人民銀行研究院-衍生金融工具講義-interestratefutures.

1、Lecture #4: Interest Rate Futures We will explore one of the most successful and exciting innovations in the history of futures market - the interest rate futures contract. We first introduce concepts of terms structures of spot and forward rates. We then discuss futures on T-bills, on Eurodollar de

2、posits, and T-bonds. 1. Disasters of 94 record six Fed increases in short-term interest rate and the associated collaps in the bond market. a number of fixed income derivatives related disasters, including April 12: Porcter Call (804) 697-8000 for a free copy. Major Categories of Primary Fixed Incom

3、e Securities Entries are year-end values for 1993, billions of dollars, as reported in the Board of Governors of the Federal Reserve System, Flow of Funds Codes Tables, Spet. 20, 1994. Updated data is available from the Fed via the internet: gopher:/town.hall:70/11/other/fed Category Amount (billion

4、) Treasury Securities 3309.9 Corporate Bonds 2066.7 Tax-Exempt Securities 1217.0 Mortgages 4209.9 US Agency Issues 1898.9 Commercial Paper 553.8 Federal Funds and Repurchase Agreements 475.8 Bond Arithmetic Prices and Yields in the US Treasury Market. Price = c1/(1+y/2) + + cn/(1+y/2)n Discount fact

5、ors and yields for zero-coupon bonds Zeros exist, most commonly in the form of STRIPS, with pries reported in Wall Street Journal. (May 19, 1995) Maturity Price Discount Factor Spot Rates (years) (dollars) (%) 0.5 97.09 0.9707 5.991 1.0 94.22 0.9422 6.045 1.5 91.39 0.9139 6.096 2.0 88.60 0.8860 6.14

6、6 (in practice, prices are quoted in 32nds of a dollar, not cents). Coupon Bonds Price = d1*c1 +d2*c2 + + dn*cn eg. Consider the 8 1/2s of May 97, a treasury note with and 8.5% coupon rate, issued on May 15, 1987, and maturing May 15, 1997. In May 1995 this is a two- year bond, with cash payments (p

7、er $100 principal or face value) of $4.25 in Nov. 1995, $4.25 in May 1996, $4.25 in Nov. 1996, and $104.25 in may 1997. Its value should be Price = 104.38 Its own yield is 6.14%. The yield on a coupon is not generally the sane as the yield on a zero with the same maturity, although for short maturit

8、y the difference is small. The reason they differ is that a coupon bond has cash flow at different dates, and each date is valued with its own discount factor and yield. If yields are higher for longer maturities, then the yield is lower on a coupon bond. Although yields on coupon bonds differ from

9、spot rates, we can compute spot rates from prices of coupon bonds. Suppose we had prices for coupon bonds with maturities n = 1,2,3: Maturity Coupon rate Price (Years) ($) 0.5 8.00 100.98 1.0 10.00 103.7 1.5 4.00 97.10 We find the discount factors by using equation (?) repeatedly for bonds of increa

10、sing maturity 100.9 8 = d1*104, implying d1=0.9707, 103.78 = 0.9709*5 +d2*105, implying d2 = 0.9422, and 97.10 = (0.9709+0.9422)*2 +d3*102, implying d3 = 0.9422. Replication and arbitrage Two of the most basic concepts of modern finance are replication and arbitrage. Replication refers to the possib

11、ility of constructing combinations of assets that reproduce or replicate the cash flow of another asset. The cash flows of a coupon bond, for example, can be a combination of zeros. Arbitrage refers to the process of buying an asset at a low price and selling an equivalent asset for a higher price,

12、to make a profit. Zeros and Coupon Bonds Bond Maturity Coupon Rate Price (Years) ($) A 0.5 none 96.00 B 1.0 none 91.00 C 0.5 8 99.84 D 1.0 6 98.00 Consider the possibility of replicating the cash flows of D from the two zeros. We buy(say) na units of A and nb units of B, generating cash flows of na*

13、100 in 0.5 year and nb*100 in a year. For these to equal the cash flows of bond D, we need 3= na*100, 103=nb*100, implying na=0.03 and nb=1.03. Thus we have replicated D with a combination of A and B. The next question is whether the cost of the synthetic version of D sells the same price as D. The

14、cost is na*96.00+nb*91.00=96.61. Since this is lower then the quoted price of D, we would buy the synthetic and sell short D pocketing the profit of 1.39. Linear Programming Problem Consider a collection of bonds, indexed by i =1,I. Each bond i can be described in terms of its price, pi, and its cas

15、h flow over T periods, (c-1i, c2i, ,cTi), with T set at the maximum maturity of I bonds. Problem: min sum(n*p, i=1,.T) (by choosing ni, i=1,I) subject to sum(ni*cti,i=1,.I) =0, for each t=1,T If this problem has a minimum zero, then these bonds are immune to arbitrage. But if the answer has negative

