1.1.5 The Future Value and Present Value of a Series of Uneven Cash Flows

Question 1

Series of Uneven Cash Flows

Answer 1

A series of uneven cash flows means that the cash flow stream is uneven over many time periods. There is no single formula available to compute the present or future value of a series of uneven cash flows.

Question 2

How to calculate Future Value and Present Value of a Series of Uneven Cash Flows?

Answer 2

Using PV + FV
When we have unequal cash flows, we must first find the present value of each individual cash flow and then the sum of the respective present values. (This is usually accomplished with the help of a spreadsheet.)

Once we know the present value of the cash flows, we can easily apply time-value equivalence by using the formula to calculate the future value of a single sum of money (LOS a).

Question 3

Example
John wants to pay off his student loan in three annual installments: $2,000, $4,000 and $6,000, respectively, in the next three years. How much should John deposit into his bank account today if he wants to use the account balance to pay off the loan? Assume that the bank pays 8% interest, compounded annually.

Answer 3

Year 1: $2000
Year 2: $4000
Year 3: $6000

FV = 2000(1+0.08)^1 + 4000(1+0.08)^2 + 6000(1+0.08)^3
= 1852 + 3429 + 4763
= $10044

Question 4

1. An investment promises to pay $1,000 one year from today, $1,500 two years from today and $2,000 three years from today. If the required rate of return is 11% compounded annually, what is the value of the investment today?

Answer 4

Correct Answer: $3,580.72
We must calculate the present value of the 3 investments separately and then add these amounts.
As usual, we use the formula PV = FV / (1 + r)N
1,000/(1.11)1+ 1,500/(1.11)2 + 2,000/(1.11)3 = 900.90 + 1217.4 + 1462.3 = $3,580.71

Question 5

2. An individual deposits $1,000 today, $1,200 one year from today, and $1,500 two years from today into an interest-earning account. The deposits earn 10% compounded annually. Find the total accumulated amount in his account three years from today.

Answer 5

Correct Answer: $4,433.00
This is not an annuity, since the cash flows change every year. This uneven cash flow is a stream of annual single cash flows. Find the future value of each cash flow and then add them up. Notice that the payments are made at the beginning of each year. Notice that PMT1 earns interest for 3 years, PMT2 earns interest for 2 years, and PMT3 earns interest for 1 year.
We need to calculate the future value of the 3 investments separately and then add them together. We use the future value formula FVN = PV (1 + r)N
1,000 (1.1)3 + 1,200 (1.1)2 + 1,500 (1.1)1= 1,331 + 1,452 + 1,650 = 4,433.00
This is probably the best method to use, since the NPV functions will only give present value, not future value.

Question 6

2. An individual deposits $1,000 today, $1,200 one year from today, and $1,500 two years from today into an interest-earning account. The deposits earn 10% compounded annually. Find the total accumulated amount in his account three years from today.

Answer 6

Correct Answer: $4,433.00
This is not an annuity, since the cash flows change every year. This uneven cash flow is a stream of annual single cash flows. Find the future value of each cash flow and then add them up. Notice that the payments are made at the beginning of each year. Notice that PMT1 earns interest for 3 years, PMT2 earns interest for 2 years, and PMT3 earns interest for 1 year.
We need to calculate the future value of the 3 investments separately and then add them together. We use the future value formula FVN = PV (1 + r)N
1,000 (1.1)3 + 1,200 (1.1)2 + 1,500 (1.1)1= 1,331 + 1,452 + 1,650 = 4,433.00
This is probably the best method to use, since the NPV functions will only give present value, not future value.

Question 7

3. The following end-of-month payments of $400, $700, and $300, respectively, are due. Given a stated annual interest rate of 3.60 percent, the minimum amount of money needed in an account today to satisfy these future payments is closest to ______.
   A. $1,391
   B. $1,327
   C. $1,368

Answer 7

Correct Answer: A The monthly interest rate is 3.6/12 = 0.3. The present value is $1,391.48 = $400.00/(1 + 0.3%) + $700.00/(1 + 0.3%)2 + $300.00/(1 + 0.3%)3. Using a financial calculator: CF1 = 400, CF2 = 700, CF3 = 300, I= 0.3 Compute PV, PV = 1,391.48