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CHAPTER 8 Interest Rate Futures Refinements In this chapter, we extend the discussion of interest rates futures. This chapter is organized into the following sections: 1. The T-Bond Futures Contract 2. Seller’s Options for T-Bond Futures 3. Interest Rate Futures Market Efficiency 4. Hedging with T-Bond Futures Chapter 8 1 T-Bond Futures Contract In this section, the discussion of T-bond futures is extended by analyzing the cheapest-to-deliver bond. Recall that a number of candidate bonds can be delivered against a T-bond future contract. Recall further that short traders choose when to deliver and which combination of bonds to deliver. Some bonds are cheaper to obtain than others. In this section, we learn techniques to identify the cheapest-todeliver bond, including: 1. Cheapest-to-deliver bond with no intervening coupons. 2. Cheapest-to-deliver bond with intervening coupons. 3. Cheapest-to-deliver and the implied repo rate. Chapter 8 2 Cheapest-to-Deliver with No Intervening Coupons Assume today, September 14, 2004, a trader observes that the SEP 04 T-bond futures settlement price is 107-16 and thus decides to deliver immediately. That is, the trader selects today, September 14, as her Position Day. Therefore, she will have to deliver on September 16. The short is considering the following bonds with $100,000 face value each for delivery. The short wishes to determine if delivering one or the other bond will produce a larger profit for her. How much should the short receive? Which bond should the short deliver? Maturity Coupon Price November 15, 2028 5.25 November 15, 2021 8.00 93-15 127-13 SEP 04 CF 0.9052 1.2113 Days May-Nov 184 184 Days May-Set 122 122 To answer these two questions, we need to determine the invoice amount and then which bond is cheapest-todeliver. Chapter 8 3 Cheapest-to-Deliver with No Intervening Coupons Recall that the total price of a bond depends upon the quoted price plus the accrued interest (AI). Invoice Amount DSP ($ 100 , 000 )( CF ) AI Where: DSP = decimal settlement price the decimal equivalent of the quoted price CF = conversion factor the conversion factor as provided by the CBOT AI = accrued interest the Interest that has accrued since the last coupon payment on the bond Pi = cash market price Chapter 8 4 Cheapest-to-Deliver with No Intervening Coupons The accrued interest (AI) is computed as follows: Days Since Last Coupon 1 AI # Coupons per year Coupon Rate Days in Half Years FaceValue The days in half-year can be obtained from Table 8.1. Table 8.1 Days in HalfBYears Days in HalfBYear Interest Period January to July February to August March to September April to October May to November June to December July to January August to February September to March October to April November to May December to June 1 year (any 2 consecutive halfByears) Interest Paid on 1st or 15th Regular Year 181 181 184 183 184 183 184 184 181 182 181 182 365 Leap Year 182 182 184 183 184 183 184 184 182 183 182 183 366 Interest Paid on Last Day Regular Year 181 184 183 184 183 184 184 181 182 181 182 181 365 Leap Year 182 184 183 184 183 184 184 182 183 182 183 182 366 Source: Treasury Circular No. 300, 4th Rev. Chapter 8 5 Cheapest-to-Deliver with No Intervening Coupons Step 1: compute the cash price and invoice price. 5.25% Bond AI = (122/184) (0.5) (0.0525) ($100,000) = $1,740.49 Invoice Amount= 1.0750 ($100,000) (0.9052) + $1,740.49 Invoice Amount = $99,049.49 8.00% Bond AI = (122/184) (0.5) (0.08) ($100,000) = $2,652.17 Invoice Amount= 1.0750 ($100,000) (1.2113) + $2,652.17 Invoice Amount = $132,866.92 The 8% bond has an invoice amount 34% greater than the 5.25% bond. Chapter 8 6 Cheapest-to-Deliver with No Intervening Coupons Sept 2: compute the cheapest-to-deliver bond. The bond that is most profitable to deliver is the cheapest-to-deliver bond. The short’s profit is the difference between the invoice amount and the cash market price. For a particular bond I, the profit πi is: πi = Invoice Amount - (Pi + AIi) Recall that the invoice amount is: Invoice Amount DSP ($ 100 , 000 )( CF ) AI Substituting the formula for the invoice amount into the profit equation gives: πi = (DFPi) ($100,000) (CFi) + AIi - (Pi + AIi) And simplifying: πi = DFPi ($100,000) (CFi) - Pi Chapter 8 7 Cheapest-to-Deliver with No Intervening Coupons The cheapest-to-deliver is: 5.25% Bond π = 1.0750 ($100,000) (0.9052) - $93,468.75 = $3,840.25 8.00% Bond π = 1.0750 ($100,000) (1.2113) - $127,093.75 = $3,121.00 Thus, in this case the cheapest-to-deliver bond is the 5.25% bond. Chapter 8 8 Cheapest-to-Deliver with No Intervening Coupons General rules based on interest rates: 1. When interest rates are below 6%, there is an incentive to deliver short maturity/high coupon bonds. 2. When interest rates exceed 6%, there is an incentive to deliver long maturity/low coupon bonds. General rules based on duration: 1. A trader should deliver low duration bonds when interest rates are below 6%. 2. A trader should deliver high duration bonds when interest rates are above 6%. Chapter 8 9 Cheapest-to-Deliver with Intervening Coupons This section examines, cheapest-to-deliver bonds when a bond pays a coupon between the beginning of the cashand-carry holding period and the futures expiration. To find the cheapest-to-deliver bond before expiration, the cash-and-carry strategy is used. The bond with the greatest profit at delivery from following the cash-and-carry strategy will be the cheapest-to-deliver bond. For this analysis Assume that: 1. A trader buys a bond a today and carries it until delivery. 2. Interest rates and futures prices remain constant. 3. Consider the estimated invoice amount plus the estimate of the cash flows associated with carrying the bond to delivery. Chapter 8 10 Cheapest-to-Deliver with Intervening Coupons The estimated invoice amount depends on three factors: 1. Today's quoted futures price. 2. The conversion factor for the bond we plan to deliver. 3. The accrued interest on the bond at the expiration date. Acquiring and carrying a bond to delivery involves three cash flows as well: 1. The amount paid today to purchase the bond. 2. The finance cost associated with obtaining money today to buy a bond in the future. 