CHAPTER 6 Interest Rate Problems 1 Spot rates, forward rates and the yield curve 1.1 Basic idea. The interest rate offered on investments usually depends on the term or length of the invest- ment. The term structure of interest rates is concerned with the analysis of the dependence of interest rates on the length of the investment. 1.2 Spot rates and zero rates; zero coupon bonds. A ￡1 zero coupon bond of term n is an agreement to pay ￡1 at the end of n years; zero coupon means that no interest payments are made. A zero coupon bond is also called a pure discount bond. The term structure of interest rates can be assessed from the prices of zero coupon bonds as follows. Let Pt denote the price at issue of a ￡1 zero coupon bond which matures after a time period of length t years. Let yt denote the yield to maturity of this bond—this is another name for the internal rate of return. This yield, yt, is sometimes called the t-year spot rate of interest . Clearly (1 + yt)tPt = 1 (1.2a) and hence yt = P 1/t t 1 Usually, yt 6= yv when t 6= v. The function y : [0,1)! R with y(t) = yt is called the yield curve. An example of a yield curve is given in figure 1.2a. maturity year yi el d 2000 2005 2010 2015 2020 2025 2030 0.05 0.06 0.07 0.08 Figure 1.2a. A Yield Curve. Yield curves are plotted against the maturity date. Yield curves usually rise gradually because long bonds have higher yields than short bonds. Note that yt, the t-year spot rate of interest or the t-year zero rate of interest is the effective rate of interest per annum for an investment which lasts for t years. Recall that , the nominal rate of interest compounded continuously or the force of interest, is defined by i = e 1 or = log(1 + i) where i is the effective rate of interest per annum. With continuous compounding, equation (1.2a) becomes etYtPt = 1 and hence Yt = 1 t logPt (1.2b) where Yt is the t-year spot force of interest. Clearly, yt = eYt 1. Thus Yt corresponds to , the nominal rate of interest compounded continuously, but now the dependence on the length of time of the investment is considered. Every fixed interest investment can be regarded as a combination of zero coupon bonds as the following exam- ple illustrates. ST334 Actuarial Methods cR.J. Reed Sep 27, 2016(9:51) Section 1 Page 101 即期利率 远期利率 收益率曲线 Page 102 Section 1 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed Example 1.2a. Suppose a bond pays a coupon of size x at times 1, 2, . . . , n and has final redemption value C at time n. (a) Express P , the current price of the bond in terms of x, C and P1, P2, . . . , Pn where P1, P2, . . . , Pn are the prices of ￡1 zero coupon bonds with different terms. (b) Express the current price of the bond in terms of x, C and y1, y2, . . . , yn where y1, y2, . . . , yn are the correspond- ing yields of the bonds. (c) Show that the gross redemption yield, i, satisfies min{y1, y2, . . . , yn} i max{y1, y2, . . . , yn}. Solution. The bond can be considered as a combination of n zero coupon bonds all with maturity value x and lengths 1, 2, . . . , n and one zero coupon bond with length n and maturity value C. Hence the answer to (a) is: P = x(P1 + P2 + · · · + Pn) + CPn and the answer to (b) is P = x 1 (1 + y1) + 1 (1 + y2)2 + · · · + 1 (1 + yn)n + C (1 + yn)n (c) This follows immediately from x (1 + y1) + x (1 + y2)2 + · · · + x (1 + yn)n + C (1 + yn)n = x (1 + i) + x (1 + i)2 + · · · + x (1 + i)n + C (1 + i)n Example 1.2b. Suppose we know the spot rate yk for k = 1, 2, . . . , 10. Find an expression for the price of a bond with redemption value ￡100, annual coupon rate 8% and maturing in 10 years in terms of y1, y2, . . . , y10. Solution. The cash flow is as follows: Time 0 1 2 3 4 5 6 7 8 9 10 Cash flow 0 8 8 8 8 8 8 8 8 8 108 The price is 10X k=1 8 (1 + yk)k + 100 (1 + y10)10 1.3 Determining the spot rate. There are few zero coupon bonds and so the zero rate must also be calculated from the prices of other financial