the vice-president desired 17 percent rate, the market's expected rate was 18 percent, and by entering the futures contract the vice-president locked in the 18 percent rate. Therefore, the opportunity loss of Table 5.15 is only apparent. The vice-president's expectation of issuing the commercial paper at 17 percent was completely unwarranted. Instead, the vice-president should have expected to issue the commercial paper at the market's expected bank discount rate of 18 percent. Then he would have expected to net \$955,000 for the firm, which is exactly what happened in the example.

### A Cross-Hedge with Faulty Expectations

In the preceding example, the vice-president misunderstood the nature of the futures market. If the vice-president had understood everything correctly, Table 5.15 would have shown a zero total wealth change. Thus far, all of the examples have been of perfect hedges - hedges leaving total wealth unchanged. Sometimes, however, even when the hedge is properly initiated with the appropriate expectations, those expectations can turn out to be false. In such cases, the hedge will not be perfect; total wealth will either increase or decrease.

Table 5.15 A cross-hedge with faulty expectations

 Date Cash Market Futures Market t = 0 The financial vice-president plans to sell 90-day commercial paper in three months in the amount of \$1 billion, at an expected yield of 18%, which should net the firm \$955 million. The vice-president sells 1,000 T-bill futures contracts to mature in three months with a futures yield of 16%, a futures price per contract of \$960, 000, and a total futures price of \$960 million. t = 3 months The spot commercial paper rate was expected to be 18% at this time, but is really 18.5%; consequently, the sale of the \$1 billion of commercial paper nets \$953.75 million, not the expected \$955 million. The T-bill futures contract is about to mature, so the T-bill futures rate = spot rate = 16.25%; the futures price is still \$953,375 per contract, so there is gain per contract of \$625,000, and a total gain the 1,000 contracts of \$625,000. Opportunity loss = -\$1,250,000 Gain = \$625,0000. Net wealth change = -\$625,000

To illustrate this possibility, assume the same basic hedging problem as in the cross-hedge example. In particular, assume that the vice-president wishes to hedge the same issuance of commercial paper and that the yield curves are as we shown in Figure 5.3. The actions and expectations of the vice-president, shown in Table 5.16, are exactly correct. The yield curve implies that, in 90 days, the 90-day T-bill and commercial paper rates will stand at 16 and 18 percent respectively.

However, in this instance, assume that these expectations formed are incorrect. During the 90-day period before the commercial paper was issued, the market came to view the commercial paper as being riskier than was previously thought, and the economy experienced a higher rate of inflation than anticipated. Historically, assume that the yield premium of commercial paper has been 2 percent above the T-bill rate, consistent with Figure 5.3. But now, due to the perception of increased risk for commercial paper, the yield differential widens to 2.25 percent. Then assume that in three months the T-bill rate happens to be 16.25 percent, rising due to greater than anticipated inflation. Under these assumptions, the commercial paper rate is 18.5 percent, not the originally expected 18 percent.

As Table 6.16 reveals, the total gain on the futures position is \$625,000. Due to the commercial paper rate being 18.5 percent, not the originally anticipated 18 percent, there is a loss on the commercial paper of -\$1.25 million. Since the error on expectation was 0.5 percent on the commercial paper, but only 0.25 percent on the T-bills, the gain on the futures does not off-set the total loss of the commercial paper. This results in a net wealth change of -\$625,000. However, the loss would have been -\$1.25 million without the futures hedge.

Table 6.16 A cross-hedge between corporate bonds and T-bill futures

 Date Cash Market Futures Market March 1 A portfolio manager learns will receive \$5 million to invest in 5%, ten-year AAA bonds in three months, with an expected yield of 7.5% and a price of \$826.30; the manager expects to buy 6,051 bonds. The portfolio manager buys \$5 million face value of T-bill futures (five contracts) to mature on June 1 with a futures yield of 6.0% and a futures price, per contract of \$985,000. June 1 AAA yields have fallen to 7.08%, causing the price of the bonds to be \$852.72 and representing a loss, per bond, of \$26.42; since the plan was to buy 6,051 bonds, the total loss is 6,051 x \$26.42 = -\$159,867. The T-bill futures yield has fallen to 5.58%, so the futures price = spot price = \$986,050 per contract; since five contracts were traded, the total profit is \$5,250. Loss = -\$159,897 Gain = \$5,250 Net wealth change = -\$154,617

In general, real-world hedges will not be perfect. Rates on both sides of the hedge tend to move in the same direction, but by uncertain amounts. On occasion, rates can even move in opposite directions, generating enormous gains or losses. In the example just discussed, assume that the commercial paper rate tuned out to be 18.5 percent, but that the T-bill rate was 15.75 percent - below the expected 16 percent. In this case, the loss on the commercial paper would be -\$1.25 million, and the loss on the futures would be -\$625,000 for a total loss of -\$1.875 million, because the firm loses on both sides of the hedge. Such an outcome is unlikely, but it is a possible result, of which hedgers should be aware of.

### Eurodollar and T-bill Futures

Previously, we considered some hedging strategies. In this section we explore alternative risk management strategies using short-term interest rate futures. We proceed by considering a series of examples. Taken together, these examples provide a handbook of techniques for a variety of risk management strategies. All of these strategies turn on protection against shifting interest rates.

