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MATH 177– Theory of Interest and Applications

September 2022

Question  1

On January 1, a fund is worth 100, 000. On June 1, the value has increased to 120, 000 and then 30, 000 of new principal is deposited. On October 1, the value has declined to 130, 000 and then 50, 000 is withdrawn. On January 1 of the following year, the fund is again worth 100, 000. Calculate the dollar- weighted rate of return using the simple interest approximation.         Total:

10 marks

Question  2

Let rt  is the spot rate for a t−year maturity zero coupon bond.

(i) Find an expression for f[k 1,k]  in terms of the rt ’s.                 (5 marks)

(ii)  Show that (1 + f[0,1] )(1 + f[1,2] ) ··· (1 + f[k 1,k]) = (1 + rk )k     (5 marks)

(iii)  Show that f[ k − 1,k] > 0 and f[ k − 1,k] < 0.            (5 marks)

(iv)  Show that if rk  > rk 1 , then f[ k − 1,k] > rk 0.                       (5 marks)

Total: 20 marks

Question  3

the following term structure is given as effective annual rates of interest on zero coupon bonds:

1–year maturity: 6%, 2–year maturity: 7%, 3–year maturity: 11%

(i) Find (1) the 1–year forward effective annual interest rate for a 1–year period, f[1,2]  and (2) the 2–year forward effective annual interest rate for a 1–year period, f[2,3] .                                                      (10 marks)

(ii) The effective annual rate of interest for a 4–year zero coupon bond is r4 .  Find the minimum value of r4  needed so that f[3,4]  ≥ f[2,3] , where f[3,4]  is the 3–year forward effective annual interest rate for a 1–year period and f[2,3] , is found in part (i)                                       (5 marks)

Total: 15 marks

Question  4

Lability payments of 100 each are due to be paid in 2, 4 and 6 years from now. Asset cashflow consists of A1 in 1 year and A5 in 5 years. The yield for all pay- ments is 10%. an attempt is made to have the asset cash flow immunize the liability cashflow by matching present value and duration.                    Total:

20 marks