instruments. Here are two possibilities: Determine y1 from the interest rate on one year Treasury bills. To find y2, suppose we observe the price P of a two year bond with redemption value C, face value f and coupon rate r payable annually. Now the price of the bond should equal the net present value of the cash flow stream, which is: Time 1 2 Cash flow fr C + fr Hence P = fr 1 + y1 + fr + C (1 + y2)2 As y1 is known, the value of y2 can be determined. Then the value of y3 can be determined from the price of a three year bond, and so on. A zero coupon bond can also be constructed by subtracting bonds with identical maturity dates but different coupons. Suppose bond A is a ten year bond with price PA, coupon rA, face value and redemption value C. Suppose bond B is a ten year bond with price PB , coupon rB , face value and redemption value C. Without loss of generality, assume rA > rB . Consider a portfolio of rBC of bond A and rAC of bond B. This portfolio will have a face value of (rA rB)C, a price of P = rAPB rBPA and zero coupon. Hence (1 + y10)10P = (rA rB)C and y10 can be determined. 总赎回收益率 6 Interest Rate Problems Sep 27, 2016(9:51) Section 1 Page 103 1.4 Par yield of a bond. The n-year par yield of a bond is the coupon rate that causes the current bond price to be equal to its face value, assuming the bond to be redeemed at par. As usual, let f denote the face value, r the coupon rate, P the current price. Assume the coupon is payable annually for n years. The cash flow is as follows: Time 0 1 2 . . . n 1 n Cash flow P fr fr . . . fr f + fr We want P = f . This implies that the n-year par yield, r, satisfies 1 = r nX k=1 1 (1 + yk)k + 1 (1 + yn)n (1.4a) where yk is the k-year spot rate of interest. The par yields give an alternative measure of the relationship between the yield and the term of an investment. The difference between the n-year par yield and the n-year spot rate is called the n-year coupon bias. Thus the n-year coupon bias is r yn where r is the n-year par yield. 1.5 Forward interest rates. Suppose fT,k denotes the interest rate per annum for money which will be borrowed at some time T in the future for a period of time of length k. This is called a forward interest rate. Clearly f0,k = yk. ………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………..
…. … …. . …. … …. . …. … …. . ….. ….. 0 T start of investment T + k end of investment time …………………………………………………………………………………………………………………..
……. …….. period of investment Now ￡1 invested at time 0 grows to ￡(1 + yT+k)T+k at time T + k. Also ￡1 grows to ￡(1 + yT )T at time T . If this amount of ￡(1 + yT )T is reinvested then it grows to ￡(1 + yT )T (1 + fT,k)r at time T + k. Hence (1 + yT+k)T+k = (1 + yT )T (1 + fT,k)k In particular (1 + y2)2 = (1 + y1)(1 + f1,1) (1 + y3)3 = (1 + y2)2(1 + f2,1) = (1 + y1)(1 + f1,1)(1 + f2,1) and so on. Hence (1 + yT )T = (1 + f0,1)(1 + f1,1) · · · (1 + fT1,1) (1.5a) where f0,1 = y1 is the rate per annum for 1 year starting at time 0; f1,1 is the rate per annum for 1 year starting at time 1, etc.1 Forward rates calculated from equations such as equation (1.5a) are called implied forward rates as opposed to the market forward rates. Example 1.5a. Suppose the one year zero rate is 0.07, or 7%, and the two year zero rate is 0.08 or 8%. Find the implied one year forward rate for a period of length 1 year. Solution. Recall that the “two year zero rate” is the rate per annum for an investment of length two years. So we are given that y1 = 0.07 and y2 = 0.08. Using (1 + y2)2 = (1 + y0)(1 + f1,1) gives f1,1 = 1.082/1.07 1 = 0.090 or 9%. 1.6 Continuous compounding. For continuous compounding, we have PT PT+k = 1 + fT,k k = ekFT,k where the forward force of interest is FT,k = ln(1 + fT,k). Hence FT,k = lnPT lnPT+k k (1.6a) = (T + k)YT+k TYT k Note that F0,k = Yk where Yk is defined in equation (1.2b) on page 101. 1 If there is no possibility of confusion, the quantity fk,1 is often abbreviated to fk. 