### Changing the Maturity of an Investment

Many investors find themselves with an existing portfolio that may have undesirable maturity characteristics. For example, a firm might hold a six-month T-bill and realize that it will have a need for funds in three months. By the same token, another investor might hold the same six-month T-bill and fear that those funds might have to face lower reinvestment rates upon maturity in six months. This investor might prefer a one-year maturity. Both the firm and the investor could sell the six-month bill and invest for the preferred maturity. However, spot market transactions costs are relatively high, and many investors prefer to alter the maturities of investments by trading futures. The two examples that follow show how to use futures to accomplish both a shortening and lengthening of maturities.

### Shortening the Maturity of a T-bill Investment

Consider a firm that has invested in a T-bill. Now, on March 20, the T-bill has a maturity of 180 days, but the firm learns of a need for cash in 90 days. Therefore, it would like to shorten the maturity so it can have access to its funds in 90 days, around mid-June.

For simplicity, we assume that the short-term yield curve is flat with all rates at 10 percent on March 20. For convenience, we assume a 360-day year to match the pricing conventions for T-bills. The face value of the firm's T-bill is \$10 million. With 180 days to maturity and a 10 percent discount yield, the price of the bill is given by

P = FV - (DY x FV x DTM) / 360 [=10000000-(0.1*10000000*180)/360] =\$9,500,000

where P is the bill price, FV is the face value, DY is the discount yield, and DTM is the days until maturity. Therefore, the 180-day bill is worth \$9.5 million. If the yield curve is flat at 10 percent, the futures yield must be 10 percent, and the 90-day T-bill futures price must be \$975,000 per contract (=1000000-(0.1*1000000*90)/360 = \$975,000). Starting from an initial position of a six-month T-bill, the firm of our example can shorten the maturity be selling T-bill futures for expiration in three months, as shown in Table 6.10. On March 20, there was no cash flow, because the firm merely sold futures. On June 20, the six month bill is now a three-month bill and can be delivered against the futures. In table 6.10, the firm delivers the bills and receives the futures invoice amount of \$9.75 million, (Although we have assumed that the futures price did not change, this does not limit the applicability of our results. No matter how the futures price changed from March to June, the firm would still receive a total of \$9.75 million. We assume that this occurs in June instead of over the period.) The firm has effectively shortened the maturity from six months to three months.

Table 6.10 Transactions to shorten maturities

 Date Cash Market Futures Market March 20 Hold a six month T-bill with a face value of \$10 million, worth \$9.5 million; wish for a three-month maturity. Sell ten JUN T-bill futures contracts at 90.0, reflecting the 10% discount yield. June 20 Deliver cash market T-bills against futures; receive \$9.75 million.

### Lengthening the Maturity

Consider now, on August 21, an investor who holds a \$100 million face value T-bill that matures in 30-days on September 20. She plans to reinvest for another three months after the T-bill matures. However, she fears that interest rates might fall unexpectedly. If so, she would be forced to reinvest at a lower rate than is now reflected in the yield curve. The SEP T-bill futures yield is 9.8 percent, as is the rate on the current investment. She finds this rate attractive and would like to lengthen the maturity of the T-bill investment. She knows that she can lengthen the maturity by buying a September futures contract and taking delivery. She will then hold the delivered bill until maturity in December.

With a 9.8 percent discount futures yield, the value of the delivery unit is \$975,500. With \$100 million being available on September 20, the investor knows she will have enough funds to take delivery of \$100,000,000/\$975,500 = 102.51 futures contracts. Therefore, she initiates the strategy presented in Table 6.11.

On August 21, she held a bill worth \$99,183,333, assuming a yield of 9.8 percent. With the transaction of Table 6.11, she had no cash flow on August 21. With the maturity of the T-bill in September, the investor received the \$100 million and used almost all of it to pay for the futures delivery. We assume she invested the remainder, \$499,000, at a bank discount rate of 9.8% for three months. In December, the total proceeds will be \$102,511,533, from an investment that was worth \$99,183,333 on August 21. This gives her discount yield of about 9.8 percent over the four month horizon from August to December.

Table 6.11 Transactions to lengthen maturities

 Date Cash Market Futures Market August 21 Hold a 30-day T-bill with a face value of \$100 million; wish to extend the mat urity for 90-days. Buy 102 SEP T-bill futures contracts, with a yield of 9.8%. September 20 The 30-day T-bill matures; receive \$100 million; invest \$499,000 in a money mar ket fund at 9.8%. December 19 T-bills received on SEP futures mature for \$102 million; receive proceeds of \$511,533 from investment =((1+(0.098*90/360))) *\$499,000 = \$511,225.50. Accept delivery on 102 SEP futures, paying \$99,501,000.

Notice that the transaction "locked-in" the 9.8 percent on the futures contract. In this example, this happens to match the spot rate of interest. However, the important point to recognize is that lengthening the maturity involves locking into the futures yield, no matter what that yield may be. Thus, for the period covered by the T-bill delivered on the futures contract, the investment will earn the futures yield at the time of contracting.

### Hedging with T-bond Futures

This section begins with an example of a cross-hedge of AAA corporate bonds. The sample shows that a single hedging rule of using \$1 of futures per \$1 of bonds can lead to horrible hedging results. This example leads to a discussion of alternative hedging techniques, focusing on hedging with T-bonds.

In all previous hedging examples, the hedged and hedging instruments were very similar. Often, however, the need arises to hedge an investment that is very different from those underlying in the futures contract. The effectiveness of a hedge depends on the gain or loss on both the spot and futures side of the transaction. But the change in the price of any bond depends on the shifts in the level of interest rates, changes in the slope of the yield curve, the maturity of the bond, and its coupon rate.

