Interest Rate Markets Chapter 5

1. Interest Rate MarketsChapter 5

2. Types of Interest Rates • Treasury rates • LIBOR rates • Repurchase rates

5. The market for Repurchase Agreements An integral part of trading T-bills and T-bill futures is the market for repurchase agreements, which are used in much of the arbitrage trading in T-bills. In a repurchase agreement -- also called an RP or repo -- one party sells a security (in this case, T-bills) to another party at one price and commits to repurchase the security at another price at a future date. The buyer of the T-bills in a repo is said to enter into a reverse repurchase agreement., or reverse repo. The buyer’s transactions are just the opposite of the seller’s. The figure below demonstrates the transactions in a repo.

6. Transactions in a Repurchase Agreement Date 0 - Open the Repo: T- Bill Party A Party B PO Date t - Close the Repo T-Bill Party A Party B Pt= P0(1+r0,t )

7. Example: T-bill FV = $1M. P0 = $980,000. The repo rate = 6%. The repo time: t = 4 days. P1= P0 [(repo rate)(n/360) + 1] = 980,000[(.06)(4/360) + 1] = 980,653.33

8. A repurchase agreement effectively allows the seller to borrow from the buyer using the security as collateral. The seller receives funds today that must be paid back in the future and relinquishes the security for the duration of the agreement. The interest on the borrowing is the difference between the initial sale price and the subsequent price for repurchasing the security. The borrowing rate in a repurchase agreement is called the repo rate. The buyer of a reverse repurchase agreement receives a lending rate called the reverse repo rate. The repo market is a competitive dealer market with quotations available for both borrowing and lending. As with all borrowing and lending rates, there is a spread between repo and reverse repo rates.

9. The amount one can borrow with a repo is less than the market value of the security by a margin called a haircut. The size of the haircut depends on the maturity and liquidity of the security. For repos on T-bills, the haircut is very small, often only one-eighth of a point. It can be as high as 5% for repurchase agreements on longer-term securities such as Treasury bonds and other government agency issues.Most repos are held only overnight, so those who wish to borrow for longer periods must roll their positions over every day. However, there are some longer-term repurchase agreements, called term repos, that come in standardized maturities of one, two, and three weeks and one, two, three, and six months.Some other customized agreements also are traded.

10. Zero Rates A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T.

11. Example (Table 5.1, page 95)

12. Bond Pricing • To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate • In our example, the theoretical price of a two-year bond (FV = $100) providing a 6% coupon semiannually is:

13. Bond Yield • The bond yield is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond • Suppose that the market price of the bond in our example equals its theoretical price of 98.39 • The bond yield is given by solving to get y=0.0676 or 6.76%.

14. Par Yield • The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value. • In our example we solve

15. Par Yield continued In general if mis the number of coupon payments per year, d is the present value of $1 received at maturity and A is the present value of an annuity of $1 on each coupon date

16. Sample Data for Determining the Zero Curve(Table 5.2, page 97) Bond Time to Annual Bond Principal Maturity Coupon Price (dollars) (years) (dollars) (dollars) 100 0.25 0 97.5 100 0.50 0 94.9 100 1.00 0 90.0 100 1.50 8 96.0 100 2.00 12 101.6

17. The Bootstrapping the Zero Curve • An amount 2.5 can be earned on 97.5 during 3 months. • The 3-month rate is 4 times 2.5/97.5 or 10.256% with quarterly compounding • This is 10.127% with continuous compounding • Similarly the 6 month and 1 year rates are 10.469% and 10.536% with continuous compounding

18. The Bootstrap Method continued • To calculate the 1.5 year rate we solve to get R = 0.10681 or 10.681% • Similarly the two-year rate is 10.808%

19. Zero Curve Calculated from the Data (Figure 5.1, page 98) Zero Rate (%) 10.808 10.681 10.469 10.536 10.127 Maturity (yrs)

24. r = the bond’s yield to maturity. That is, if the investor buys the bonds at the market price and holds it to its maturity, r is the annual rate of return on this investment.