债券的票 收益 Page 104 Section 1 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed Example 1.6a. Suppose the spot force of interest for an investment of 1 year is a nominal 7% per annum com- pounded continuously and the spot force of interest for an investment of 6 months is a nominal 6% per annum compounded continuously. Find the implied force of interest for an investment starting in 6 months’ time and lasting for 6 months. Solution. The general result is e(t+k)F0,t+k = etF0,t ekFt,k or (t + k)F0,t+k = tF0,t + kFt,k In our case, t = k = 0.5, F0,1 = 0.07 and F0,0.5 = 0.06. Hence F0.5,0.5 = 0.08. So the answer is a nominal 8% per annum compounded continuously. 1.7 A more sophisticated model with allowance for interest rates varying with time. Yields which depend on time and the length of the investment. The above equations are too simplistic: we should really treat ￡P , the price of a ￡1 zero coupon bond, as a function of two variables instead of just one. So, for 0 t T , let P (t, T ) denote the value at time t of a zero coupon bond with face value 1 and maturing at time T . Clearly P : {(x, y) 2 R R : 0 x y}! R with P (x, x) = 1 for all x 0. This function P of two variables will be complicated: for example, the prices of 10 year bonds and 15 year bonds will tend to move together in the short term. Hence the functions t! P (t, t + 10) and t! P (t, t + 15) will be related. fs Corresponding to this function of two variables, we have the yield to maturity, y(t, T ) defined by [1 + y(t, T )]Tt P (t, T ) = 1 for all 0 t T . (1.7a) Note that y(t, T ) denotes the yield to maturity from a zero coupon bond that starts at time t and ends at time T . We can now plot the yield curve at time t: this is a plot of the function T ! y(t, T ) for T t. Figure (1.2a) on page 101 is such a plot with t = 2000. This model reduces to the previous model if we assume the price just depends on the length of time between issue and maturity and not on the date of issue. This is equivalent to assuming P (t, T ) = P (0, T t) for all t. Then set Pt = P (0, t). As P (t, T ) determines Y (t, T ) and conversely, it follows that if P (t, T ) = P (0, T t) for all t, then y(t, T ) = y(0, T t) for all t, and conversely. Set yt = y(0, t). And so we arrive at the simpler model. With continuous compounding, we have e(Tt)Y (t,T )P (t, T ) = 1 where eY (t,T ) = 1 + y(t, T ) and so Y (t, T ) = lnP (t, T ) T t Forward rates. A forward rate is an interest rate agreed now for an investment which starts at some time in the future. ………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………….
…. … …. . …. … …. . …. … …. . …. … …. . ….. ….. 0 t time of agreement T start of investment T + k end of investment time …………………………………………………………………………………………………………………………………………
……. …….. period of investment Let ft,T,k denote the effective interest rate per annum agreed at time t for an investment starting at time T for a period of length k. Hence the investment matures at time T + k. This means that ￡1 invested at time T will accumulate to ￡(1 + ft,T,k)r at time T + k. The discrete time forward rate can be related to the spot rate as follows: suppose we invest ￡1 at time t for T t years and agree to invest the accumulated amount at time T for a further k years. Now the accumulated amount at time T will be ￡ [1 + y(t, T )]Tt. This amount will accumulate to ￡ [1 + y(t, T )]Tt 1 + ft,T,k k at time T + k. But ￡1 invested at time t for T + k t years will accumulate to ￡ [1 + y(t, T + k)]T+kt. Hence [1 + y(t, T + k)]T+kt = [1 + y(t, T )]Tt 1 + ft,T,k k (1.7b) and so, by equation (1.7a) 1 + ft,T,k k = [1 + y(t, T + k)]T+kt [1 + y(t, T )]Tt = P (t, T ) P (t, T + k) 6 Interest Rate Problems Sep 27, 2016(9:51) Section 1 Page 105 More generally, repeated applications of equation (1.7b) gives: [1 + y(t, T )]Tt = [1 + y(t, T 1)]T1t 1 + ft,T1,1 = (1 + ft,t,1)(1 + ft,t+1,1) · · · (1 + ft,T1,1) (1.7c) Note that ft,t,1 = y(t, t+ 1) is the one-year spot interest rate—it