To illustrate the effect of the maturity and coupon rate on hedging performance, consider the following example. A portfolio manager learns on March 1 that he will receive \$5 million on June 1 to invest in AAA corporate bonds that pay a coupon rate of 5 percent and have ten years to maturity. The yield curve is flat and assumed to remain so over the period from March 1 to June 1. The current yield on AAA bonds is 7.5 percent. Since the yield curve is flat, the forward rates are also all 7.5 percent, so the portfolio manager expects to acquire the bonds at that yield. However, fearing a drop in rates, he decides to hedge in the futures market to lock in the forward rate of 7.5 percent.

The next step is to select the appropriate hedging instrument. The manger considers two possibilities. T-bills or T-bonds. However, the AAA bonds have a 5 percent coupon rate and a ten-year maturity, which does not match the coupon and maturity characteristics of either the T-bills or T-bonds deliverability on the respective futures contracts. The deliverable T-bills have a zero coupon and a maturity of only 90 days, while the deliverable T-bonds have a maturity of a least 15 years and an assortment of semiannual coupons. For this example, we assume that the cheapest-to-deliver T-bond will have a 20-year maturity at the target date of June 1, and that this cheapest-to-deliver bond has a 6 percent coupon.

To explore fully the potential difficulties of the situation, we consider hedging the AAA position with T-bill and T-bond futures. We ignore T-notes to dramatize the need to match coupon and maturity characteristics. For the bills and bonds, we assume the yields are 6 and 6.5 percent, respectively. Assuming that yields remain at 7.5 percent, the bond in which the manager plans to invest will have a price of \$826.30. With \$5 million to invest, the manager anticipates buying 6,051 bonds.

Table 6.16 present the hedging transaction and results for the T-bill hedge. Because \$5 million is becoming available for investment, assume the manager buy \$5 million face value of T-bill futures contracts. Time passes, and by June 1 yields have fallen by 42 basis points on both the AAA and the T-bills, respectively. The price of the corporate bond is \$852.72, or \$26.42 higher then the anticipated price of \$826.30. Since the manager expected to buy 6,051 bonds, this means that the total additional outlay will be 6,051 x \$26.42 = \$159,867, and this represents the loss in the cash market. In the futures market, rates have also fallen by 42 basis points, generating a futures price increase of \$1,050 per contract. Because five contracts were bought, the futures profit is \$5,250. However, the loss in the cash market exceeds the gain in the futures market, for a net loss of \$154,617. Note that this loss results even though rates changed by the same amount on both investments.

Table 6.16 A cross-hedge between corporate bonds and T-bill futures

 Date Cash Market Futures Market March 1 A portfolio manager learns will receive \$5 million to invest in 5%, ten-year AAA bonds in three months, with an expected yield of 7.5% and a price of \$826.30; the manager expects to buy 6,051 bonds. The portfolio manager buys \$5 million face value of T-bill futures (five contracts) to mature on June 1 with a futures yield of 6.0% and a futures price, per contract of \$985,000. June 1 AAA yields have fallen to 7.08%, causing the price of the bonds to be \$852.72 and representing a loss, per bond, of \$26.42; since the plan was to buy 6,051 bonds, the total loss is 6,051 x \$26.42 = -\$159,867 The T-bill futures yield has fallen to 5.58%, so the futures price = spot price = \$986,050 per contract; since five contracts were traded, the total profit is \$5,250. Loss = -\$159,897 Gain = \$5,250 Net wealth change = -\$154,617

Consider now the same hedging problem, but assume that we implement the hedge using a \$5 million face value of T-bond futures. Table 6.17 presents the transaction and results. Again yields fall by 42 basis points on both instruments. Consequently, the effect on the cash market is the same, but the total futures gain is \$231,650, more than offsetting the loss in the cash market and generating a net wealth change equal to \$71,783.

If the goal of the hedge is to secure a net wealth change of zero, a gain is approximately viewed as no better than a loss. It is only by accident of rates moving in appropriate direction that the gain was not a loss anyway. Recall that all of the simplifying assumptions were in place - a flat yield curve, with rates on both instruments moving in the same direction and by the same amount. However, as noted earlier, the coupon and maturity of the hedged and hedging instruments do not match. All three instruments - the bond, the T-bill futures, and the T-bond futures - have different durations, reflecting different sensitivities to interest rates. Consequently, for a given shift in yields (e.g., 42 basis points), the prices of the three instruments will change by different amounts. Therefore, a simple hedge of \$1 in the futures market per \$1 in the cash market is unlikely to produce satisfactory results.

Table 6.17 A cross-hedge between corporate bonds and T-bond futures

 Date Cash Market Futures Market March 1 A portfolio manager learns will receive \$5 million to invest in 5%, ten-year AAA bonds in three months, with an expected yield of 7.5% and a price of \$826.30; the manager expects to buy 6,051 bonds. The portfolio manager buys \$5 million face value of T-bond futures (50 contracts) to mature on June 1 with a futures yield of 6.5% and a futures price, per contract of \$94,448. June 1 AAA yields have fallen to 7.08%, causing the price of the bonds to be \$852.72 and representing a loss, per bond, of \$26.42; since the plan was to buy 6,051 bonds, the total loss is 6,051 x \$26.42 = -\$159,867. The T-bond futures yield has fallen to 6.08%, so the futures price = spot price = \$99,081 per contract, for a pro0 of \$4,633 per contract; since 50 cont racts were traded, the total profit is \$231,650. Loss = -\$159,897 Gain = \$231,650 Net wealth change = +\$71,783

### Alternative Hedging Strategies [viewable here in Excel]

We have seen that simple approaches to hedging interest rate risk often give unsatisfactory results, due to mismatches of coupon and maturity characteristics. For the best possible hedges, we need strategies that take these coupon and maturity mismatches into consideration. This section chronicles some of the major strategies for hedging interest rate risk, starting from simple models and going on to more complex models.