25. A PURE DISCOUNT BOND DOES NOT PAY CUPONS UNTIL ITS MATURITY; C = 0:

27. Result: The bond sells at par if P = FV. If CR = r the bond sells at par. If CR > r the bond sells at a premium. If CR < r the bond sells at a discount.

29. DURATION IS THE WIEGHTED AVERAGE OF COUPON PAYMENTS’ TIME PERIODS, t, WEIGHTED BY THE PROPORTION THAT THE DISCOUNTED CASH FLOW, PAID AT EACH PERIOD, IS OF THE CURRENT BOND PRICE.

30. Duration in continuous time • Duration of a bond that provides cash flow c iat time t i is where Bis its price and y is its yield (continuously compounded) • This leads to

31. DURATION INTERPRETED AS A MEASURE OF THE BOND PRICE SENSITIVITY

32. The negative sign merely indicates that D changes in opposite direction to the change in the yield, r. Next we present a closed form formula to calculate duration of a bond:

33. Coupon Rate

36. Duration Continued • When the yield y is expressed with compounding m times per year • The expression is referred to as the “modified duration”

37. DURATION OF A BOND PORTFOLIO V = The total bond portfolio value Pi = The value of the i-th bond Ni = The number of bonds of the i-th bond in the portfolio Vi = Pi Ni = The total value of the i-th bond in the portfolio V = ΣPiNi The total portfolio value. We now prove that: DP = ΣwiDi .

41. is the weighted average of the durations of the bonds in the portfolio. The weights are the proportions the bond value is of the entire portfolio value.

42. Example: a portfolio of two T-bonds: D = (.1673)(10.4673) +(.8327)(12.4674) = 12.1392

43. Example: a two bond portfolio: D = (0,5013)(10,4673) +(0,4987)(12,4674) = 11,45.

44. APPLICATION OF DURATION 2. IMMUNIZING BANK PORTFOLIO OF ASSETS AND LIABILITIES TIME 0ASSETSLIABIABILITIES $100,000,000 $100,000,000 (LOANS) (DEPOSITS) D = 5 D = 1 r = 10% r = 10% TIME 1 r => 12%

45. BUT IF DA = DL THEY REACT TO RATES CHANGES IN EQUAL AMOUNTS. THE BANK PORTFOLIO IS IMMUNIZED , i.e., IT’S VALUE WILL NOT CHANGE FOR A “small” INTEREST RATE CHANGE, IF THE PORTFOLIO’S DURATION IS ZERO or: DP = DA - DL = 0.

46. APPLICATIONS OF DURATION. 3. EXAMPLE: A 5-YEAR PLANNING PERIOD CASE OF IMMUNIZATION IN THE CASH MARKET BONDCFVMrDP A $100 $1,000 5 yrs 10% 4.17 $1,000 B $100 $1,000 10 yrs 10% 6.76 $1,000 4.17WA + 6.76WB = 5 WA + WB = 1 WA = .677953668. WB = .322046332. VP = $200M implies: Hold $135,590,733.6 in bond A, And $64,409,266.4 in bond B. Next, assume that r rose to 12%. The portfolio in which bonds A and B are held in equal proportions will change to:

47. [1 - 4.17 (.02/1.1)] 100M = $92,418,181.2 [1 - 6.76 (.02/1.1)] 100M = $87,709,090.91 TOTAL = $180,127,272.7 INVEST THIS AMOUNT FOR 5 YEARS AT 12%, CONTINUOUSLY COMPOUNDED YIELDS: $328,213,290. ANNUAL RETURN OF:

48. The weighted average portfolio changes to: AFTER 5 YEARS AT 12%: $331,267,162. ANNUAL RETURN OF:

50. S = The bond’s spot value. F = The futures price. n = The number of futures used in the hedge.