is the one-year riskless rate prevailing at time t for repayment at time t + 1. Forward rates calculated from equations such as equation (1.7c) are called implied forward rates as opposed to the market forward rates. Relation to previous model. Suppose ft,T,k = f0,T,k for all t; i.e. the forward rate just depends on the start and length of the investment and not when the agreement is made. Denote the common value ft,T,k for all t by fT,k; hence fT,k is the forward rate for money which will be borrowed at some time T in the future for the period of time of length k. The model then reduces to the simpler model at the start of this chapter. Continuous compounding. For continuous compounding, we have P (t, T ) P (t, T + k) = 1 + ft,T,k k = ekFt,T,k where the forward force of interest is Ft,T,k = ln(1 + ft,T,k). Hence Ft,T,k = lnP (t, T ) lnP (t, T + k) k (1.7d) = (T + k t)Y (t, T + k) (T t)Y (t, T ) k by equation (1.7a). Note that Ft,t,k = Y (t, t + k). Instantaneous forward rates. The instantaneous forward rate at time t for an investment commencing at time T is defined by Ft,T = lim k!0 Ft,T,k Using equation (1.7d) shows that Ft,T = lim k!0 logP (t, T ) logP (t, T + k) k = lim k!0 logP (t, T + k) logP (t, T ) k = @ @T logP (t, T ) (1.7e) = 1 P (t, T ) @P (t, T ) @T Conversely, using P (T, T ) = 1 and integrating equation (1.7e) shows that P (t, T ) = exp Z T t Ft,u du ◆ The instantaneous forward rate is a theoretical concept which is useful in modelling bond prices. Note that any one of the functions P (t, T ), Y (t, T ) or Ft,T determines the other two. The short rate at time t or the instantaneous rate at time t is defined to be rt = Y (t, t) = lim k!0 Y (t, t + k) = lim k!0 Ft,t,k = Ft,t = @ logP (t, T ) @T T=t where the third equality follows from Ft,t,k = Y (t, t + k). Hence, rt, which is equal to Ft,t, is the instantaneous forward rate at time t for an investment commencing at time t. There is less information in the function rt than in the functions P (t, T ), Y (t, T ) or Ft,T . Page 106 Section 1 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed term in years yi el d 0 5 10 15 20 25 30 0.05 0.06 0.07 0.08 Figure 1.8a. An Increasing Yield Curve. term in years yi el d 0 5 10 15 20 25 30 0.03 0.04 0.05 0.06 0.07 Figure 1.8b. A Decreasing Yield Curve. term in years yi el d 0 5 10 15 20 25 30 0.03 0.04 0.05 0.06 0.07 Figure 1.8c. A Humped Yield Curve. 1.8 Explaining the term structure of interest rates. The three common types of yield curves are as follows. Decreasing yield curve. Long term bonds have lower yields than short term bonds. Increasing yield curve. Long term bonds have higher yields than short term bonds. This is the most common shape. Humped yield curve. Long term bonds have lower yields than short term bonds, but the very short term bonds (less than 1 year) have lower yields than 1 year bonds. There are three popular explanations of the term structure of interest rates: (1) Expectations Theory. This theory postulates that the relative attraction of long and short term bonds depends on the expectations of future movements in interest rates. If investors expect interest rates to fall, then long term bonds will become more attractive; this will cause the yields on long term bonds to fall and lead to a decreasing yield curve. Similarly, if investors expect interest rates to rise, then long term bonds will become less attractive and so lead to an increasing yield curve. (2) Li