16、 cost, we have found a pure arbitrage opportunity: we sell the overpriced bonds(ni T, and r* and r are the rates (a.c.c) related to T, and T* respectively. The zero rate curve is usually obtained using the bootstrap method or spline function fitting method. Year (n) Spot rate for an n-year Investmen

17、t (% per annum) Forward rate for nth year (% per annum) 1 2 3 4 5 10.0 10.5 10.8 11.0 11.1 11.0 11.4 11.6 11.5 Data for Bootstrap Method Bond Principal (dollars) Time to Maturity (years) Annual Coupon (dollars) Bond Price (dollars) Zero rate 100 100 100 100 100 100 0.25 0.50 1.00 1.50 2.00 2.75 0 0

18、0 8 12 10 97.5 94.9 90.0 96.0 101.6 99.8 10.13% 10.47% 10.54% 10.68% 10.81% 10.87% The last bond provides cash flow as follows: The rate for the first cash-flow is 10.13%. Using linear interpolation, the discount rate for the next three cash flows are 10.505%, 10.61%, and 10.745%. The present value

19、for the first four cash-flows is 18.081. Suppose that the 2.75-year rate is R. Using liner interpolation, the 2.25-year rate is 0.1081*2/3 +R/3. Cash flow $5 $5 $5 $5 $5 $105 Time 0.25 Y 0.75 Y 1.25 Y 1.75 Y 2.25 Y 2.75 Y Cubic spline function Maturity (year) Spot rate Forward rate (t-1,t) Discount

20、function 1 2 3 4 5 6 7 8 9 10 5.61 5.95 6.27 ? 6.68 6.78 ? 6.86 6.87 6.87 6.29 6.91 ? ? 7.28 ? ? 6.95 6.87 0.945445 0.887808 0.828532 ? 0.716054 0.665777 ? 0.577643 0.538860 0.503083 Use D(t) = a0+a1t+a2t2+a3t3 Thus D(4)=0.77125384, D(7)=0.61964212 r(4) = -ln(D(4)/4 = 6.4934% For bond in general, we

21、 view them as portfolios of zeros. Use one coupon bond to illustrate the point. Bond #1 is a 2-year 6% coupon (semiannual) and its market price is 101. a0 a1 a2 a3 1.008209-0.062087280.0004086957.585e-005 101 =3D(0.5) + 3D(1) + 3D(1.5) + 103D(2)= = 3(a0+0.5a1+0.25a2+0.125a3)+ 3(a0+a1+a2+a3)+ 3(a0+1.

22、5a1+2.25a2+3.375a3)+ 103(a0+2a1+4a2+8a3)+ =112a0+215a1+422.5a2+837.5a3 Tuckman(Fixed Income Securities, Wiley, 1995) is a good book to read. There are a number of online sources of bond information, including Bloomberg and Reuters. Measuring Interest Rate Risk Prudent management and the fear induced

23、 by periodic derivatives disasters have spurred the development and widespread use of tools for managing financial risk. One of these tools is a system for quantifying risk - for answering the question: How much can we lose over the next day, week, or year? Consider contexts the risk management tool

24、s may be used: Investing pension funds. Some pension funds are committed to paying benefits that are relatively easy to forecast, and that must be financed by investing current contributions. The question is how they invest these contributions to guarantee that theyll be able to satisfy claims for b

25、enefits. Managing interest rate risk at a financial institution. Commercial banks, for example, often borrow short and lend long. What kind of risk does this expose them to? Managing a bond fund. How should a fund manager trade off the risk and return of various combinations of bonds? Treasury manag

26、ement at an industrial corporation. Should it issue long debt or short? Long debt has less uncertainty over future interest payments, but short debt is cheaper on average. Price Value of a Basis Point (PVBP) PVBP = -(P P0) P is the price associated with a rise of one basis point in the yield. eg. (t

27、wo-year 10% par bond). At y0=1.100 the bond sells for P0 = 100. At y=0.1001 the bond sells for 99.9823. PVBP = 0.0177. Duration D = - (slope of price-yield relation)/ price = -(dp/dy)/p. Thus a bond with duration D declines in value by D percent if the yeild rises by 1 percent or 100 basis point. By

28、 doing a simple calculation D = -(1+y/2)-1sum(j/2)*wj, j=1,n), with wj = (1+y/2)-j * ci/p Example 1(two-year 10% par bond). Payment Cash flow Present Weight Number (j) (cj) value (wj) 1 5 4.762 0.04762 2 5 4.535 0.04535 3 5 4.319 0.04319 4 105 86.384 0.86384 D = 1.77 years. Example 2(five-year 10% p