### The Face Value Naïve Model (FVN)

According to the face value naive (FVN) model, the hedger should hedge \$1 face value of the cash instrument with \$1 face value of the futures contract. For example, a hedger wishing to hedge \$100,000 face value of the bonds would use one T-bond futures contract. The example we just considered used this strategy. The FVN strategy neglects two critically important factors:

(1) By focusing on face value, the FVN model completely neglects potential differences in market values between the cash and futures positions. Therefore, keeping face value amounts equal between the cash and futures market can result in poor hedges because the market values of the two positions differ.

(2) The FVN model neglects the coupon and maturity characteristics that affect duration for both the cash market good and the futures contract.

Because of these deficiencies, we will not consider the FVN model further.

### The Market Value Naive Model (MVN)

The market value naïve model (MVN) resembles the FVN model, except that it recommends hedging \$1 of market value in the cash good with \$1 of market value in the futures market. For example, if a \$100,000 face value bond has a market value of \$90,000 and the \$100,000 face value T-bond futures contract is priced at 80-00, the MVN model would recommend hedging the cash bonds with 1.125 = 90/80 futures contracts.

Because it considers the differences between market and face value, the MVN model escapes the first criticism lodged against the FVN model. However, the MVN model still makes no adjustment for the price sensitivity of the two goods. Therefore, we dismiss the MVN model with-out further consideration.

### The Conversion Factor Model (CF)

The conversion factor model (CF) applies only to futures contracts that use conversion factors to determine the invoice amount, such as T-bond and T-note futures. The intuition of this model is to adjust for differing price sensitivities by using the conversion factor as an index of the sensitivity.

In particular, the CF model recommends hedging \$1 of face value of a cash market security with \$1 of face value of the futures good times the conversion factor. As we have seen for T-bond and T-note futures, there are many deliverable instruments with different conversion factors. To apply the CF model, we must determine which instrument is cheapest to deliver and use the conversion factor for that instrument. Assuming we have identified the cheapest-to-deliver security, the hedge ratio (HR) is given by the following:

HR = -(cash market principal / futures market principal) x conversion factor

The negative sign indicates that one must take a futures market position opposite to the cash market position. For example, if the hedge is long in the cash market, the hedger should sell futures.

As an example, assume that a bond manager wishes to hedge a long position of \$500,000 face value of bonds with T-bonds futures. We assume the cheapest-to-deliver bond has a conversion factor of 1.2. In this situation, the manager should sell \$600,000 worth of T-bond futures (\$500,000 x 1.2) or six contracts, since the notional principal of each T-bond futures contract is \$100,000. The CF model attempts to secure the same amount of principal value of bonds on both the cash and futures sides of the hedge. This method is useful principally when on contemplates delivering a cash market bond against a futures contract.

### The Basis Point Model (BP)

The basis point model (BP) focuses on the price effect of a one basis point change in yields. For example, we have seen that a change of one basis point causes a \$25 change in the futures price of a T-bill or Eurodollar contract. Assume that today is April 2 and that a firm plans to issue \$50 million of 180-day commercial paper in six weeks. For a one basis point yield change, the price of 180-day commercial will change twice as much as the 90-day T-bill futures contract, assuming equal face value amounts. In other words, on \$1 million of 180-day commercial paper, a one basis point yield change causes a \$50 price change. In an important sense, the commercial paper will be twice as sensitive to a change in yields.

To reflect this greater sensitivity, we can use the BP model to compute the following hedge ratio:

HR = BPCS / BPCF

where BPCS is the dollar price change for a one basis point change in the spot instrument, and BPCF is the dollar price change for a one basis point change in the futures instrument. The ratio BPCS/BPCF, indicates the relative number of contracts to trade. In our commercial paper example, the cash basis price change (BPCS) is twice as great as the futures basis price change (BPCF), so the hedge ratio is -2.0.

To explore the effect of this weighting, consider the following BP model hedge of the commercial paper. Planning to issue commercial paper, the firm will lose if rates rise, because the firm will receive less cash for its commercial paper. As it needs to sell the commercial paper, it is now long commercial paper and the firm must hedge by selling futures. With a -2.0 hedge ratio and a \$50 million face value commitment in the cash market, the firm should sell 100 T-bill futures contracts.

Table 6.18 present the BP model transactions. After rates on both sides of the contract move by 45 basis points, we have the following result. In the cash market, the firm receives \$112,500 less than anticipated for its commercial paper. This loss, however, is exactly offset by the price movement on the \$100 million of T-bills underlying the futures position. The BP model helped identify the correct number of futures to trade for each unit in the cash market. By contrast, the FVN model would have suggested trading only 50 futures, which would have hedged only half of the loss.