quidity Preference. This theory asserts that investors usually prefer short term investments over long term ones. The reason for this is that investors anticipate they may need to sell their bonds soon and they know that long term bonds are more sensitive to interest changes than short term bonds. Hence purchasing a short term bond lessens the risk. This implies that longer-term bonds must offer higher yields as an inducement. 6 Interest Rate Problems Sep 27, 2016(9:51) Exercises 2 Page 107 (3) Market Segmentation. Banks are interested in short term bonds because their liabilities are short term. Pension funds are interested in long term bonds because their liabilities are long term. The market segmentation theory postulates that the term structure arises from the different forces of supply and demand for bonds of different lengths. 1.9 Summary. The simple model: interest rates constant in time. yt denotes the t-year spot rate of interest. This is the effective interest rate p.a. for an investment which lasts for t years. Pt denotes the price in ￡ at issue of a ￡1 zero coupon bond which matures after t years. Hence Pt(1 + yt)t = 1 ft,k denotes the forward rate for an investment starting at time t for a period of k years. Then: (1 + yt+k)t+k = (1 + yt)t(1 + ft,k)k and (1 + ft,k)k = (1 + yt+k)t+k (1 + yt)t = Pt Pt+k (1 + yt)t = (1 + f0,1)(1 + f1,1) · · · (1 + ft1,1) where ft,1, sometimes written as ft, is the 1-year forward rate for an investment starting at time t for a period of 1 year. Ft,k denotes the forward force of interest for an investment starting at time t for k years. Hence Ft,k = ln(1 + ft,k) = ln(Pt) ln(Pt+k) k Also F0,k = ln(1 + yk) and e(t+k)F0,t+k = etF0,t ekFt,k Ft denotes the instantaneous forward rate at time t. By definition Ft = lim k!0 Ft,k = lim k!0 ln(1 + ft,k) = lim k!0 ln(Pt) ln(Pt+k) k = d dt lnPt = 1 P dP dt Conversely Pt = exp Z t 0 Ft du ◆ The more sophisticated model: interest rates varying with time. y(t, T ) denotes the effective rate of interest per annum for an investment starting at time t and maturing at time T . P (t, T ) denotes the price in ￡ at time t of a ￡1 zero coupon bond which matures at time T . Hence P (t, T ) [1 + y(t, T )]Tt = 1. ft,T,k denotes the forward rate per annum at time t of an investment starting at time T and maturing at time T + k. Explaining the term structure of interest rates: expectations theory, liquidity preference, market segmentation. Par yield of a bond. 2 Exercises (exs6-1.tex) 1. At time 0 the 1-year spot rate is 8% p.a., the 2-year spot rate is 9% p.a. and the 3-year spot rate is 91/2% p.a. What is the value of the 2-year forward rate from time 1 (Institute/Faculty of Actuaries Examinations, April 1997) [2] 2. The following n-year spot rates apply at time t = 0: 1 year spot rate of interest: 41/2% per annum effective 2 year spot rate of interest: 5% per annum effective 3 year spot rate of interest: 51/2% per annum effective Calculate the 2 year forward rate of interest from time t = 1 expressed as an annual effective rate of interest. (Institute/Faculty of Actuaries Examinations, April 2000) [2] 市场细分 Page 108 Exercises 2 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed 3. The 1-year forward rates for transactions beginning at times t = 0, 1, 2 are fi, where f0 = 0.06 f1 = 0.065 f2 = 0.07 Find the par yield for a 3-year bond. (Institute/Faculty of Actuaries Examinations, September 1997) [3] 4. The n year forward rate for transactions beginning at time t and maturing at time t + n is denoted as ft,n. You are given: f0,1 = 6.0% per annum; f0,2 = 6.5% per annum; f1,2 = 6.6% per annum. Determine the 3-year par yield. (Institute/Faculty of Actuaries Examinations, September 1998) [3] 5. (i) The one-year forward rate applying at a particular point in time t is defined as ft,1. If ft,1 = 4%, calculate the continuous time forward rate Ft,1 applying over the same period of time. (ii) Define the instantaneous forward rate Ft. (Institute/Faculty of Actuaries Examinations, September 2004) [2+2=4] 6. In a particular bond market, the two-year par yield at time t = 0 is 4.15% and the issue price at time t = 0 of a two-year fixed interest stock, paying coupons of 8% annually in arrears and redeemed at 98, is ￡105.40 per ￡100 nominal. Calculate: (a) the one-year spot rate; (b) the two-year spot rate. (Institute/Faculty of Actuaries Examinations, April 2004) [6] 7. At time t = 0 the n-year spot rate of interest is equal