3. The receipt and reinvesting of coupon payment. Figure 8.1 brings all these factors together. Chapter 8 11 Cheapest to Delivery and Bond Yield Insert Figure 8.1 here Chapter 8 12 Cheapest-to-Deliver with Intervening Coupons Estimated Invoice Amount = DFP0 $100,000 (CF) + AI2 Estimated Future Value of the Delivered Bond = (P0 + AI0)(1 + C0,2) - COUP1(1 + C1,2) For bond I, the expected profit from delivery is the estimated invoice amount minus the estimated value of what will be delivered: π = DFP0 ($100,000) (CF) + AI2 - {(P0 + AI0)(1 + C0,2) COUP1(1 + C1,2)} where: P0 AI0 C0,2 COUP1 C1,2 DFP0 CF AI2 = = = = = = = quoted price of the bond today, t = 0 accrued interest as of today, t = 0 interest factor for t = 0 to expiration at t = 2 coupon to be received before delivery at t = 1 interest factor from t = 1 to t = 2 decimal futures price today, t = 0 conversion factor for a particular bond and the specified futures expiration = accrued interest at t = 2 Chapter 8 13 Cheapest-to-Deliver with Intervening Coupons To illustrate these computation consider the following situation. Suppose that today is Sept 14, 2004, and you want to find the cheapest-to-deliver bond for the DEC 04 futures expiration. The bond has a $100,000 face value and a target delivery date of Dec 31, 2004. The futures contract is the DEC 04. The T-bond contract had a settlement price of 106-23 today. The coupon invested repo rates is 7%. Summary Today = Bond face value = Target delivery date = Futures contract = Coupon invested repo rate = Settlement price Sept 14 = Sept 14 $100,000 Dec 31 DEC 04 T-bond 7% 106-23 You are considering two bonds for delivery. The bonds are as follows: Chapter 8 14 Cheapest-to-Deliver with Intervening Coupons Maturity Coupon Price November 15, 2028 5.25 November 15, 2021 8.00 SEP 04 CF 93-15 0.9056 127-13 1.2094 Accrued Interest days 122 122 Accrued interest $1,740.49 $2,652.17 Step 1: estimate the value of AI. 5.25% Bond P0 + AI0 = $93,468.75 + $1,740.49 = $95,209.24 8% Bond P0 + AI0 = $ 127,093.75 + $2,652.17 = $129,745.92 Step 2: estimate the accrued interest that will accumulate from the next coupon date, Nov 15, 2004 if the planned delivery date is Dec 31, 2004 (46 days). 5.25% Bond AI2 = (46/181) (0.5) (0.0525) ($100,000) = $667.13 8% Bond AI2 = (46/181) (0.5) (0.08) ($100,000) = $1,016.57 Chapter 8 15 Cheapest-to-Deliver with Intervening Coupons Step 3: compute the estimated invoice amounts. 5.25% Bond 1.0671875 ($100,000) (0.9056) + $667.13 = $97,311.63 8% Bond 1.0671875 ($100,000) (1.2094) + $1,016.57 = $130,082.23 Step 4: compute financing rates. Period: Sept 15 until Dec 31 (108 days) C0,2 = 0.07 (108/360) = 0.0210 Period: Nov 15 until Dec 31 (46 days) C1,2 = 0.07 (46/360) = 0.008944 Table 8.2 summarizes these calculations. Chapter 8 16 Cheapest-to-Deliver with Intervening Coupons Table 8.2 Data for CheapestBtoBDeliver Bonds Bond P0 AI0 C0,2 5.25% 8.00% $93,468.75 $127,093.75 $1,740.49 $2,652.17 .0210 .0210 C1,2 DFP0 .008944 1.0671875 .008944 1.0671875 CF (DEC 04) AI2 0.9056 1.2094 $667.13 $1,016.57 Step 5: Compute expected profit for each bond. 5.25% Bond π = (1.06718750) ($100,000) (0.9056) + $667.13- [($ 93,468.75 + $1,740.49) (1.0210) - ($2,625) (1.008944)] π = $2,751.48 8% Bond π= (1.06718750) ($100,000) (1.2094) + $1,016.57[($127,093.75 + $2,652.17) (1.0210) - $4,000 (1.008944)] π =$1,647.43 The profit from the 5.25% bond is higher, so it is the cheapest- to-deliver. Chapter 8 17 Cheapest-to-Deliver Bond and The Implied Repo Rate We can analyze the same situation using the implied repo rate. The implied repo rate for a given period equals the net cash flow at delivery divided by the net cash flow when the carry starts. Repo Rate General Rules 1. A cash-and-carry arbitrage nets a zero profit if the actual borrowing cost equals the implied repo rate. 2. If the effective borrowing rate is less than the implied repo rate, one can earn an arbitrage profit by using cash-and-carry arbitrage (i.e., buy a cash bond and sell a futures). 3. If the effective borrowing rate exceeds the implied repo rate and if one can sell bonds short, then one can earn an arbitrage profit by using a reverse cash-and-carry arbitrage ( i.e., sell a bond short, buy the futures, and cover the short position at the expiration of the futures). Chapter 8 18 Cheapest-to-Deliver Bond and The Implied Repo Rate Implied Repo Rate = Net Cash Flow Over Horizon Net Cash Flow at Inception The numerator consists of cash inflows of the Invoice Amount, plus the future value of the coupons at the time of delivery, less the cost of acquiring the bond initially. The denominator consists of the cost of buying the bond. Thus, the Implied repo rates is: Implied Repo Rate DFP 0 ($100,000) (CF ) + AI 2 + funcCOUP ( P 0 + AI 0 1 (1 + C 1,2 ) - ( P 0 + AI 0 ) ) For the 5.25% bond, we have: Implied Repo Rate 1 . 0671875 $ 100 , 000 0 . 9056 $ 667 . 13 $ 2 , 625 1 . 008944 $ 93 , 468 . 75 $ 93 , 468 . 75 $ 1, 740 . 49 $ 1, 740 . 49 Implied Repo Rate = 0.0499 Annualized, the implied repo rate is: 0.0499(360/180) = 16.63% Chapter 8 19 Cheapest-to-Deliver Bond and The Implied Repo Rate For the 8% bond, we have: Implied Repo Rate (1.0671875 ) ($100,000) (0.9056) + $667.13 ($93,468.7 + $2,625 (1.008944) - ($93,468.7 5 + $1740.49) 5 + $1,740.49) Implied Repo Rate = 0.0337 Annualized, the implied repo rate is: 0.0337(360/180) = 13.23% The cheapest-to-deliver bon has the highest repo rate in a cash-and-carry arbitrage, so we should deliver the 5.25% bond. Chapter 8 20 Cheapest-to-Deliver Bond and Implied Repo Rate Table 8.3 shows how financing a cash-and-carry arbitrage at the implied repo rate yields a zero profit. Table 8.3 Transactions Showing Implied Repo Rates September 14, 2004 Borrow $129,745.92 for 108 days at implied repo rate of 11.23 percent. Buy $100,000 face value of 8.00 T-bonds maturing on Nov. 15, 2021, for a total price of $129,745.92 including accrued interest. Sell one DEC 04 T-bond futures contract at the current price of 106-23. November 15, 2004 Receive coupon payment of $4,000 and invest for 46 days at 7.00 percent. December 31, 2004 (Assuming futures is still at 106-23) Deliver the bond and receive invoice amount of $130,082.23 From the invested coupon receive $4,000 + $4,000 (0.07) (46/360) = $4,035.78 Repay debt: $129,745.92 + $129,745.92 (0.1123) (108/360) = $134,117.06 Net Profit = $130,082.23 + $4,035.78 B $134,117.06 = $.95 0 (given rounding error) Chapter 8 21 T-Bond Risk Arbitrage Arbitrage in the T-bond futures market is really risk arbitrage. Risks stem from three sources: 1. Intervening coupon payments that must face reinvestment. 2. The use of conversion factors. 