## December 11, 2011

### Time Value of Money Calculations

This topic explains the various concepts used to calculate future values or present values of a series of cash flows that result from engineering decisions to buy new equipment or replace old equipments.

Introduction

In one has money in his hand, he can invest it in a bank deposit and after one year he gets back his principal amount and in addition some interest. Therefore \$1 today, deposited in a bank at 3% interest per annum will become \$1.03 after one year. This is the concept of time value of money. Over time, money increases due to accumulation of interest. Compound interest formula A = P(1 +i)n represents the future value of money.
P = A (1+i)-n represents the present value of money received after n years.

This topic explains the various concepts used to calculate future values or present values of a series of cash flows that result from engineering decisions to buy new equipment or replace old equipments.

Time Value can be present value of a series of future cash flows or future value of series of future cash flows.

Single Payment Cashflow

For a single payment now made, one can calculate a future value. This is done by compounding using the formula A = P(1 +i)n

For a payment to be received in the future, one can calculate the present value. This is done by using discounting formula P = A (1+i)-n

Uniform Periodic Payments

For uniform periodic payments, one can calculate the present value or future value. The payments are assumed to be made at the end of period.

Compounding of uniform series of cash payments

S = R [(1+i)n - 1]/i

where S = Compound amount or future amount at the end of n periods
R = Periodic cash payment at the end of the period
i = rate of interest or required rate of return
n = number of periods for payments are made

Discounting of uniform series of cash payments

P = R [(1+i)n - 1]/[i 1+i)n ]

Where

P = Present Value

R = Uniform series of periodic payments

i = interest rate

n = number of periods of payments

These time value formulas are expressed in factors

A = P*Single payment future worth factor =P*Spfwf

P = A* Single payment present worth factor = A*Sppwf

S = R* Uniform series future worth factor = R*Usfwf

P = R*Unform series present worth factor = R*Uspwf

Two More Factors

Sinking Fund Deposit Factor

Sfdf = 1/Usfwf

Sinking fund is fund accumulated with periodic payments for incurring a lumpsum expenditure at the end of a long period. Sfdf gives the amount to be deposited at the end of each period for n period to accumulate one dollar at the end n periods.

Capital Recovery Factor

Crf = 1/Uspwf

Capital recovery factor gives the uniform payment to be received by you at the end period of n years to get recover back the investment you made today.

The factor tables are available and factors depend on interest rate i and term n.

Factors for a required rate of return of 10% and 5 years term.

Spfwf - 1.6105

Sppwf - .62092

Usfwf - 6.1051

Uspwf - 3.7908

Sfdf - 0.16380

Crf - 0.26380

References

Engineering Economics, 4th Edition, James L. Riggs, David D. Bedworth, and Sabah U. Randhawa, McGraw Hill, New York, 1996