Sometimes the yields may not change by the same amount as they did in Table 6.18. In that case, the hedger may wish to incorporate the relative volatility of the yields into the hedge ratio. For example, assume that the commercial paper rate is 25 percent more volatile than the T-bill futures rate. In other words, a 100 basis point rise in the T-bill futures rate normally might be accompanied by a 125 basis point rise in the commercial paper rate. To give the same total price change in the futures market as in the cash position, we would need to consider the difference in volatility in determining the hedge ratio. In that case, the hedge ratio becomes:

Table 6.18 Hedging results with the BP model for the issuance of commercial paper

 Date Cash Market Futures Market April 2 Firm anticipates issuing \$50 million in 180-day commercial paper in 45 days at a yield of 11% Firm sells 100 T-bill June futures contracts yielding 10% with an index value of \$90.00 May 15 Spot market and futures market have both risen 45 basis points; the spot rate is now 11.45% and the futures market yield is 10.45% Cash market effect Futures market effect Each basis point move causes a price change of \$50 per million-dollar face value; firm will receive \$112,500 less for commercial paper, due to the change in rates (45 basis points x -\$50 x 50 = -\$112,500). Each basis point increase give a futures market profit of \$25 per contract Net wealth change = 0 Futures profit = 45 basis points x \$25 x 100 contracts = +\$125,000

HR = (BPCS / BPCF) RV

where RV is the volatility of the cash market yield relative to the futures yield, normally found by regressing the yield of the cash market instrument in the futures market yield.

If we incorporate RV, assumed to be 1.25, into our commercial paper hedge, the transactions would appear as shown in the top portion of Table 6.19. Now the hedge ratio is as follows:

HR = (\$50 / \$25) x 1.25 = -2.5

Consequently, the hedger sells 125 T-bill futures contracts. Assume again that the T-bills yields rise by 45 basis points. Also, true to its greater relative volatility, the commercial paper yield moves 56 basis point, 1.25 times as much. Because more T-bill futures were sold, the T-bill futures profit still almost exactly offsets the commercial paper loss.

### The Regression Model (RGR)

One way of calculating a hedge ratio for interest rate futures is the regression technique that we considered in "Hedging-Futures" Risk minimization hedging. The hedge ratio found by regression minimizes the variance of the combined futures-cash position during the estimation period. This estimated ratio is applied to the hedging period.

For the regression model (RGR), the hedge ratio is as follows:

HR = -COVSF / σ2F

where COVSF is the covariance between cash and futures, and σ2F is the variance of futures. As noted previously, this hedge ratio is the negative of the regression coefficient found by regressing the change in the cash position on the change in the futures position. These changes can be measured as dollar price changes or a percentage price changes:

ΔSt ∝ + ΒΔFt + Εt

The RGR model finds the hedge ratio that gives the lowest sum-of-squares error for the data used in the estimation. Using the estimated hedge ratio for an actual hedge assumes that the relationship between the price changes on the futures and cash instrument does not change dramatically between the sample period and the actual hedging period.

This is a practical assumption. If the relationship is basically unchanged, then the estimated hedge ratio will perform well in the actual hedging situation. Fundamental shifts in the relationship between the price of the futures contract and the cash market good can lead to serious hedging errors. This danger is present in all hedging situation, but may be exacerbated in interest rate hedging. Without doubt, the RGR Model has proven usefulness in the market for the traditional futures contracts, and it has been adapted for use in the interest rate futures market by Louis Ederinton, Charles Franckle, Joanne Hill and Thomas Schneeweis.

However, there are some problems in applying the RGR Model to interest rate hedging. First, since it involves statistical estimation, the technique requires a dataset for both cash and futures prices. This data may sometimes be difficult to acquires, particularly for an attempt to hedge a new security. In such a case, no cash market data would even exist and a proxy would have to be used. Second, the RGR Model does not explicitly consider the differences in the sensitivity of different bond prices to changes in interest rates. As the example of Table 6.16 and 6.17 indicate, this can be a very important factor. The regression approach does include the different price sensitivities indirectly, however, since their differential sensitivities will be reflected in the estimation of the hedge ratio. Third, any cash bond will have a predictable price movement over time. The price of any instrument will equal its par value at maturity. The RGR Model does not consider this change in the cash bond's price explicitly, but the sample data should reflect this price movement tendency. Forth, the hedge ratio is found by minimizing the variability in the combined futures-cash position over the life of the hedge. Since the regression hedge ratio depends crucially on the planned hedge length, one might reasonably prefer a hedging technique that focuses on the wealth position of the hedge when the hedge ends. After all, the wealth change from the hedge depends on the gain or loss when the hedge is terminated, not on the variability of the cash-futures position over the life of the hedge. In spite of these difficulties, the RGR Model is a useful way to estimate hedge ratios, both for traditional commodities and, to a lesser extent, for interest rate hedging.

Table 6.19 Hedging results with the BP model adjusted for relative value yield variance for the issuance of commercial paper

 Date Cash Market Futures Market April 2 Firm anticipates issuing \$50 million in 180-day commercial paper in 45 days at a yield of 11%. Firm sells 125 T-bill June futures contracts yielding 10% with an index value of \$90.00. May 15 Spot market and futures market have both risen 56 basis points; the spot rate is now 11.45% and the futures rates have risen to 10.45% Cash market effect Futures market effect Each basis point move causes a price change of \$50 per million-dollar face value; firm will receive \$112,500 less for commercial paper, due to the change in rates (56 basis points x -\$50 x 50 = -\$140,500). Each basis point increase give a futures market profit of \$25 per contract. Net wealth change = \$625 Futures profit = 45 basis points x \$25 x 125 contracts = +\$140,625

### The Price Sensitivity Model (PS)

The price sensitivity model (PS) has been designed explicitly for interest rate hedging. The PS model assumes that the goal of hedging is to eliminate unexpected wealth changes at the hedging horizon defined as follows:

Equation (6.9) dPi = dPF(N) = 0

where dPi is the unexpected change in the price of the cash market instrument, dPF is the unexpected change in the price of the futures instrument, and N is the number of futures to hedge a single unit of the cash market asset. Equation 6.9 expresses the goal that the unexpected change in the value of the spot instrument denoted by ¡, and that in the futures position, denoted by F, should together equal zero. If this is achieved, the wealth change, or hedging error, is zero. Instead of focusing on the variance over the period of the hedge, the PS Model uses a hedge ratio to achieve a zero net wealth change at the end of the hedge.