3. The seller options. Chapter 8 22 T-Bond Risk Arbitrage Table 8.3 Transactions Showing Implied Repo Rates September 14, 2004 Borrow $129,745.92 for 108 days at implied repo rate of 11.23 percent. Buy $100,000 face value of 8.00 T-bonds maturing on Nov. 15, 2021, for a total price of $129,745.92 including accrued interest. Sell one DEC 04 T-bond futures contract at the current price of 106-23. November 15, 2004 Receive coupon payment of $4,000 and invest for 46 days at 7.00 percent. December 31, 2004 (Assuming futures is still at 106-23) Deliver the bond and receive invoice amount of $130,082.23 From the invested coupon receive $4,000 + $4,000 (0.07) (46/360) = $4,035.78 Repay debt: $129,745.92 + $129,745.92 (0.1123) (108/360) = $134,117.06 Net Profit = $130,082.23 + $4,035.78 B $134,117.06 = $.95 0 (given rounding error) A closer examination of Table 8.3 shows some potentially risky elements of the cash-and-carry arbitrage. Notice that: 1. The debt was financed at a constant rate throughout the 108-day carry period. 2. The trader was able to invest the coupon at the reinvestment rate of 7%. 3. The futures price did not change over the horizon. Chapter 8 23 T-Bond Risk Arbitrage Cash-and-Carry Strategy Changes in the futures price can affect the cash flow from the cash-and-carry strategy, as Table 8.4 illustrates. In this case, the futures price drops from 106-23 to 104-23 over the life of the contract. Table 8.4 Transactions Showing Implied Repo Rates September 14, 2004 Borrow $129,745.92 for 108 days at implied repo rate of 11.23 percent. Buy $100,000 face value of 8.00 T-bonds maturing on Nov. 15, 2021, for a total price of $129,745.92 including accrued interest. Sell one DEC 04 T-bond futures contract at the current price of 106-23. November 15, 2004 Receive coupon payment of $4,000 and invest for 46 days at 7.00 percent. December 31, 2004 (Assuming futures has fallen to 104-23) From September 14 to December 31, the futures price has fallen from 106-23 to 104-23, generating cash inflows of $2,000. Deliver the bond and receive invoice amount of $127,663.43 From the invested coupon receive $4,000 + $4,000 (0.07) (46/360) = $4,035.78 Repay debt: $129,745.92 + $129,745.92 (0.1123) (108/360) = $134,117.06 Net Profit = $2,000 + $127,663.43 + $4,035.78 B $134,117.06 = B$417.85 Thus, the cash-and-carry strategy now produces a negative profit. Chapter 8 24 T-Bond Risk Arbitrage Reserve Cash-and Carry Here we examine an attempt to earn a profit using a reserve cash-and-carry strategy. Recall that the trades used in a reverse cash-and-carry are as follows: Buy futures Sell bond short Invest proceeds until futures exp. Realize profit Repay short sale obligation Take delivery Chapter 8 25 T-Bond Risk Arbitrage Reserve Cash-and Carry Utilizing the same information from Table 8.3, we have: Table 8.5 Transactions Showing Implied Repo Rates September 14, 2004 Sell short $100,000 face value of 8.00 T-bonds maturing on Nov. 15, 2021, for a total price of $129,745.92 including accrued interest. Buy one DEC 04 T-bond futures contract at the current price of 106-23. Lend $129,745.92 for 108 days at implied repo rate of 11.23 percent. November 15, 2004 Borrow $4,000 for 46 days at 7 percent and make coupon payment of $4,000. December 31, 2004 (Assuming futures is still at 106-23) Collect investment: $129,745.92 + $129,745.92 (0.1123) (108/360) = $134,117.06 Accept delivery of the bond and pay invoice amount of $130,082.23 Pay debt from funds borrowed to make coupon payment: $4,000 + $4,000 (0.07) (46/360) = $4,035.78 Net Profit = $134,117.06 B $130,082.23 B $4,035.78 = -$.95 0 (given rounding error) As expected ,there is no arbitrage profit in this case. Chapter 8 26 T-Bond Futures Seller’s Options The structure of T-bond futures contract gives sellers timing and quality options. 1. Timing option The seller’s right to choose the time of delivery. 2. Quality option The seller’s right to select which bond to deliver. These two main seller's options become entangled in the actual T-bond futures contract. The timing and quality options are commonly present in what the futures markets refers to as: 1. The wildcard option. 2. The end-of-the-month option. Chapter 8 27 Wildcard Option The settlement price is determined at 2:00 PM. However, the short seller has until 8:00 PM to notify the exchange of his/her intent to deliver. Thus, the seller can observe what happens between 2:00 PM and 8:00 PM before making his/her decision. If interest rates jump between 2:00 PM and 8:00 PM, the short trader notifies the exchange his/her intent to deliver at the 2:00 PM price. If interest rates stay the same or go down, the short seller waits for the next day to notify the exchange of an intent to deliver. Chapter 8 28 The End-of-the-Month Option Recall that the last trading day for T-bond futures is the 8th of the month. The settlement price established on the final trading day is the settlement price used in all invoice calculations for all deliveries in the month. Thus, the seller can still make two choices: 1. The seller can choose the delivery date. 2. The seller can choose the bond to deliver. Assuming that interest rates are stable, then the seller may apply the following general rules: 1. If the coupon yield on the bond exceeds the financing rate to hold the bond, the seller should deliver on the last day. 2. If the financing rate exceeds the coupon yield, the seller should deliver immediately. Chapter 8 29 Value of The Seller’s Options Recall that under perfect market conditions, the Cost-ofCarry Model concludes that the futures price is equal to: F = S (1 + C) If the seller's options have value, then market equilibrium requires that the following equation holds: F + SO = S (1 + C) where: SO = value of seller's options This implies that: F = S ( 1 + C ) - SO This implies that the futures price observed in the market should be below the cost of carry by an amount equal to the seller’s options. Chapter 8 30 Interest Rate Futures Market Efficiency There are three commonly distinguished forms of the market efficiency hypothesis: – The weak form. – The semi-strong form. – The strong form. While many studies neglect the full magnitude of transaction charges, more recent studies find potential for arbitrage even after transaction costs. Pure Arbitrage For a pure arbitrage, the yield discrepancy must be large enough to cover all transaction costs faced by a market outsider. Quasi-Arbitrage Occurs when a trader with an initial portfolio can successfully engage in an arbitrage. For quasi-arbitrage, the trader faces less than full transaction costs. Chapter 8 31 Pure Arbitrage Table 8.9 is from a famous study on the efficiency of the Tbill futures market conducted by Elton, Gruber and Rentzler. They found large arbitrage profits exist, many with single contract profits in excess of $800. Table 8.9 Pure Arbitrage Results for TBBill Futures Immediate Execution Size of Filter $ 0 100 200 300 400 500 600 700 800 Number Expected of Trades Profit 2,304 2,093 1,902 1,738 1,595 1,469 1,332 1,190 1,063 $ 894 