Question

In: Finance

An investment opportunity has the following characteristics: payments of $5,000 will be made to you and...

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?

Solutions

Expert Solution

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


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