The problem for the hedger is to choose the correct number of contracts, denoted by N in Equation 6.9, to achieve a zero hedging error. Modified duration, MD, is defined as Macaulay's duration divided by 1 + r. Thus, for a debt instrument x, with Macaulay's duration Dx, and yield rx:

Equation (6.10) MD = Dx / 1 + rx

The correct number of contracts to trade (N) per spot market bond is as follows:

N = PiMDi / FPFMDF x RYC

where FPF is the futures contract price; Pi is the price of asset i expected to prevail at the hedging horizon; MDi is the modified duration of asset i expected to prevail at the hedging horizon; MDF is the modified duration of the asset underlying futures contract F expected to prevail at the hedging horizon; and RYC is for a given change in the risk-free rate, the change in the cash market yield relative to the change in the futures yield, often assumed to be 1.0 in practice. In nontechnical terms, Equation 6.10 say's that the number of futures contracts to trade for each cash market instrument to be hedged is the number that should give a perfect hedge, assuming that yields on the cash and futures instrument change by the same amount. To explore the meaning and application of this technique, consider again the AAA bond hedges of Table 6.16 and 6.17. The large hedging errors resulted from the different price sensitivities of the futures instrument and the AAA bonds.

Table 6.20 present the data need to calculate the hedge ratios for hedging the AAA bonds with T-bill or T-bond futures. Here, we assume that the cash and futures market assets have the same volatilities, so that RYS = 1.0. For the T-bill hedge,

N = - \$826.30 x 7.207359 / \$94,448 x 10.946953 = -0.005760

With 6,051 bonds to hedge, the portfolio manager should sell 34.85 T-bond futures =6051*0.00576 = 34.85.

With either of these hedges, the same shift in yields on the AAA bonds and the futures instrument should give a perfect hedge. Table 6.21 present the performance of these two hedges for the same 42 basis point drop in rates used in Tables 6.16 and 6.17.

Table 6.20 Data for the price sensitivity hedge

 Cash instrument T-bill futures T-bond futures Pi \$826.30 FPi \$985,000 FPF \$94,448 MDi 7.207538 MDf 0.235849 MDF 10.946953 N -0.025636 N -0.00576 Number of contracts to trade. -155.62 Number of contracts to trade. -34.85

With the given hedges and the same drop in yields, the T-bill hedge gave a futures gain of 155.12 x 42 basis point x \$25 per basis point = \$162,876 to offset the loss on the AAA bonds of \$159,867. The futures gain on the T-bond hedge is 34.85 contracts x \$4,633 per contract = \$161,460. The next line of Table 6.21 shows the size of the hedging error for the T-bill and T-bond hedges. While the final line gives the percentage hedging error. The hedges worked quite will, both restricting the hedging error to less than 2 percent, and the hedging error on the T-bond hedge being just under 1 percent The hedging error is largely due to the large discrete change in interest rates. Also, for the T-bill hedge, part of the error is due to the difference between the bank discount yield on the T-bill and the bond yields on the corporate bond.

Table 6.21 A performance analysis of the price sensitivity futures hedge

 Cash Market T-bill hedge T-bond Hedge Gain/loss -\$159,867 \$162,876 \$161,460 Hedging error 0 \$3,009 \$1,593 Percentage hedging error 1.8822% 0.9965%

### Immunization with Interest Rate Futures [viewable here in Excel]

In bond investing, duration mismatches result in exposure to interest rate risk. For example, a financial institution, such as a bank or savings and loan association, might have an asset portfolio with a duration greater than its liability portfolio. A sudden rise in interest rates will cause the value of the assets to fall more than the value of the liability portfolio. As another type of risk, a bond portfolio might be managed to a certain future date, perhaps when a firm's pension liabilities become due.

If the duration of the bond portfolio exceeds the time until the horizon date, a swing in interest rates will cause the present value of the bond portfolio to change to interest rate risk. By matching the duration of the assets and liabilities, it is possible for the financial institution to immunize itself against interest rate risk, which we call the bank immunization case. For the bond portfolio being managed to a horizon date, a similar immunization can be achieved by setting the duration of a bond portfolio equal to the length of the planning period. We call this the planning period case.

Often, such immunization is very difficult to achieve. For example, banks cannot simply turn away depositors because they wish to lengthen the duration of their liabilities. With the development of interest rate futures markets, financial managers have a valuable new tool to use in futures, one for planning period case and one for the bank immunization case. Table 6.23 present data on three bonds we will use in the immunization examples, along with data for T-bill and T-bond futures contracts. The table reflects the assumption of a flat yield curve and instruments of the same risk level.