980 1,064 1,142 1,212 1,279 1,352 1,437 1,519 Delayed Execution Actual Profit $ 889 975 1,058 1,135 1,206 1,271 1,346 1,432 1,516 Standard Number Expected Error of Trades Profit $15 16 16 16 17 17 18 18 18 1,725 1,569 1,428 1,301 1,206 1,107 1,005 890 789 $ 893 977 1,059 1,137 1,199 1,267 1,339 1,429 1,517 Actual Profit $ 880 964 1,041 1,117 1,176 1,244 1,315 1,401 1,490 Standard Error $18 19 19 20 20 21 22 23 24 Source: E. Elton, M. Gruber, and J. Rentzler, AIntraBDay Tests of the Efficiency of the Treasury Bill Futures Market,@The Review of Economics and Statistics, February 1984, 66, pp. 129B137. Chapter 8 32 Pure Arbitrage in T-Bond Futures Kolb, Gay and Jordan conducted a study on T-bond futures. They investigated the possibility of a pure arbitrage for all T-bond contracts from December 1977 through June 1981. Figure 8.4 shows the profitability of deliverable bond for 15 contracts maturities. Insert Figure 8.4 here Chapter 8 33 Alternative Risk Management Strategy In this section, alternative risk management strategies using short-term interest rate futures are explored, including: 1. Changing the Maturity of an Investment Shortening the maturity of a T-bill investment Lengthening the maturity 2. Fixed and Floating Loan Rates 3. Strip and Stack Hedges 4. Tailing Hedge Chapter 8 34 Changing The Maturity of an Investment Shortening the Maturity Many investors find themselves holding a portfolio with undesirable maturity characteristics. Spot market transaction costs are relatively high, and many investors prefer to alter the maturities of investment by trading futures. Consider a firm that has invested in a T-bill with a $1,000,000 face value. Today, March 20, the T-bill has a maturity of 180 days. The firm’s manager learns that the company will need cash in 90 days. Assume that the shortterm yield is flat with all rates at 10% and a 360-day year. Chapter 8 35 Changing The Maturity of an Investment Shortening the Maturity Table 8.10 illustrate the process of shortening the maturity. Table 8.10 Transactions to Shorten Maturities Date March 20 Cash Market Futures Market Holds six-month T-bill with a face value of $10,000,000, worth $9,500,000. Wishes a threemonth maturity. June 20 Sell 10 JUN T-bill futures contracts at 90.00, reflecting the 10% discount yield. Deliver cash market T-bills against futures; receive $9,750,000. The price of a bill is given by: P = FV - [DY(FV)(DTM)]/360 P= $1,000,000- [(.10)($10,000,000(180)]/360 P= $1,000,000 - $50,000 = $9,500,000,000 By making the above trades, the firm has effectively shortened the maturity from 6 months to three months. Chapter 8 36 Changing The Maturity of an Investment Lengthening the Maturity On August 21, an investor holds a T-bill with a $100 million face value. The T-bill matures in 30 days (September 20). The investor plans to reinvest for another 3 months after the T-bill matures. The investor fears that interest rates might fall. The investor finds the current SEP T-Bill futures yield of 9.8% attractive and would like to lengthen the maturity of the T-bill investment. The transaction necessary to do so are presented in Table 8.11. Table 8.11 Transactions to Lengthen Maturities Date Cash Market Futures Market August 21 Holds 30Bday TBbill with a face value of $100,000,000. Wishes to extend the maturity for 90 days. Buy 102 SEP TBbill futures contracts, with a yield of 9.8%. September 20 30Bday TBbill matures and investor receives $100,000,000. Invest $499,000 in money market fund at 9.8% Accept delivery on 102 SEP futures, paying $99,501,000. December 19 TBbills received on SEP futures mature for $102,000,000. Receive proceeds of $511,533 from investment. These transactions locked in a 9.8% rate over the four months (Aug-Dec). Thereby, lengthening the maturity of the individuals investments. Chapter 8 37 Fixed and Floating Loan Rates This section examines: 1. Converting a Floating Rate to a Fixed Rate Loan – How a borrower holding a floating rate loan can effectively convert this loan into a fixed rate loan. 2. Converting a Fixed Rate to a Floating Rate Loan – How a lender who feels compelled to offer fixed rate loans can use the futures markets to make the investment perform like a floating rate loan. Chapter 8 38 Converting a Floating Rate to a Fixed Rate Loan Converting a floating rate loan to a fixed rate loan, also known as creating a synthetic fixed rate loan, occurs when you start with a floating rate loan and transact to fix the interest rate. Today is Sept 20th, assume that a construction company has planned a project which will take 6 months to complete. The cost of the project is $100,000,000. The firm’s bank offers the following conditions on a loan. Rates First 3 months= Last 3 months= LIBOR + 200 basis point DEC 20 LIBOR + 200 basis point The bank insists that the second 3-month rate be based on the LIBOR prevailing 3 months from today. This is a risky preposition for the construction company. Chapter 8 39 Converting a Floating Rate to a Fixed Rate Loan The construction company wishes to lock in a fixed rate loan for the entire period. The company has accumulated the following information: Sept 20 DEC Eurodollar futures LIBOR Loan 7% 7.3%. 9.0% 9.3% These rates give the following cash flows on the loan: Sept 20 Receive principal + $100,000,000 Dec 20 Pay interest - 2,250,000 Mar 20 Pay interest and principal - 102,325,000 The cash flows for September and December are certain but the cash flow for March is unknown. Using the above information, construct a synthetic fixed rate loan. Chapter 8 40 Converting a Floating Rate to a Fixed Rate Loan To convert to a variable rate loan to a fixed rate loan, the following transaction are completed. Table 8.12 Synthetic Fixed Rate Borrowing Date Cash Market Futures Market September 20 Borrow $100,000,000 at 9.00% for three months and commit to extend the loan for three additional months at a rate 200 basis points above the threeB month LIBOR rate prevailing at that time. Sell 100 DEC Eurodollar futures contracts at 92.70, reflecting the 7.3% yield. December 20 Pay interest of $2,250,000. LIBOR is now at 7.8%, so borrow $100,000,000 for three months at 9.8%. Offset 100 DEC Eurodollar futures at 92.20, reflecting the 7.8% yield. Produces profit of $125,000 = 50 basis points $25 per point 100 contracts. March 20 Pay interest of $2,450,000 and repay principal of $100,000,000. Total Interest Expense: $4,700,000 Futures Profit: $125,000 Net Interest Expense After Hedging: $4,575,000 By engaging in the above transactions, the company knows with certainty the interest expense that it will pay over the life of the loan. As such, it has created a fixed