to (2.25 + 0.25n)% per annum effective (1 n 5). (a) Calculate the 2-year forward rate of interest from time t = 3 expressed as an annual effective rate of interest. (b) Calculate the 4-year par yield. (c) Without explicitly calculating the gross redemption yield of a 4-year bond paying annual coupons of 3.5%, explain how you would expect this yield to compare with the par yield calculated in (b). (Institute/Faculty of Actuaries Examinations, April 2006, adapted) [7] 8. The following n-year spot rates were observed at time t = 0: 1 year spot rate of interest: 4% per annum 2 year spot rate of interest: 5% per annum 3 year spot rate of interest: 6% per annum 4 year spot rate of interest: 7% per annum 5 year spot rate of interest: 71/2% per annum 6 year spot rate of interest: 8% per annum (i) Define what is meant by an n-year spot rate of interest. (ii) Calculate the two-year forward rate of interest at time t = 3. (iii) Using the above n-year spot rates calculate the 6-year par yield at time t = 0. (Institute/Faculty of Actuaries Examinations, April 1998) [2+2+4=8] 9. The force of interest (t) is a function of time and at any time, measured in years, is given by the formula (t) = 0.04 + 0.001t (i) (a) Calculate the accumulated value of a unit sum of money accumulated from time t = 0 to time t = 8. (b) Calculate the accumulated value of a unit sum of money accumulated from time t = 0 to time t = 9. (c) Calculate the accumulated value of a unit sum of money accumulated from time t = 8 to time t = 9. (ii) Using your results from (i), or otherwise, calculate: (a) The 8-year spot rate of interest from time t = 0 to time t = 8. (b) The 9-year spot rate of interest from time t = 0 to time t = 9. (c) f8,1, where f8,1 is the one-year forward rate of interest from time t = 8. (Institute/Faculty of Actuaries Examinations, September 2003) [5+3=8] 10. Three bonds paying annual coupons in arrears of 7% and redeemable at 105 per ￡100 nominal reach their redemption dates in exactly one, two and three years’ time, respectively. The price of each of the bonds is ￡98 per ￡100 nominal. (i) Determine the gross redemption yield of the 3-year bond. (ii) Calculate all possible spot rates implied by the information given. (Institute/Faculty of Actuaries Examinations, September 2002) [3+5=8] 11. An individual is investing in a market in which a variety of spot rates and forward contracts are available. If at time t = 0 he invests ￡1,000 for 2 years, he will receive ￡1,118 at time t = 2. Alternatively, if at time t = 0 he agrees to invest ￡1,000 at time t = 1 for 2 years, he will receive ￡1,140 at time t = 3. However, if at time t = 0 he agrees to invest ￡1,000 at time t = 1 for one year, he will receive ￡1,058 at time t = 2. (i) Calculate the following rates per annum effective, implied by this data: (a) The one year spot rate at time t = 0. (b) The two year spot rate at time t = 0. (c) The three year spot rate at time t = 0. (ii) Calculate the three year par yield at time t = 0 in this market. (Institute/Faculty of Actuaries Examinations, April 2003) [5+3=8] 6 Interest Rate Problems Sep 27, 2016(9:51) Exercises 2 Page 109 12. The n-year spot rate of interest, yn is given by: yn = 0.04 + n 1000 for n = 1, 2 and 3. (i) Calculate the implied one-year forward rates applicable at times t = 1 and t = 2. (ii) Assuming that coupon and capital payments may be discounted using the same discount factors, and that no arbitrage applies, calculate: (a) The price at time t = 0 per ￡100 nominal of a bond which pays annual coupons of 3% in arrears and is redeemed at 110% after 3 years. (b) The 2-year par yield. (Institute/Faculty of Actuaries Examinations, April 2001) [9] 13. The forward rate from time t 1 to time t has the following values: f0,1 = 4.0%, f1,2 = 4.5%, f2,3 = 4.8% (i) Assuming no arbitrage, calculate: (a) the price per ￡100 nominal of a 3-year bond paying an annual coupon in arrears of 5%, redeemed at par in