29、ar bond). D =3.86 years Example 3(two year zero). D = (1+.1/2)-1(n/2) = 1.90 years. Duration and PVBP PVBP = dp/dy*0.01%. Duration of Portfolios Consider a collection of m bonds selling a t (p1,p2,pm), with total p=sum(pi, i=1,m).Let Dj be the duraton for bond j. Then the duration of the portfolio i

30、s D = sum(Dj*wj, j=1m), with wj = pj/p. Immunization Cash flow matching Payments Time(Yrs) Payment 0.5 2000 1.0 1900 1.5 1800 2.0 1700 List of possible investments consists the following bonds: Bond Coupon Maturity Price(Ask) Yield(%) 1 0 0.5 96.618 7.000 2 0 1.0 92.456 8.000 3 0 1.5 87.630 9.000 4

31、0 2.0 83.856 9.000 5 8 1.0 96.750 7.432 6 6 2.0 97.500 9.400 If we do matching with zeros, we would buy 20 six-month zeros, 19 one-year zeros, 18 eighteen-month zeros, and 17 twenty-month zeros, for a total cost of 6691.91. We can use linear programming to find a cheaper solution: Bond Units 1 19.35

32、 2 18.35 3 17.35 4 0.000 5 0.000 6 16.35 The total cost is 6679.19. Duration Matching The difficulty with cash flow matching is that it applies to an idea world. We often find that we do not have all the required bond maturities, or that the number of maturities is so great that we incur enormous tr

33、ansaction costs to match them all exactly. An alternative to do duration matching. Consider the case of Foresi Bank of Mahopac (FBM). FBM has assets of $25b, liabilities of $20b, and shareholders equity $5b. The market value of both assets and liabilities are sensitive to movements in interest rates

34、. Assets(largely business loans) have an overall duration of 1.0 years. Of the $20b of liabilities, currently $10b is in six-month commercial paper and $10b in two-year notes just issued at par. With a flat spot rate curve at 10%, the total commercial paper has duration of 0.48 and the notes have a

35、duration of 1.77. The DL = 1.12. Thus the bank is exposed to increase in interest risk: an increase in rates will reduce the value of both assets and liabilities, but since the former is larger and the bank loses overall. What the bank should d? One possibility is to convert some of the commercial p

36、aper to notes. But how much? Change in asset value = -DA*25b*dy Change in liability value = - DL*20b*dy with DA=1.0 the two change will be equal if DL=1.25. To get this we need 0.48*w+1.77*(1-w)=1.25, w=0.4, implying 0.4*20=$8b in paper. Statistical Measures of Price Sensitivity The premise behind d

37、uration analysis is that yields of all maturities changed by the same amount. In statistical language, the yields (1) are perfectly correlated and (2) have the same standard deviation. We see that the standard deviation declines with maturity, and that correlations are generally strong but less than

38、 one. As a result, most quantitative risk analysis is based on the statistical properties of returns. Comparison with Duration One example is a comparison of the risk in two- and ten-year zeros. With duration, we have dp/p = _D*dy so the standard deviation is std(dp/p) = D*std(dy). In duration analy

39、sis, we assume that changes in yields are the same for all maturities, so the only thing that differs across positions is duration. Say D10(9.5) =D2(1.9)*5. Statistical measures tell us that the ten-year zero is more risky than the two-year, but not by a factor of five. If std(dy(10)=0.309%, std(dp/

40、p(10)=2.8%. If std(dy(2)=0.49%, then std(dp/p(2)=0.94%. JP Mogans RiskMetrics They supply daily and monthly standard deviations and correlations for returns in more than twenty countries and covering fixed income securities, foreign exchange, and equities. Their approach shares a number of features

41、of the statistical methods just outlines, but a number of differences are worth noting: Standard deviations and correlations are updated frequently to reflect changes in market volatilities. Yield volatilities pertain to proportional changes. (log(yt/yt-1). Yields are reported for a more extensive s

42、et of maturities: 1 day; 1 week; 1,3,and 6 month; and 1-5,7,9,10,15,30, and 30 years. Shifts and Twists Yield changes are 90% parallel shifts and 9% twists, with about 1% left over. Forward rate agreements (FRAs) It is a contract written on LIBOR which requires a cash payment at maturity based on th

43、e difference between a realized spot rate of interest rate and a pre-specified forward rate. - R will be earned for the period T to T*. It is agreed at time 0. The cash-flows for the agreement: Time T: -100 Time T*: 100eR(T*-T) The value of FRA at time zero, V(0) = 100eR(T*-T)e-r*T*-100e-rT= 100e- r

44、TeR(T*-T)-r*T*+rT-1. V(0) =0 -R= r(0) =(r*T*-rT)/(T*-T) - If R(T,T*) is the spot rate for the period of T to T*, V(T), the value of FRA at Time T is 100(eR-R(T ,T*)T*-T-1). - What is V(t) 0tT? V(t) = 100eR(T*-T)e-r*(t,T*)(T*-t)-100e-r(t,T)(T-t) = 100e-r(t,T)(T-t)(eR-r(t)(T ,T*)T*-T-1). Where r(t) is