rate loan. Chapter 8 41 Converting a Fixed Rate to a Floating Rate Loan From the bank’s perspective, it can grant the fixed rate loan. However, doing so exposes the bank to risks. The bank expects to obtain money to make to the loan by borrowing at LIBOR 7% today and 7.3% for the next quarter, for an average of 7.15%. The bank makes a fixed rate loan at 9.15%. The bank sources of funds are as follows: BANK Sep 20 Dec 20 Mar 20 Borrow principal + $100,000,000 Make loan to - $100,000,000 Pay interest - $ 1,750,000 Receive principal & interest + $104.575,000 Pay principal & interest - $101,825,000 Chapter 8 42 Converting a Fixed Rate to a Floating Rate Loan To reduce its risks and lock in a profit, the bank trades as follows: Table 8.13 Synthetic Floating Rate Lending Date September 20 December 20 Cash Market Futures Market Borrow $100,000,000 at 7.00% for three months and lend it for six months at 9.15%. Pay interest of $1,750,000. LIBOR is now at 7.8%, so borrow $100,000,000 for three months at 7.8%. Sell 100 DEC Eurodollar futures contracts at 92.70, reflecting the 7.3% yield. Offset 100 DEC Eurodollar futures at 92.20, reflecting the 7.8% yield. Produces profit of $125,000 = 50 basis points $25 per point 100 contracts. March 20 Pay interest of $1,950,000 and repay principal of $100,000,000. Total Interest Expense: Futures Profit: $125,000 $3,700,000 Net Interest Expense After Hedging: $3,575,000 Thus, the bank has locked in a profit fo $1,000,000 ($4,575,000 - $3,575,000). The bank has also effectively crated a fixed rate loan. Chapter 8 43 Strip and Stack Hedges Using the same example. Now assume that the construction company needs a one year loan instead of 6month loan. The bank sets the rates to be LIBOR plus 200 basis points. The rate will be adjusted every 3 months to reflect any LIBOR rate changes. On September 15, the construction company observes the following rates: Three-month LIBOR DEC Eurodollar MAR Eurodollar JUN Eurodollar 7.00% 7.30 7.60 7.90 The company estimates that it can finance $100,000,000 at the following rates, for an average rate of 9.45%. I Quarter II Quarte r III Quarter IV Quarter 9.0% 9.3 9.6 9.9 Chapter 8 44 Stack Hedges A stack hedge occurs when futures contracts are concentrated or stacked in a single future expiration. The construction company enters into a stacked hedge by transacting as shown in Table 8.14. Table 8.14 Results of a Stack Hedge Date Cash Market Futures Market September 20 Borrow $100,000,000 at 9.00, for three months and commit to roll over the loan for three quarters at 200 basis points over the prevailing LIBOR rate. Sell 300 DEC Eurodollar futures contracts at 92.70, reflecting the 7.3% yield. December 20 Pay interest of $2,250,000. LIBOR is now at 7.8 percent, so borrow $100,000,000 for three months at 9.8 percent. Offset 300 DEC Eurodollar futures at 92.20, reflecting the 7.8% yield. Produces profit of $375,000 = 50 basis points $25 per point 300 contracts. March 20 Pay interest of $2,450,000 and borrow $100,000,000 for three months at 10.10 percent. June 20 Pay interest of $2,525,000 and borrow $100,000,000 for three months at 10.40 percent. September 20 Pay interest of $2,600,000 and principal of $100,000,000. Total Interest Expense: $9,825,000 Futures Profit: $375,000 Interest Expense Net of Hedging: $9,450,000 The hedge worked perfectly by locking in the cost of borrowing regardless of the future course of interest rates. Chapter 8 45 Stack Hedges Notice that in the above example all interest rates change by 50 basis points. Stack hedges may perform poorly if interest rates change in differing amounts. That is, the yield curve shifts. Figure 8.5 illustrates this situation. Insert Figure 8.5 here Chapter 8 46 Strip Hedge A strip hedge uses an equal number of contracts for each futures expiration over the hedging horizon. By doing so, the futures market hedge is aligned with the actual risk exposure. The transactions necessary to implement a strip hedge are demonstrated in Table 8.15. Table 8.15 Results of a Strip Hedge Date Cash Market Futures Market September 20 Borrow $100,000,000 at 9.00% for three months and commit to roll over the loan for three quarters at 200 basis points over the prevailing LIBOR rate. Sell 100 Eurodollar futures for each of: DEC at 92.70, MAR at 92.40, and JUN at 92.10. December 20 Pay interest of $2,250,000. LIBOR is now at 7.4%, so borrow $100,000,000 for three months at 9.4%. Offset 100 DEC Eurodollar futures at 92.60. Produces profit of $25,000 = 10 basis points $25 per point 100 contracts. March 20 Pay interest of $2,350,000 and borrow $100,000,000 for three months at 10.30%. Offset 100 MAR Eurodollar futures at 91.70. Produces profit of $175,000 = 70 basis points $25 per point 100 contracts. June 20 Pay interest of $2,575,000 and borrow $100,000,000 for three months at 10.60%. Offset 100 JUN Eurodollar futures at 91.40. Produces profit of $175,000 = 70 basis points $25 per point 100 contracts. September 20 Pay interest of $2,650,000 and principal of $100,000,000. Total Interest Expense: $9,825,000 Futures Profit: $375,000 Interest Expense Net of Hedging: $9,450,000 The performance of a strip hedge is superior to the stack hedge because the interest rates adjust every quarter. Chapter 8 47 Advantages of Stacked and Striped Hedge Advantages of Stack Hedges 1. Works better when the cash position has a single horizon. 2. Requires trading a single contract. Advantage of Strip Hedges 1. Can provide a more aligned hedge and better results with a multiple-maturity cash position. Chapter 8 48 Tailing The Hedge In a tailing hedge the trader slightly adjusts the hedge to compensate for the interest that can be earned from daily resettlement profits or paid on daily resettlement losses. Thus, the tail of the hedge is the slight reduction in the hedge position to offset the effect of daily resettlement interest. Tail Factor The tail factor is the present value of $1 at the hedging horizon discounted to the present (plus one day) at the investment rate for the resettlement cash flows. Tailed Hedge = Untailed Hedge Tailing Factor . Chapter 8 49 Hedging with T-Bond Futures The effectiveness of a hedge depends on the gain or loss on both the spot and futures sides of the transaction. The change in the price of any bond depends on the shifts in the levels of: – Interest rates – Changes in the shape of the yield curve – The maturity of the bond – Bond coupon rate Table 8.16 and 8.17 illustrate to effect of maturity and coupon rates on hedging performance. Chapter 8 50 Hedging with T-Bond Futures A manager learns on March 1 that he will receive $5 million on June 1 to invest in AAA corporate