Question

Gloria borrows 100,000 to be repaid over 30 years. You are given:

(i) Her first payment is X at the end of year 1.

(ii) Her payments increase at the rate of 100 per year for the next 19 years and remain level for the following 10 years.

(iii) The effective rate of interest is 5% per annum.

Calculate X

Ans: 5505

Answer 1

Total PV	100000
Time	30 years
Interest	5%
There are 3 annuities
First level annuity of (X-100) for 20 years
2nd, Increasing annuity of 100 for 20 years
3rd, a level annuity of (X+1900) for the next 10 years
PV of annuity for making the payment
P = PMT x (((1-(1 + r) ^- n)) / r)
Where:
P = the present value of an annuity stream
PMT = the dollar amount of each annuity payment
r = the effective interest rate (also known as the discount rate)
n = the number of periods in which payments will be made
PV of annuity for increasing annuity
P= PV of level annuity as explained above + Arithmatic gradient * ((PV factor of level annuity * (1+r) - n*(1/(1+r)^n))/r)
PV of 3rd annuity at t 20=	= (X+1900)* (((1-(1 + 5%) ^- 10)) / 5%)
PV of 3rd annuity at t 20=	= (X+1900)* 7.721735
PV of 3rd annuity at t 0=	= (X+1900)* 7.721735/(1+5%)^20
PV of 3rd annuity at t 0=	= (X+1900)* 2.9102407
PV of 1st annuity at t0	= (X-100)* (((1-(1 + 5%) ^- 20)) / 5%)
PV of 1st annuity at t0	= (X-100)* 12.4622
PV of 2nd annuity at t0	Arithmatic gradient * ((PV factor of level annuity * (1+r) - n*(1/(1+r)^n))/r)
PV of 2nd annuity at t0	100 * ((12.4622 * (1+5%) - 20*(1/(1+5%)^20))/5%)
PV of 2nd annuity at t0	11,095
Sum of all 3 annuities should be equal to 100000
(X-100)* 12.4622 + 11,095 + (X+1900)* 2.9102407	=100000
12.4622 X-1246.22 + 11,095 + 2.9102 X+5529.38	=100000
12.4622 X + 2.9102 X	=100000-5529.38-11095+1246.22
12.4622 X + 2.9102 X	84621.84
15.3724 X=	84621.84
X=	84621.84/15.3724
X=	5,505
So first payment should be 5,505