1.  (Interest rate)

(a)  [3 marks].  Find the annual rate of simple interest that over 5 years accrues the

same interest as i(4)  = 3% p.a. over the same period.

(b)  [3 marks]. Find the semi-annual effective interest rate that pays f150 in interest

over f1,000 invested for 4 years.

2.  (Annuities)

(a)  [3 marks].  Compute the present value of an annuity immediate that pays f100

at the end of each month for 5 years at a rate i = 7.5% p.a. effective.

(b)  [3  marks].   Compute the accumulated value after  10 years of an  annuity due

that pays f10,000 per year in equal quarterly installments at a rate i = 7.5% p.a. effective.

(c)  [3 marks]. Prove that a25 i  = a5 i + v5 a20 i  with v = 1/(1 + i).

3.  (Bonds)

(a)  [3 marks]. A Zero Coupon Bond with face value f100 and maturity in 4 years is

sold for f85. Find its (gross) yield.

(b)  [3 marks].  The same bond as in (a) pays capital gains tax at 20%.  What is its

net yield?

(c)  [3 marks]. Find the price of a bond whose face value is f100, it pays semi-annual coupons at a rate of 1.5% p.a., it has maturity in 3 years and produces a yield of 3% p.a. effective. The bond is redeemable at par.

4.  (Investment funds) An  investment fund contains  f100,000 at time zero.   After six months its value has grown to f105,000 and further f10,000 are added.  At the end of the year the fund is worth f125,000.  In the first six months of the second year its value grows to f126,000 and at the end of the year it is worth  f128,000.  At that point f50,000 are withdrawn. At the end of the following six months the fund’s value is f80,000 and the fund is liquidated.

(a)  [3 marks].  In which semester did the fund have its best performance? (To avoid

any confusion we notice that a ‘semester’ is a 6-month period.)

(b)  [3 marks]. Compute the time-weighted rate of return per semester.

5. An individual borrows f15,000 to be repaid in 10 years with monthly payments at the

end of each month. The initial interest rate applied to the loan is 3% p.a. effective.

(a)  [4 marks]. Find the monthly repayment P1 .

After 5 years the terms of the loan are modified:  the interest rate increases to 4% p.a. effective (the date of the final payment instead remains unchanged).

(b)  [5 marks]. Find the revised monthly payment P2 .

The capital borrowed is invested in the stock market and produces returns according to a time-dependent force of interest:

, 0.01,                   0 < t < 3,

6(t) = ．   0.01 t + 0.02,     3 < t < 7,

( 0.007 t - 0.02,   7 < t < 10,

where time t is measured in years.

(c)  [4 marks].  Compute the variable annual effective rate of interest equivalent to the force of interest above.  In particular find i1  for the first 3 years, i2  from the end of year 3 until year 7, and i3  for the final 3 years.

(d)  [2 marks]. In which periods the rate of return on the investment in stock exceeds the interest rate paid on the loan?  (Notice that period 1 is t e [0, 3), period 2 is t e [3, 7) and period 3 is t e [7, 10].)

(e)  [5 marks].  Compute the value at time t = 10 years of all the loan repayments

(allow for the change of interest rate on the loan from (b) above).  Compute the accumulated value of the investment in stock. What is the profit/loss made by the investor?

6. Assume the annual effective interest rate is i = 2%.  An individual wants to invest a capital of f100,000 and is contemplating two projects:

Project A. She buys a property for exactly f100,000 at time t = 0 that will generate income from rent.

(a)  [4 marks].  If rent is paid monthly in advance and each payment is f700, how

long will it take before the present value of rental payments exceeds the initial cost of purchasing the property?

(b)  [6 marks].  To be more realistic assume that rent increases every three years at

a rate of 1% effective per annum compound, and taxes are paid at 20% on rental income. Moreover, maintenance work for the property will cost f1,500 to be paid at the end of every 5 years (and these are not subject to tax relief).   If rental payments stop after 15 years, what is the net present value (at time t = 0) of all cash flows for 15 years? (Include maintenance costs paid at t = 15 years.)

(c)  [5 marks].  With the same assumptions as in (b), the property is sold after 15 years for f130,000. What is the yield of this project?

Project  B.  She  invests  f100,000  in  bonds  with  maturity  in  20  years,  face  value F =f100 and paying semiannual coupons at a nominal rate D = 2% p.a.  The bonds are redeemed at par and, for simplicity, we assume no taxes are paid on bonds.

(d)  [4 marks].  Assume the bonds produce a yield of 5% p.a. effective.  How many bonds will the investor be able to purchase?  (We assume the bonds are infinitely divisible, i.e. the investor can purchase non-integer amounts of it)

(e)  [4 marks]. Assume the bond price is f50 each. What is the yield of project B?

(f)  [2 marks].  If the investor were to choose between project A and B based on the

respective yields obtained in (c) and (e), which should she choose and why?

7.  On 1st of January 2018 an individual knows that she will retire in 10 years and will receive a state pension at a rate of f10,000 per year, paid in equal monthly instalments at the beginning of each month, and starting on 1st of January 2028.  The annual effective rate is 3% p.a.

(a)  [5 marks]. Assuming that the pension is paid for 20 years (i.e. the last payment is

on 1st of December 2047), compute the value on 1st of January 2018 of the state pension.

In order to increase the overall retirement income the individual also decides to purchase an annuity on 1st of January 2018, as a form of private pension.

(b)  [2 marks].  Without lengthy calculations, explain how much the investor would

need to pay, on 1st January 2018, towards the private pension, if she wants to double her future retirement income until 1st December 2047 (retirement income includes the state pension from above).

Let us assume that the annuity purchased by the individual will pay at a rate of f8,000 per year, in equal monthly instalments paid at the beginning of each month, starting from 1st of January 2028 for 20 years.  (We will refer to this annuity as ‘retirement-annuity’ for the sake of clarity below.)

(c)  [5 marks]. Assume the individual purchases the retirement-annuity by equal quar- terly payments, made at the beginning of each quarter for 10 years and starting on 1st of January 2018. What is the amount C of each payment?

(d)  [6  marks].   Assume the individual  purchases the  retirement-annuity  by annual instalments of f5,000 at the end of each year for 10 years, plus two lump payments of equal amount L, made on 1st of January 2023 and 1st of January 2028. Compute the amount L of the lump payments.

Assume the individual has opted for quarterly payments of amount C as in (c). However, after the first payment she decides to increase the amount of contribution in order to earn a higher retirement income.  Each subsequent payment increases at a rate of 1% effective per quarter.  This results in a retirement-annuity rate P per annum which is larger than the original f8,000 and which will be paid from 1st January 2028. (Notice: if you have not answered (c) you may still derive all equations formally and gain some marks for the reasoning.)

(e)  [7 marks]. What is the new value of P?