bonds with a 5% coupon rate and 10 years to maturity. The yield curve is flat and will remain so. The current yield on AAA bonds as well as forward rates are 7.5%. So the manager expects to acquire the bonds at 7.5%. However, fearing a drop in rates, he decides to hedge in the futures market to lock-in the forward rate of 7.5%. The manager considers hedging with T-bills or T-bonds. The AAA bonds have a 5% coupon rate and a 10-year maturity, which do not match the characteristics of either the T-bill or T-bond futures contracts. The deliverable Tbills have a zero coupon and a maturity of only 90 days, and the T-bonds have a maturity of at least 15 years and an assortment of semi-annual coupons. Assume that the cheapest-to-deliver T-bond will have a 20-year maturity at the target date of June 1, and a 6% coupon. The manager will hedge the AAA position with T-bill or Tbond futures with yields of 6 and 6.5%, respectively. The manager plans to invest in 6,051 bonds each with a price of $826.30. Chapter 8 51 Hedging with T-Bond Futures Table 8.16 illustrates the transactions and results of hedging with T-bill futures. Table 8.16 A CrossBHedge Between Corporate Bonds and TBBill Futures Date Cash Market Futures Market March 1 A portfolio manager learns he will receive $5 million to invest in 5%, 10-year AAA bonds in 3 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 (5 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. This represents a loss, per bond, of $26.42. Since the plan was to buy 6,051 bonds, the total loss is (6,051 $26.42) = B$159,867. The T-bill futures yield has fallen to 5.58%, so the futures price = spot price = $986,050 per contract, for a profit of $1,050 per contract. Since 5 contracts were traded, the total profit is $5,250. Loss = B$159,867 Gain = $5,250 Net wealth change = B$154,617 Notice that this loss occurs despite the fact that rates changed by the same amount on both investments. Chapter 8 52 Hedging with T-Bond Futures Table 8.17 illustrates the results of hedging with Tbond futures. Table 8.17 A Cross Hedge Between Corporate Bonds and TBBond Futures Date Cash Market Futures Market March 1 A portfolio manager learns he will receive $5 million to invest in 5%, 10-year AAA bonds in 3 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. This represents a loss, per bond, of $26.42. Since the plan was to buy 6,051 bonds, the total loss is (6,051 $26.42) = B$159,867. The T-bond futures yield has fallen to 6.08%, so the futures price = spot price = $99,081 per contract, for a profit of $4,633 per contract. Since 50 contracts were traded, the total profit is $231,650. Loss = B$159,867 Gain = $231,650 Net wealth change = +$71,783 Again the hedge did not produce the desired results of isolating the portfolio. Chapter 8 53 Hedging with T-Bond Futures Simple approaches to hedging interest rate risk often give unsatisfactory results due to mismatches of coupon and maturity characteristics, as demonstrated in the previous examples. This section examines some of the major alternative strategies for hedging interest rate risk: – Face Value Naive (FVN) Model – Market Value Naive (MVN) Model – Conversion Factor (CF) Model – Basis Point (BP) Model – Regression (RGR) Model – Price Sensitivity (PS) Model Chapter 8 54 Face Value Naive (FVN) Model According to FVN Model, the hedger should hedge $1 of face value of the cash instrument with $1 face value of the futures contract. Disadvantages Neglects potential differences in market values between the cash and futures positions. Neglects the coupon and maturity characteristics that affect duration for both the cash market good and the futures contract. Chapter 8 55 Market Value Naïve (MVN) Model The MVN Model recommends hedging $1 of market value in the cash good with $1 of market value in the futures market. Disadvantages Neglects to make adjustments for price sensitivity. Advantages Consider potential differences in market values between cash and futures positions. Chapter 8 56 Conversion Factor (CF) Model The CF Model applies only to futures contracts that use conversion factors to determine the invoice amount, such as T-bond and T-note futures. The intuition behind this model is to adjust for differing price sensitivities by using the conversion factor as an index of the sensitivity. 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. Cash Market Principal HR = _ Futures Market Principal (Conversio Chapter 8 n Factor) 57 Basic Point (BP) Model The BP Model focuses on the price effect of a one basis point change in yields on different financial instruments. To correct for the differences in sensitivity, the BP Model can be used to compute the following hedge ratio: HR = Where BPCS BPCF BPC S BPC F = dollar price change for a 1 basis point change in the spot instrument. = dollar price change for a 1 basis point change in the futures instrument. Chapter 8 58 Basic Point (BP) Model Today, April 2, a firm plans to issue $50 million of 180-day commercial paper in 6 weeks. For a one basis point yield change, the price of 180-day commercial paper will change twice as much as the 90-day T-bill futures contract, assuming equal face value amounts. The cash basis price change (BPCS) is twice as great as the futures basis price change (BPCF), so the hedge ratio is -2.0. With a -2.0 hedge ratio and a $50 million face value commitment in the cash market, the firm should sell 100 Tbill futures contracts. Table 8.18 illustrates the hedging results. Table 8.18 Hedging Results with the BP Model for the Commercial Paper Issuance April 2 Cash Market Futures Market 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 rates 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 the commercial paper, due to the change in rates. (45 basis points B$50 50 = B$112,500) Each basis point increase gives a futures market profit of $25 per contract. Futures Profit = 45 basis points +$25 100 contracts = +$112,500 Net wealth change = 0 Chapter 8 59 Basic Point (BP) Model Sometimes rates do not change by the same amounts. In our previous example, suppose that the commercial paper rate is 25% more volatile than the T-bill futures rate. To consider differences in volatility in determining the hedge ratio. The hedge ratio is recomputed as: BPC HR = - BPC where: RV = RV F S volatility of cash market yield relative to futures yield. Normally found by regressing the yield of the cash market instrument on the futures market yield. Assume a RV equal to 1.25. Now the hedge ratio is: $50 HR = - 1.25 = - 2.5 $25 Table 8.19 shows these transactions. Chapter 8 60 Basic Point (BP) Model Table 8.19 Hedging Results with the BP Model Adjusted for Relative Yield Variances for the Commercial Paper Issuance April 2 Cash Market Futures Market Firm anticipates issuing $50 million in 180Bday 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 rates have risen 56 basis points to 11.56% and futures rates have risen 45 basis points 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 $140,000 less for the commercial paper, due to the change in rates. 