Question

(Related to Checkpoint​ 6.6)  

​(Present value of annuities and complex cash flows​)

You are given three investment alternatives to analyze. The cash flows from these three investments are as​ follows:

Investment Alternatives
End of Year	A		B		C
1	​$	15,000			​$	15,000
2		15,000
3		15,000
4		15,000
5		15,000	​$	15,000
6				15,000		75,000
7				15,000
8				15,000
9				15,000
10				15,000		15,000

Assuming an annual discount rate of 23 ​percent, find the present value of each investment.

a.  What is the present value of investment A at an annual discount rate of 23 percent?

​ ​(Round to the nearest​ cent.)

b.  What is the present value of investment B at an annual discount rate of

23 ​percent?  ​(Round to the nearest​ cent.)

c.  What is the present value of investment C at an annual discount rate of 23 ​percent? 

​(Round to the nearest​ cent.)

Answer 1

Answer (a):

Discount rate = 23%

Annual cash flows (year 1 to year 5) = $15,000

Present value = Annual cash flows * PV of $1 annuity for 5 years at 23% rate

= 15000 * (1 - 1/ (1 + 23%) ⁵) / 23%

= 42052.09467

Present value of investment A = $42,052.09

Answer (b):

There are uniform annual cash flows of $15,000 for 6 years starting from year 5.

Hence:

(i) First we calculate present value at the end of year 4:

Present value at the end of year 4 = = Annual cash flows * PV of $1 annuity for 6 years at 23% rate

= 15000 * (1 - 1/ (1 + 23%) ⁶) / 23%

= $46,383.8168

(ii) Now we calculated present value of this amount:

Present value = FV * PV of $1 for 4 years at 23% rate

= 46383.8168 / (1 + 23%) ⁴

= $20,264.97

Present value of investment B = $20,264.97

Answer (c):

Cash flows are:

Year 1 = $15,000

Year 6 = $75,000

Year 10 = $15,000

Present value = Year 1 cash flow * PV of $1 for 1 year at 23% rate + Year 6 cash flow * PV of $1 for 6 years at 23% rate + Year 10 cash flow * PV of $1 for 10 years at 23% rate

= 15000 / (1 + 23%) + 75000 / (1 + 23%) ⁶ + 15000 / (1 + 23%) ¹⁰

= $35,746.25

Present value of investment C= $35,746.25