45、 the forward rate determined at time t. We can value FARs by using current forward rate. T-bill futures The underlying asset is a 90-day T-bill. Under the terms of contract, the party with the short position must delivery $1m of T-bill. 0 T T* |-|-|- T* - T = 90 days, the current time is zero. T is

46、the maturity of T-bill futures, and T* is the maturity of T-bill say, paying $100. The value of T-bill today is V* = 100 e- r*T* F = V*erT=100erT-r*T*=100e-r(T*-T) There are two ways to by T-bill matured at T*. You can buy it either from spot market or through T-bill futures. - Example: 45-day T-bil

47、l rate is 10%, 135-day T-bill rate is 10.5%, 135*10.5-45*10/90 = 10.75%, F = 100e-10.75*1/4=97.34829 - Arbitrage opportunity If the T-bill futures price described in the above example is 97.3848, the rate implied is 10.6% 10.75% there is an arbitrage opportunity. Time: 0 Time: T - Empirical evidence

48、 Summary statistics for the difference between futures rates and implied forward rate; daily data for September 1977 though June 1982 (from Kawller and Koch (1984) Number of days to delivery of futures contract Entire Statistics Sample 60 Mean -0.314 -0.006 -0.166 -0.330 -0.434 -0.484 Std 0.365 0.23

49、2 0.288 0.296 0.316 0.387 No. of Obs. 1126 189 191 194 207 345 t-value -28.8 -0.3 -8.0 -15.5 -19.8 -23.2 - What are the explanations? Transaction cost Daily settlement for futures traders Differential tax influences - Storage costs related explanation: The arbitrage based on the comparison of implie

50、d and actual forward rates described above is not the predominant one in the market place. Instead, professional traders and government securities dealers focus on the period from present to the delivery date of the nearby futures contract and compare a holding period yield with a financing rate. Th

51、e focus of active bill traders covers the interval from the present until the delivery of the nearby futures contract. For example, a trader can effectively lock-in a return by purchasing a cash bill with T+91 days to maturity and simultaneously shorting the nearby futures, thus fixing the selling p

52、rice of the cash bill T days after purchase. This rate of return is commonly referred as the “implied repo rate” and it can be calculated as r(t,T)=1/(T-t) ln(Ft(T)/Vt(T*) Traders compared the implied repo rate with a financing rate that is typically the actual repo rate. If the implied repo rate is

53、 exceeds the actual repo rate, traders buy the deliverable bill and short the futures. If the relationship between these two rates is the other way around, traders reverse the transaction by borrowing the deliverable bill via a reverse repurchase agreement, selling it, and going long the nearby futu

54、res contract. At delivery, the deliverable T-bill is immediately returned to its lender, thereby satisfying the reverse repurchase agreement. With this arbitrage, the futures price is determined at the margin by speculative activity during the three months prior to delivery. Traders buy or sell T-bi

55、ll futures depending on the sign and magnitude of these two rates. Traditional tests of efficiency using forward rates ignore this speculative activity. Instead it is inferred that futures prices are set relative to 91 day return available at the delivery of the futures of the futures contract. It s

56、hould be noticed, this near-term arbitrage is not entirely risk-free due to the margin call of futures contract and rule governing the value adjustment of collateral under repurchase agreement. A different implied forward rate by using repo rate is calculated by (r(T*)T* - RP*T)/(T*-T), where RP is

57、the repo rate. Summary statistics for the difference between futures rates and implied forward rate using the overnight repo rate; daily data for September 1980 though June 1982 (from Kawller and Koch (1984) Number of days to delivery of futures contract Entire Statistics Sample 60 Mean 0.041 0.072

58、0.037 -0.027 0.028 0.072 Std 0.640 0.288 0.271 0.351 0.379 1.053 No. of Obs. 588 100 101 100 109 178 t-value 1.6 2.5 1.4 -0.8 0.8 0.9 - Cash price vs quoted price in the US a. The actual/360 day count convention is used for T-bill in US. T-bill price quotes are for a T-bill with a face value of $100

59、 b. T-bills (n days to maturity): Quoted price = 360/n (100-cash price) This is referred as the discount rate. If 90- day T-bills cash price is $98, the quoted price would be 8. The annualized rate of return would be 8/98=8.28% C. T-bill futures on a 90 day instrument: If Z is the quoted futures pri

60、ce and Y is the cash futures price, Z=100-4(100-Y), or 100-DY (discount yield) - Example: Suppose that the 140day interest rate is 8%(a.c.c.) and 230-day rte is 8.25% (a.c.c.). The forward rate is 8.64% (a.c.c 140-230). The futures price for 90-day T-bill futures deliverable in 140 days is 100*e-0.0