56 basis points B$50 50 = B$140,000 Each basis point increase gives a futures market profit of $25 per contract. Futures Profit = 45 basis points +$25 125 contracts = +$140,625 Net wealth change = $625 Because more T-bill futures were sold, the futures profit still almost exactly offsets the commercial paper loss. Chapter 8 61 Regression (RGR) Model 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 RGR Model the hedge ratio is: HR = - COV S , F F 2 where: COVS,F = covariance between cash and futures σF2 = variance of futures Recall from Chapter 4, the 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 as percentage price changes. S t = + Ft + t Chapter 8 62 Price Sensitivity Model The PS Model assumes that the goal of hedging is to eliminate unexpected wealth changes at the hedging horizon, as defined in the following equation: dPi + dPF (N) = 0 where: dPi = unexpected change in the price of the cash market instrument dPF = unexpected change in the price of the futures instrument N number of futures to hedge a single unit of the cash market asset = Chapter 8 63 Price Sensitivity Model The correct number of contracts (N) is calculated using the Modified Duration MD: MD x = N=- Dx 1+ rx P i MD i FP F MD where: FPF Pi MDi DF RYC RYC F = the futures contract price. = the price of asset I expected to prevail at the hedging horizon. = the modified duration of asset I expected to prevail at the hedging horizon. = the modified duration of the asset underlying futures contract F expected to prevail at the hedging horizon. = 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. Chapter 8 64 Price Sensitivity Model Suppose that you have accumulated the data in Table 8.20. Assume that the cash and futures markets have the same volatility. Table 8.20 Data for the Price Sensitivity Hedge Cash Instrument Pi MDi TBBill Futures TBBond Futures $826.30 FPi $985,000 FPF $94,448 7.207358 MDF 0.235849 MDF 10.946953 N -0.025636 N -0.005760 Number of Contracts to Trade -155.12 Number of Contracts to Trade -34.85 For the T-bill hedge, the number of contracts to trade is given by: N=- ($826.30) (7.207358) = - 0.025636 ($985,000) (0.235849) For the T-bond hedge the number of contracts to be traded is: N=- ($826.30) (7.207358) = - 0.005760 ($94,448) (10.946953 ) Chapter 8 65 Price Sensitivity Model The performance of the T-bond and T-bill hedges are presented in Table 8.21. Table 8.21 Performance Analysis of Price Sensitivity Futures Hedge Cash Market TBBill Hedge TBBond Hedge Gain/Loss Hedging Error B$159,867 B Percentage Hedging Error +$162,876 +$161,460 $3,009 $1,593 1.8822% 0.9965% The T-bond hedge is slightly more effective as it produces a lower hedging error. Chapter 8 66 Summary of Alternative Hedging Strategies Table 8.22 Summary of Alternative Hedging Strategies Hedging Model Face Value Naive (FVN) Market Value Naive (MVN) Conversion Factor (CF) Basis Point (BP) Regression (RGR) Price Sensitivity (PS) Basic Intuition Hedge $1 of cash instrument face value with $1 of futures instrument face value. Hedge $1 of cash instrument market value with $1 of futures instrument market value. Find ratio of cash market principal to futures market principal. Multiply this ratio by the conversion factor for the cheapestBtodeliver instrument. For a 1 basis point yield change, find the ratio of the cash market price change to the futures market price change. (Sometimes weighted by the relative volatility of interest rates on the cash market instrument compared to the futures instrument interest rate.) For a given cash market position, use regression analysis to find the futures position that minimizes the variance of the combined cash/futures position. Using duration analysis, find the futures market position designed to give a zero wealth change at the hedging horizon. (Sometimes weighted by the relative volatility of interest rates on the cash market instrument compared to the futures instrument interest rate.) Chapter 8 67 Immunization In bond investing, maturity mismatches result in exposure to interest rate risk. Consider the case of a bank. when the asset duration is higher than the liability duration, a sudden rise in interest rates will cause the value of the portfolio to decline. When the asset duration is less than the liability duration a sudden rise in interest rates will cause the value of the portfolio to rise. By matching the duration of asset and liabilities, it is possible for the bank to immunize itself from changes in interest rates. We consider two examples of immunization: 1. Planning Period Case 2. Bank Immunization case Chapter 8 68 Immunization with Interest Rate Futures Planning Period Case A portfolio manager has collected the following information: Table 8.23 Instruments for the Immunization Analysis Coupon Maturity Yield Price Duration Bond A: 8% 4 yrs. 12% 875.80 3.4605 Bond B: 10% 10 yrs. 12% 885.30 6.3092 Bond C: 4% 15 yrs. 12% 449.41 9.2853 6% 20 yrs. 12% 548.61 9.0401 TBBond Futures* B 12% 970.00 0.2500 TBBill Futures* 3 yr. *For comparability, face values of $1,000 are assumed for these instruments. The portfolio manager has a $100 million bond portfolio of bond C with a duration of 9.2853 years and is considering two alternatives. The manager has a 6-year planning period. The manager wants to shorten the portfolio duration to six years to match the planning period, and is considering two alternatives to do so. Chapter 8 69 Immunization with Interest Rate Futures Planning Period Case Alternative 1 The shortening could be accomplished by selling Bond C and buying Bond A until the following conditions are met: W A D A + W C D C = 6 years Subject to: WA + WC =1 Where: WI = percent of portfolio funds committed to asset I. Chapter 8 70 Immunization with Interest Rate Futures Planning Period Case Alternative 2 The manager could also adjust the portfolio's duration to match the six-year planning period by trading interest rate futures and keeping bond C. If bond C and a T-bill futures comprise the portfolio, the Tbill futures position must satisfy the condition: PP = PC NC + FPT-bill NT-bill where: Pp Pc FPT-bill Nc NT-bill = = = = = value of the portfolio price of bond C t-bill futures price number of C bonds number of T-bills Expressing the change in