exactly 3 years, and (b) the gross redemption yield from the bond. (ii) Explain why the bond with a higher coupon would have a lower gross redemption yield, for the same term to redemption. (Institute/Faculty of Actuaries Examinations, April, 2002) [7+2=9] 14. Bonds paying annual coupons of 6% annually in arrears and redeemable at par will be redeemed in exactly one year, two years and three years respectively. The price of each of the bonds is ￡96 per ￡100 nominal. (i) Determine the gross redemption yield of the 3 year bond. (ii) Determine the discount factors (1), (2) and (3) that the market is using to discount payments due in 1, 2 and 3 years respectively. (iii) Calculate f0,1, f1,2 and f2,3 where ft1,t is the forward interest rate from time t 1 to t. (Institute/Faculty of Actuaries Examinations, September 1999) [3+3+4=10] 15. For a particular bond market, zero coupon bonds redeemable at par are priced as follows: bonds redeemable in exactly one year are priced at 97; bonds redeemable in exactly two years are priced at 93; bonds redeemable in exactly three years are priced at 88; and bonds redeemable in exactly four years are priced at 83. (i) Assuming no arbitrage, calculate: (a) the one-year, two-year, three-year and four-year spot interest rates (b) the rate of return from a bond redeemable at par in 4 years’ time that pays a coupon of 4% annually in arrears. (ii) Explain why the four-year spot rate is greater than the rate of return from a bond redeemable at par in exactly 4 years’ time paying a coupon of 4% annually in arrears. (iii) Explain the shape of the yield curve indicated by the spot rates calculated in (i)(a) using the liquidity preference theory, if expectations of future short term interest rates are constant. (Institute/Faculty of Actuaries Examinations, September 2004) [8+2+2=12] 16. Suppose ft,r is the forward rate applicable over the period t to t + r and it is the spot rate over the period 0 to t. The gross redemption yield from a one year bond with a 6% annual coupon is 6% per annum effective; the gross redemption yield from a 2 year bond with a 6% annual coupon is 6.3% per annum effective; and the gross redemption yield from a 3 year bond with a 6% annual coupon is 6.6% per annum effective. All the bonds are redeemed at par and are exactly one year from the next coupon payment. (i) (a) Calculate i1, i2 and i3 assuming no arbitrage. (b) Calculate f0,1, f1,1 and f2,1 assuming no arbitrage. (ii) Explain why the forward rates increase more rapidly with term than the spot rates. (Institute/Faculty of Actuaries Examinations, September 2000) [12] 17. (i) (a) Explain what is meant by the “expectations theory” explanation for the shape of the yield curve. (b) Explain how expectations theory can be modified by both “liquidity preference” and “market segmenta- tion” theories. (ii) Short-term, one-year annual effective interest rates are currently 10%; they are expected to be 9% in one year’s time, 8% in two year’s time, 7% in three years’ time and to remain at that level thereafter indefinitely. (a) If bond yields over all terms to maturity are assumed only to reflect expectations of future short-term interest rates, calculate the gross redemption yields from 1-year, 3-year, 5-year and 10-year zero coupon bonds. (b) Draw a rough plot of the yield curve for zero coupon bonds using the data from part (ii)(a). (Graph paper is not required.) (c) Explain why the gross redemption yield curve for coupon paying bonds will slope down with a less steep gradient than the zero coupon yield curve. (Institute/Faculty of Actuaries Examinations, September 2003) [6+8=14] Page 110 Exercises 2 Sep 27, 2016(9:51) ST334 Actuarial Methods cR.J. Reed 18. In a particular bond market, n-year spot rates per annum can be approximated by the function 0.08 0.04e0.1n. Calculate: (i) The price per unit nominal of a zero coupon bond with term 9 years. (ii) The 4-year forward rate at time 7 years. (iii) The 