1. Problem 5
At a constant interest rate of 15%, compounded annually, what is the present value of an income
stream paying $50 next quarter and growing at 2% per quarter until the end of the third year?
From that point on it grows at 1% per quarter indefinitely.
Solution
The income grows by 2% for first 12 quarters after which it grows perpetually at 1% per quarter.
Interest rate, r = 15% p.a. = 15/4 = 3.75% quarterly
The income stream looks as shown below
Quarter
Income(I)
Remarks
1
50.00
2
51.00
2% increase
3
52.02
2% increase
4
53.06
2% increase
5
54.12
2% increase
6
55.20
2% increase
7
56.31
2% increase
8
57.43

2. 2% increase
9
58.58
2% increase
10
59.75
2% increase
11
60.95
2% increase
12
62.17
2% increase
13
62.79
1% increase
Present Value,PV = I1/(1+r) + I2/(1+r)2 + I3/(1+r)3+ ........ + I12/(1+r)12 + I13/(1+r)13 + ....
constant growth
So PV = I1/(1+r) + I2/(1+r)2+ I2/(1+r)2+ .........+ I12/(1+r)12+ I13/((1+r)13*(r-g)) {Infinite
GP series formula used} where g = 1% or 0.01
So PV = 48.19 + 47.38 + 46.58 + 45.80 + 45.02 + 44.26+ 43.52+ 42.78+
42.06+41.35+40.65+39.97+1414.87
So Present Value, PV = $1942.43
Quarter
Income(I)
Remarks
1
50.00
2
51.00
2% increase
3
52.02
2% increase
4
53.06

3. 2% increase
5
54.12
2% increase
6
55.20
2% increase
7
56.31
2% increase
8
57.43
2% increase
9
58.58
2% increase
10
59.75
2% increase
11
60.95
2% increase
12
62.17
2% increase
13
62.79
1% increase