dP dt Conversely
P
t= exp (
−
∫
t0
F
tdu )
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 )]
T−t= 1.
• f
t,T ,kdenotes 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]
Page 104 Exercises 6.2 Nov 4, 2014(9:33) ST334 Actuarial Methods c⃝R.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,1applying 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 δ(t) is a function of time and at any time, measured in years, is given by the formula δ(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 f8,1is 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]
Interest Rate Problems Nov 4, 2014(9:33) Exercises 6.2 Page 105 12. The n-year spot rate of interest, ynis 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 ν(1), ν(2) and ν(3) that the market is using to discount payments due in 1, 2 and 3 years respectively.
(iii) Calculate f0,1, f1,2and f2,3where ft−1,tis 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,ris the forward rate applicable over the period t to t + r and itis 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, i2and i3assuming no arbitrage.
(b) Calculate f0,1, f1,1and f2,1assuming 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 106 Exercises 6.2 Nov 4, 2014(9:33) ST334 Actuarial Methods c
⃝
R.J. Reed 18. In a particular bond market, n-year spot rates per annum can be approximated by the function 0.08− 0.04e−0.1n.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,rwhere 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− 0.02e−0.1n.
(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 propro-vides 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, δ(t), is a function of time and at any time t, measured in years, is given by the formula δ(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 f6,1is the one year forward rate of interest per annum from time t = 6.
(iii) Explain why your answer to part (ii)(c) is higher than your answer to part (ii)(a).
(iv) Calculate the present value of an annuity that is paid continuously at a rate of 30e−0.01t+0.001t2units per annum from t = 3 to t = 10. (Institute/Faculty of Actuaries Examinations, September 2013)[5+3+2+5=15]
Interest Rate Problems Nov 4, 2014(9:33) Section 6.3 Page 107