3-year par yield. (Institute/Faculty of Actuaries Examinations, April 2007) [2+3+3=8] 19. The annual effective forward rate applicable over the period t to t + r is defined as ft,r where t and r are measured in years. We have f0,1 = 4%, f1,1 = 4.25%, f2,1 = 4.5% and f2,2 = 5%. Calculate the following: (i) f3,1 (ii) All possible zero coupon (spot) yields that the above information allows you to calculate. (iii) The gross redemption yield of a 4-year bond redeemable at par with a 3% coupon payable annually in arrears. (iv) Explain why the gross redemption yield from the 4-year bond is lower than the 1-year forward rate up to time 4, f3,1. (Institute/Faculty of Actuaries Examinations, September 2007) [1+4+6+2=13] 20. The n-year spot rate of interest, in, is given by in = a bn for n = 1, 2 and 3, and where a and b are constants. The one-year forward rates applicable at time 0 and at time 1 are 6.1% per annum effective and 6.5% per annum effective respectively. The 4-year par yield is 7% per annum. Stating any assumptions: (i) calculate the values of a and b; (ii) calculate the price per ￡1 nominal at time 0 of a bond which pays annual coupons of 5% in arrear and is redeemed at 103% after 4 years. (Institute/Faculty of Actuaries Examinations, April 2008) [4+5=9] 21. Three bonds, paying annual coupons in arrears of 6% are redeemable at ￡105 per ￡100 nominal and reach their redemption dates in exactly one, two and three years’ time respectively. The price of each bond is ￡103 per ￡100 nominal. (i) Calculate the gross redemption yield of the three-year bond. (ii) Calculate, to 3 decimal places, all possible spot rates implied by the information given, as annual effective rates of interest. (iii) Calculate, to 3 decimal places, all possible forward rates implied by the information given, as annual effective rates of interest. (Institute/Faculty of Actuaries Examinations, September 2008) [3+4+4=11] 22. In a particular bond market, n-year spot rates can be approximated by the function 0.06 0.02e0.1n. (i) Calculate the gross redemption yield for a 3-year bond which pays coupons of 3% annually in arrear, and is redeemed at par. Show all workings. (ii) Calculate the 4-year par yield. (Institute/Faculty of Actuaries Examinations, April 2012) [6+3=9] 23. (i) State the characteristics of a Eurobond. (ii) (a) State the characteristics of a certificate of deposit. (b) Two certificates of deposit issued by a given bank are being traded. A one-month certificate of deposit pro- vides a rate of return of 12% per annum convertible monthly. A two-month certificate of deposit provides a rate of return of 24% per annum convertible monthly. Calculate the forward rate of interest per annum convertible monthly in the second month, assuming no arbitrage. (Institute/Faculty of Actuaries Examinations, September 2012) [4+4=8] 24. Three bonds each paying annual coupons in arrear of 6% and redeemable at ￡103 per ￡100 nominal reach their redemption dates in exactly one, two and three years’ time respectively. The price of each bond is ￡97 per ￡100 nominal. (i) Calculate the gross redemption yield of the 3-year bond. (ii) Calculate the one-year and two-year spot rates implied by the information given. (Institute/Faculty of Actuaries Examinations, April 2013) [3+3=6] 25. The force of interest, (t), is a function of time and at any time t, measured in years, is given by the formula (t) = 0.05 + 0.002t. (i) Calculate the accumulated value of a unit sum of money: (a) accumulated from time t = 0 to time t = 7; (b) accumulated from time t = 0 to time t = 6; (c) accumulated from time t = 6 to time t = 7. (ii) Calculate, using your results from part (i) or otherwise: (a) the seven spot rate of interest per annum from time t = 0 to time t = 7; (b) the six spot rate of interest per annum from time t = 0 to time t = 6; (c) f6,1, where f6,1 is the one year forward rate of interest per annum from time t = 6. (iii) Explain why your an