6.3 Uneven Cash Flows

Calculate the present value of these three cash flows: $100 in Year 1, $200 in Year 2, $300 in Year 3, and an assumed interest rate of 10%.

(100 / 1.1) + (200 / 1.1)^2 + (300 / 1.1) ^3 = $481.59

Calculate the future value of these three cash flows: $100 in Year 1, $200 in Year 2, $300 in Year 3, and an assumed interest rate of 10%.

100(1.1)2 + 200(1.1) + 300 = $641

Suppose you have a $1,000 ordinary annuity earning an 8% return. How many annual end-of-year $150 withdrawals can be made?

I/Y = 8; PMT = 150; PV = -1,000; CPT → N = 9.9 years

How many $100 end-of-year payments are required to accumulate $920 if the discount rate is 9%?

I/Y = 9%; FV = $920; PMT = -$100; CPT → N = 7 years

What rate of return will you earn on an ordinary annuity that requires a $700 deposit today and promises to pay $100 per year at the end of each of the next 10 years?

N = 10; PV = -700; PMT = 100; CPT → I/Y = 7.07%

Suppose you are considering applying for a $2,000 loan that will be repaid with equal end-of-year payments over the next 13 years. If the annual interest rate for the loan is 6%, how much will your payments be?

N = 13; I/Y = 6; PV = -2,000; CPT → PMT = $225.92

At an expected rate of return of 7%, how much must be deposited at the end of each year for the next 15 years to accumulate $3,000?

N = 15; I/Y = 7; FV = +$3,000; CPT → PMT = -$119.38 (ignore sign)

Compute the FV of $2,000 today, five years from today using an interest rate of 12%, compounded quarterly.

N = 5 × 4 = 20; I/Y = 12 / 4 = 3; PV = -$2,000; CPT → FV = $3,612.22

Suppose you have the opportunity to invest $100 at the end of each of the next five years in exchange for $600 at the end of the fifth year. What is the annual rate of return on this investment?

N = 5; FV = $600; PMT = -100; CPT → I/Y = 9.13%

+300 +600 +200 I------I------I------I Compute the present value of this three-year uneven cash flow stream described previously using a 10% rate of return.

PV1: FV = 300; I/Y = 10; N = 1; CPT → PV = PV1 = -272.73 PV2: FV = 600; I/Y = 10; N = 2; CPT → PV = PV2 = -495.87 PV3: FV = 200; I/Y = 10; N = 3; CPT → PV = PV3 = -150.26 PV of cash flow stream = ΣPVindividual = $918.86

A security will make the following payments at the end of the next four years: $100, $100, $400, and $100. Calculate the present value of these cash flows using the concept of the present value of an annuity when the appropriate discount rate is 10%.

The additivity principle tells us that to get the present value of the original series, we can just add the present values of series #1 (a 4-period annuity) and series #2 (a single payment three periods from now). For the annuity: N = 4; PMT = 100; FV = 0, I/Y = 10; CPT → PV = -$316.99 For the single payment: N = 3; PMT = 0; FV = 300; I/Y = 10; CPT → PV = -$225.39 The sum of these two values is 316.99 + 225.39 = $542.38. OR (100 / 1.1) + (100 / 1.1^2) + (400 / 1.1^3) + (100 / 1.1^4) = 542.38

Suppose you must make five annual $1,000 payments, the first one starting at the beginning of Year 4 (end of Year 3). To accumulate the money to make these payments, you want to make three equal payments into an investment account, the first to be made one year from today. Assuming a 10% rate of return, what is the amount of these three payments?

The first step in this type of problem is to determine the amount of money that must be available at the beginning of Year 4 (t = 3) in order to satisfy the payment requirements. This amount is the PV of a 5-year annuity due at the beginning of Year 4 (end of Year 3). To determine this amount, set your calculator to the BGN mode, enter the relevant data, and compute PV. N = 5; I/Y = 10; PMT = -1,000; CPT → PV = PV3 = $4,169.87 PV3 becomes the FV that you need three years from today from your three equal end-of-year deposits. To determine the amount of the three payments necessary to meet this funding requirement, be sure that your calculator is in the END mode, input the relevant data, and compute PMT. N = 3; I/Y = 10; FV = -4,169.87; CPT → PMT = $1,259.78 The second part of this problem is an ordinary annuity. If you changed your calculator to BGN mode and failed to put it back in the END mode, you will get a PMT of $1,145, which is incorrect.

+300 +600 +200 I------I------I------I Using a rate of return of 10%, compute the future value of the three-year uneven cash flow stream described above at the end of the third year.

This three-year cash flow series is not an annuity since the cash flows are different every year. In essence, this series of uneven cash flows is nothing more than a stream of annual single sum cash flows. Thus, to find the PV or FV of this cash flow stream, all we need to do is sum the PVs or FVs of the individual cash flows. The FV for the cash flow stream is determined by first computing the FV of each individual cash flow, then summing the FVs of the individual cash flows. FV1: PV = -300; I/Y = 10; N = 2; CPT → FV = FV1 = 363 FV2: PV = -600; I/Y = 10; N = 1; CPT → FV = FV2 = 660 FV3: PV = -200; I/Y = 10; N = 0; CPT → FV = FV3 = 200 FV of cash flow stream = ΣFVindividual = 1,223

More Frequent Compounding Does What to FV & PV

increase in the frequency of compounding increases the effective rate of interest, it also increases the FV of a given cash flow and decreases the PV of a given cash flow.

cash flow additivity principle

refers to the fact that present value of any stream of cash flows equals the sum of the present values of the cash flows.