the price of a bond as a function of duration and the yield on the asset: dP = -D{d(1 + r)/(1 + r)}P Chapter 8 71 Immunization with Interest Rate Futures Planning Period Case Applying the equation to the portfolio value, bond C, and the T-bill futures, the following immunization condition is obtained: d(1 + r) d(1 + r) d(1 + r) P c N c + - D T- bill -Dp P P = - D c (1 + r) 1+ r (1 + r) FP T- bill N T- bill This can be simplified to: DP PP = DC PC NC + DT-bill FPT-bill NT-bill Chapter 8 72 Immunization with Interest Rate Futures Planning Period Case Because immunization requires mimicking alternative 1, which has a total value of $100,000,000 and a duration of six years, it must be that: Pp = Dp = DC = PC = NC= DT-bill = FPT-bill = $100,000,000 6 9.2853 $449.41 222,514 0.25 $970.00 Solving by: DP PP = DC PC NC + DT-bill FPT-bill NT-bill Or alternatively for a T-bond: DP PP = DC PC NC + DT-bond FPT-bond NT-bond Table 8.24 shows the relevant data for each of the three scenarios. Chapter 8 73 Immunization with Interest Rate Futures Planning Period Case Table 8.24 Portfolio Characteristics for the Planning Period Case Portfolio Weights Number of Instruments Value of Each Instrument Portfolio Value WA WC WCash NA NC NT-bill NT-bond NAPA NCPC NT-billFPT-bill NT-bondFPT-bond Cash NAPA + NCPC + Cash Portfolio 1 (Bonds Only) Portfolio 2 (Short T-Bill Fut.) Portfolio 3 (Short T-Bond Fut.) 56.39% 43.61% ~0 64,387 97,038 B B B B 100% ~0 0 222,514 (1,354,764) B B 100% ~0 B 222,514 B (66,243) B 56,390,135 43,609,848 B B 100,000,017 1,314,121,080 B 17 (17) 36,341,572 (17) 100,000,000 100,000,000 100,000,000 100,000,017 B Source: Adapted from R. Kolb and G. Gay, AImmunizing Bond Portfolios with Interest Rate Futures,@ Financial Management, Summer 1982, pp. 81-89. Chapter 8 74 Immunization with Interest Rate Futures Planning Period Case 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: PA = $904.98 PC = $491.32 FPT-bill = $972.50 FPT-bond = $598.85 Table 8.25 shows the effect of the interest rate shift on portfolio values, terminal wealth at the horizon (year 6), and on the total wealth position of the portfolio holder. Chapter 8 75 Immunization with Interest Rate Futures Planning Period Case Table 8.25 Effect of a 1% Drop in Yields on Realized Portfolio Returns Portfolio 1 Original Portfolio Value New Portfolio Value Gain/Loss on Futures Total Wealth Change Terminal Value of all Funds at t = 6 Annualized Holding Period Return over 6 Years 100,000,000 105,945,674 B0B 5,945,674 $201,424,708 1.120180 Portfolio 2 Portfolio 3 100,000,000 100,000,000 109,325,562 109,325,562 (3,386,910) (3,328,048) 5,938,652 5,997,514 $201,411,358 $201,523,27 1.120168 1.120266 Source: Adapted from R. Kolb and G. Gay, AImmunizing Bond Portfolios with Interest Rate Futures,@ Financial Management, Summer 1982, pp. 81-89. Terminal values and holding period returns assume semi-annual compounding at 11 percent. Notice that each portfolio responds similarly. Chapter 8 76 Transaction Costs for Planning Period Case While each of the portfolios are equally effective in immunizing, the cost of obtaining the immunization varies as demonstrated in Table 8.28. Table 8.28 Transaction Costs for the Planning Period Case Portfolio 1 Portfolio 2 Portfolio 3 64,387 B B 125,476 B B B B B 1,355 B Bond A @ $5 321,935 B B Bond C @ $5 627,380 B B B B B 27,100 B 13,240 $949,315 $27,100 $13,240 Number of Instruments Traded Bond A Bond C TBBill Futures Contracts TBBond Futures Contracts 662 One Way Transaction Cost TBBill Futures $20 TBBond Futures @ $20 Total Cost of Becoming Immunized Source: Adapted from R. Kolb and G. Gay, AImmunizing Bond Portfolios with Interest Rate Futures,@Finan- cial Management, Summer 1982, pp. 81-89. Notice that the cost of becoming immunized varies from $949,315 to $13,240 depending upon the strategy selected. Chapter 8 77 Immunization with Interest Rate Futures Bank Immunization Case Assume that a bank holds a $100,000,000 liability portfolio in Bond B, the composition of which is fixed. The bank wishes to hold an asset portfolio of Bonds A and C that will protect the wealth position of the bank from any change as a result of a change in yields. Five different portfolio combinations illustrate different means to achieve the desired result: Portfolio 1: Hold Bond A and Bond C (the traditional approach) Portfolio 2: Hold 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; Sell T-bond futures The portfolios are presented in Table 8.26. For each portfolio, the full $100,0000,000 is put into a bond portfolio and is balanced out by cash. Chapter 8 78 Immunization with Interest Rate Futures Bank Immunization Case Table 8.26 Liability Portfolio and Five Alternative Immunizing Portfolios Portfolio 2 Portfolio 3 Portfolio 4 (Short (Long (Long Liability Portfolio 1 TBBill TBBond TBBill Portfolio (Bonds Futures) Futures) Futures) Only) Portfolio Weights WA WB WC WCash Number of NA InstruNB ments NC NTBbill NTBbond NAPA NBPB NCPC Cash NTBbillPTBbill NTBbond PTBbond Portfolio Value Portfolio 5 (Short TBBond Futures) 0 100% 0 ~0 51.0936% 0 48.9064% ~0 0 0 100% ~0 100% 0 0 ~0 100% 0 0 ~0 0 0 100% ~0 0 112,956 58,339 0 0 0 114,181 0 114,181 0 0 0 0 0 0 108,824 0 0 222,514 (1,227,258) 0 0 0 57,440 0 1,174,724 0 222,514 0 (60,008) 0 99,999,947 0 53 0 0 51,093,296 0 48,906,594 110 0 0 0 0 100,000,017 (17) 99,999,720 0 0 280 0 31,512,158 99,999,720 0 0 280 100,000,000 100,000,000 (1,190,440,260) 0 100,000,000 0 0 100,000,017 (17) 1,139,482,280 0 0 (32,920,989) 100,000,000 100,000,000 100,000,000 Source: Adapted from R. Kolb and G. Gay, AImmunizing Bond Portfolios with Interest Rate Futures,@ Financial Management, Summer 1982, pp. 81-89. Chapter 8 79 Immunization with Interest Rate Futures Bank Immunization Case Now consider a drop in rates from 12% to 11% for all maturities. The effect on the portfolio is presented in Table 8.27. Table 8.27 Effect of a 1% Drop in Yields on Total Wealth Liability Original Port. Value New Port. Value Profit on Futures Total Wealth Change (Port. and Futures) Total Wealth Change (AssetBLiability Port.) % Wealth Change Portfolio 1 Portfolio 2 Portfolio 3 Portfolio 4 Portfolio 5 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 106,206,932 106,263,146 109,325,578 103,331,521 103,331,521 109,325,579 0 B (3,068,145) 2,885,786 2,936,810 (3,014,802) 6,206,932 6,263,146 6,257,416 6,217,587 6,268,611 6,310,760 B 56,214 50,484 10,655 61,679 103,828 B .00056 .00050 .00011 .00062 .00104 Source: Adapted from R. Kolb and G. Gay, AImmunizing Bond Portfolios with Interest Rate Futures, Financial Management, Summer 1982, pp. 81-89. Notice that all 5 methods perform similarly. Chapter 8 80