Question

An investment opportunity has the following characteristics: payments of $5,000 will be made to you and invested into a fund at the end of each year, for the next 30 years. These payments will earn a 9% effective annual rate, and the interest payments (paid at the end of each year) are reinvested into a second account earning a 5% effective annual rate. What would the purchase price of this investment opportunity be if it had an annual yield of 8% over the 30-year life of the investment?

Answer 1

We use excel to calculate the payment and the interest component

Year	Payment	Interest component formula	Interest component	Future value factor	Future value of interest (after 30 years)
1	5000
2	5000	(Year-1)*5000*0.09	450	1/(1.05^(30-Year))	1764.058112
3	5000	(Year-1)*5000*0.09	900	1/(1.05^(30-Year))	3360.11069
4	5000	(Year-1)*5000*0.09	1350	1/(1.05^(30-Year))	4800.158129
5	5000	(Year-1)*5000*0.09	1800	1/(1.05^(30-Year))	6095.438894
6	5000	(Year-1)*5000*0.09	2250	1/(1.05^(30-Year))	7256.474873
7	5000	(Year-1)*5000*0.09	2700	1/(1.05^(30-Year))	8293.114141
8	5000	(Year-1)*5000*0.09	3150	1/(1.05^(30-Year))	9214.571268
9	5000	(Year-1)*5000*0.09	3600	1/(1.05^(30-Year))	10029.46533
10	5000	(Year-1)*5000*0.09	4050	1/(1.05^(30-Year))	10745.85571
11	5000	(Year-1)*5000*0.09	4500	1/(1.05^(30-Year))	11371.27588
12	5000	(Year-1)*5000*0.09	4950	1/(1.05^(30-Year))	11912.76521
13	5000	(Year-1)*5000*0.09	5400	1/(1.05^(30-Year))	12376.89892
14	5000	(Year-1)*5000*0.09	5850	1/(1.05^(30-Year))	12769.81634
15	5000	(Year-1)*5000*0.09	6300	1/(1.05^(30-Year))	13097.24753
16	5000	(Year-1)*5000*0.09	6750	1/(1.05^(30-Year))	13364.5383
17	5000	(Year-1)*5000*0.09	7200	1/(1.05^(30-Year))	13576.67382
18	5000	(Year-1)*5000*0.09	7650	1/(1.05^(30-Year))	13738.30089
19	5000	(Year-1)*5000*0.09	8100	1/(1.05^(30-Year))	13853.7488
20	5000	(Year-1)*5000*0.09	8550	1/(1.05^(30-Year))	13927.04906
21	5000	(Year-1)*5000*0.09	9000	1/(1.05^(30-Year))	13961.95394
22	5000	(Year-1)*5000*0.09	9450	1/(1.05^(30-Year))	13961.95394
23	5000	(Year-1)*5000*0.09	9900	1/(1.05^(30-Year))	13930.29418
24	5000	(Year-1)*5000*0.09	10350	1/(1.05^(30-Year))	13869.98988
25	5000	(Year-1)*5000*0.09	10800	1/(1.05^(30-Year))	13783.84088
26	5000	(Year-1)*5000*0.09	11250	1/(1.05^(30-Year))	13674.44531
27	5000	(Year-1)*5000*0.09	11700	1/(1.05^(30-Year))	13544.2125
28	5000	(Year-1)*5000*0.09	12150	1/(1.05^(30-Year))	13395.375
29	5000	(Year-1)*5000*0.09	12600	1/(1.05^(30-Year))	13230
30	5000	(Year-1)*5000*0.09	13050	1/(1.05^(30-Year))	13050
					327949.6275

Here, the future value of interests is calculated by compounding the interest payments by 5% for the remaining years until the 30th year

Now, we have 2 components:

1. The annuity paying 5000 at each year-end

2. The interest component, the value of which is 327949.62 at the end of year 30

PV of annuity paying $5000 at year-end, for 30 years

Using PV function in excel

PV(0.08,30,5000) =

$56,288.92

PV of interest component, the value of which is 327,949.62 at the end of year 30

= 327949.62 / (1.08^30) = $32,590.75

Hence the total PV of the investment opportunity = $56,288.92 + $32590.75 = $88,879.67

Hence the purchase price of this investment opportunity = $88,879.67