Table 6.23 Instruments for the immunization analysis

 Coupon Maturity Yield Price Duration Bond A 8 4 12 878.80 3.4605 Bond B 10 10 12 885.30 6.3092 Bond C 4 15 12 449.41 9.2853 T-bond futures 6 20 12 548.61 9.0401 T-bill futures 0 1/4 12 970.00 0.2500

### The Planning Period Case

Consider a \$100 million bond portfolio of Bond C with a duration of 9.2853 years. Assume now that a manager wants to shorten the portfolio duration to six years to match a given planning period. The shortening could be accomplished be selling Bond C and buying Bond A until the following conditions are met:

WADA + WCDC = six years WA + WC = 1 year

where W¡ is the percentage of the portfolio funds committed to asset ¡. This means that the manager must put 56.39 percent of the \$100 million in Bond A, the funds coming from the sale of Bond C. Call this portfolio 1.

Alternatively, the manager could adjust the portfolio's duration to match the six-year planning period by trading interest rate futures. In Portfolio 2, the manager will keep \$100 million in Bond C and trade futures to adjust the duration of the combined portfolio of Bond C and futures. If Bond C and T-bill futures comprise Portfolio 2, the T-bill futures position must satisfy the following condition:

PP = PCNC + FPT-billNT-bill

Where PP is the value of the portfolio, PC is the price of Bond C, FPT-bill is the T-bill futures price, NC is the number of C Bonds, and NT-bill is the number of T-bills.

The following equation expresses the change in the price of a bond as a function of duration and the yield on the asset:

Equation (6.11) dP = -D[d(1+r)/(1+r)]P

Applying Equation 6.11 to the portfolio value, Bond C and the T-bill futures we have the following immunization conditions:

-DP(d(1+r)/1+r)PP = -DC(d(1+r)/1+r)PCNC - DT-bill(d(1+r)/1+r)FPT-BillNT-bill

This can be simplified to the following:

DPPP = DCPCNC +DT-billFPT-billNT-bill

Because immunization requires mimicking Portfolio 1, which has a total value of \$100 million and a duration of six years, it must be the case that PP = \$100 million, DP = 6, DC = 9.2853, PC = \$449.41, NC = 222,514, DT-bill = 0.25 and FPT-bill = \$970.00.

Solving for NT-bill = -1,354,764 indicates that this many T-bill (assuming \$1,000 par value) must be sold short in the futures market. Because a T-bill futures contract has a \$1 million face value, this technique requires selling 1,355 contracts. The same technique used to create Portfolio 2 can be applied using a T-bond futures contract, given rise to Portfolio 3. Solving

DPPP = DCPCNC +DT-bondFPT-bondNT-bond

for NT-bond gives NT-bond = -66,243. Since T-bond futures contracts have a face value denomination of \$100,000, the trader must sell 662 T-bond futures contracts. For each of the three portfolios, Table 6.24 summarizes the relevant data.

To see how the immunized portfolio performs, assume that rates drop from 12 to 11 percent for all maturities. Assume also that all coupon receipts during the six-year planning period can be reinvested at 11 percent, compounded semiannually, until the end of the planning period. With the shift in interest rates the new prices are as follows: PA = \$904.98, PC = \$491.32, FPT-bill = \$972.50, and FPT-bond = \$598.85.

Table 6.24 Portfolio characteristics for the planning period

 Portfolio 2 Portfolio 3 Portfolio 1 (short T-bill (short T-bond (bonds only) futures) futures) Portfolio weights WA 56.39% 0 0 WC 43.61% 100% 100% WCash 0 0 0 Number of instruments NA \$64,387 \$0 \$0 NC \$97,038 \$222,514 \$222,514 NT-bill \$0 -\$1,354,764 \$0 NT-bond \$0 \$0 -\$66,243 Value of each instrument NAPA \$56,390,135 \$0 \$0 NCPC \$43,609,848 -\$100,000,017 -\$100,000,017 NT-billFPT-bill \$0 \$1,314,121,080 \$0 NT-bondFPT-bond \$0 \$0 \$36,341,572 Cash \$17 -\$17 -\$17 Portfolio Value NAPA + NCPC + Cash \$100,000,000 \$100,000,000 \$100,000,000

### Portfolio Value

Table 6.25 shows the effect of the interest rate short on portfolio values, terminal wealth at the horizon (year 6), and on the total wealth position of the portfolio holder. As Table 6.25 reveals, each portfolio responds similar to the shift in yields. The slight differences are due to either rounding errors or the fact that the duration price change formula holds exactly only for infinitesimal changes in yields. The largest difference (between terminal values for Portfolio 2 and 3) is only 0.056 percent, which reveals the effectiveness of the alternative strategies.

Table 6.25 The effect of a 1 percent drop in yields on realized portfolio returns

 Portfolio 1 Portfolio 2 Portfolio 3 Original portfolio value 100,000,000 100,000,000 100,000,000 New portfolio value 105,945,674 109,325,562 109,325,562 Gain/loss on futures - (3,386,910.00) (3,328,048.00) Total wealth change 5,945,674 5,938,652 5,997,514 Terminal value of all funds at t=6 201,424,708 201,411,358 201,523,267 Annualized holding period: Return over six years 1.12018 1.120168 1.12027

### The Bank Immunization Case

Assume that a bank holds a \$100 million liability portfolio in Bond B, the composition of which is fixed. The bank wishes to hold an asset portfolio on Bonds A and C that will protect the wealth position of the bank form any change as a result of a change in yields.

Five different portfolio combinations illustrate different means to a achieve the desired results:

Portfolio 1: hold Bond A and C (the traditional approach)
Portfolio 2: holds Bond C; sell T-bill futures
Portfolio 3: hold Bond A; buy T-bond futures
Portfolio 4: hold Bond A; buy T-bill futures
Portfolio 5: hold Bond C; buy T-bond futures

For each portfolio in Table 6.26, the full \$100 million is put in a bond portfolio (and is balanced out by cash). Portfolio 1 exemplifies the traditional approach of immunizing by holding only bonds. Portfolio 2 and 5 are composed of Bond C and a short futures position. By contrast, the low volatility Bond A is held in Portfolios 3 and 4. In conjunction with Bond A, the overall interest rate sensitivity is increased by buying interest rate futures.

Table 6.26 A liability portfolio and five alternative immunization portions

 Liability portfolio Portfolio 1 (bonds only) Portfolio 2 (short T-bill futures) Portfolio 3 (long T-bond futures) Portfolio 4 (long T-bill futures) Portfolio 5 (short T-bond futures) Portfolio weights WA 0.00% 51.0936 - 100 100 - WC 100.00% - - - - 100 WC - 48.9046 100 - - - WCash - - - - - - Number of instruments NA \$0 \$58,339 \$0 \$114,181 \$114,181 \$0 NB \$112,956 \$0 \$0 \$0 \$0 \$0 NC \$0 \$108,824 \$222,514 \$0 \$0 \$222,514 NT-bill \$0 \$0 -\$1,227,258 \$0 \$1,174,724 \$0 NT-bond \$0 \$0 \$0 \$57,400 \$0 -\$60,008 NAPA \$0 \$51,093,296 \$0 \$99,999,720 \$99,999,720 \$0 NBPB \$99,999,947 \$0 \$0 \$0 \$0 \$0 NCPC \$0 \$48,906,594 \$100,000,017 \$0 \$0 \$100,000,017 Cash \$53 \$110 -\$17 \$280 \$289 -\$17 NT-billFPT-bill \$0 \$0 -\$1,190,440,260 \$0 \$1,139,482,280 \$0 NT-bondFPT-bond \$0 \$0 \$0 \$31,512,158 \$0 \$32,920,989 Portfolio Value \$100,000,000 \$100,000,000 \$100,000,000 \$100,000,000 \$100,000,000 \$100,000,000

Now assume an instantaneous drop in rates from 12 to 11 percent for all maturities. Table 6.27 shows the effect on the portfolios. As the rows that report wealth change reveal, all five methods perform similarly. The small differences stem from rounding errors and the discrete change in interest rates.

One important concern in the implementation of immunization strategies is the transaction cost involved. In immunizing, commission charges, marketability, and liquidity of the instruments involved become increasingly important. These considerations highlight the practical usefulness on interest rate futures in bond portfolio management. Consider as an example the transaction cost associated with the different immunization portfolios for the planning period case. Starting from the initial position of \$100 million in Bond C, and wishing to shorten the duration to six years, Table 6.28 shows the trades necessary and the estimated cost involved. To implement the "bonds only" traditional approach of Portfolio 1,one must sell 125,476 bonds of type C and buy 64,387 bonds of type A. Assuming a commission charge of \$5 per bond, the total commission is \$949,315. By contrast, one could sell 1,355 T-bill futures contracts to immunize Portfolio 2, or sell 662 T-bond futures contracts for Portfolio 3, at a total costs of \$27,100 and \$13,240, respectively. (Additionally, one would have to deposit approximately \$2 million margin for the T-bill strategy or \$1 million for the T-bond strategy. But this margin deposit can be in the form of interest earning assets.) Table 6.28 presents these transaction costs calculations.

Table 6.27 The effect of a 1 percent drop in yields on realized on total wealth

 Liability Portfolio 1 Portfolio 2 Portfolio 3 Portfolio 4 Portfolio 5 Original portfolio value 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 New portfolio value 106,206,932 106,263,146 109,325,578 103,331,521 103,331,521 109,325,579 Gain/loss on futures - - (3,068,145) 2,885,786 2,936,810 (3,014,802) Total wealth change 6,206,932 6,263,146 6,257,433 6,217,307 6,268,331 6,310,777 Total wealth change - 56,214 50,484 10,655 61,679 103,828 % Wealth change - 0.00056 0.00050 0.00011 0.00062 0.00104

Table 6.28 Transaction costs for the planning period case

 Portfolio 1 Portfolio 2 Portfolio 3 Number of instruments traded Bond A 64,387 Bond C 125,476 T-bill futures 1,355 T-bond futures 662 One-way transaction cost Bond A @ \$5 321,935 Bond C @ \$5 627,380 T-bill futures @ \$20 27,100 T-bond futures @ \$20 13,240 Total cost of becoming immunized \$949,315 \$27,100 \$13,240

Clearly, there is a tremendous difference in transaction costs between trading the cash and futures instruments. In an extreme example of this type, the transaction cost for the "bonds only" case is prohibitive, amounting to almost 1 percent of the total portfolio value. It is practically impossible for another reason: the volume of bonds to be traded is enormous, exceeding any reasonable volume for bonds of even the largest issue. By contrast, today's robust futures volume makes it easy to implement the futures-based immunization strategies.

Until recently, immunization strategies for bond portfolios have focused on all bond portfolios. Here it has been shown that interest rate futures can be used in conjunction with bond portfolios to provide the same kind of immunization. The method advocated here works equally well for planning period case and the bank immunization case. Note that all of the examples assumed parallel shifting yield curves. If the change in interest rates brings about nonparallel shifts in the yield curve, then the "bonds only" and "bonds-with-futures" approaches will give different results. Which method turns out to be superior would depend upon the pattern of interest rate changes that actually occurred.

Sources: Damodaran, Hull, Basu, Kolb, Brueggeman, Fisher, Bodie, Merton, White, Case, Pratt, Agee.

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