Time Value of Money

Outline

Meaning of Time Value Concept of Future Value and Compounding (FV) Concept of Present Value and Discounting (PV) Frequency of Compounding Present Value versus Future Value Determining the Interest rate (r) Determining the Time Period (n) Future Value and Present Value of Multiple Cash Flows Annuities and Perpetuities

Time Value of Money

Basic Problem:– How to determine value today of cash flows that are expected in the

future? Time value of money refers to the fact that a dollar in hand today

is worth more than a dollar promised at some time in the future Which would you rather have -- $1,000 today $1,000 today or $1,000 in 5 $1,000 in 5

years?years? Obviously, $1,000 today$1,000 today. Money received sooner rather than later allows one to use the

funds for investment or consumption purposes. This concept is referred to as the TIME VALUE OF MONEYTIME VALUE OF MONEY!!

TIMETIME allows one the opportunity to postpone consumption and earn INTERESTINTEREST.

Future Value and Compounding

Future value refers to the amount of money an investment will grow to over some length of time at some given interest rate

To determine the future value of a single cash flows, we need: present value of the cash flow (PV) interest rate (r), and time period (n)

FVn = PV0 × (1 + r)n

Future Value Interest Factor at ‘r’ rate of interest for ‘n’ time periods

Examples on computation of future value of a single cash flow

If you invested $2,000 today in an account that $2,000 today in an account that pays 6pays 6% interest, with interest compounded annually, how much will be in the account at the end of two years if there are no withdrawals?

Future Value (Graphic)Future Value (Graphic)

0 1 2

$2,000$2,000

FVFV

6%

FVFV11 = PVPV (1+r)n

= $2,000$2,000 (1.06)2

= $2,247.20$2,247.20

Future Value (Formula)Future Value (Formula)

FV = future value, a value at some future point in timePV = present value, a value today which is usually designated as time 0r = rate of interest per compounding period n = number of compounding periods

Calculator Keystrokes: 1.06 (2nd yx) 2 x 2000 =

John wants to know how large his $5,000$5,000 deposit will become at an annual compound interest rate of 8% at the end of 5 years5 years.

Future Value (Example)Future Value (Example)

0 1 2 3 4 55

$5,000$5,000

FVFV55

8%

Future Value SolutionFuture Value Solution

Calculation based on general formula: FVFVnn = PV (1+r)n

FVFV55 = $5,000 (1+ 0.08)5

= $7,346.64$7,346.64

Calculator keystrokes: 1.08 2nd yx x 5000 =

Present Value and Discounting

The current value of future cash flows discounted at the appropriate discount rate over some length of time period

Discounting is the process of translating a future value or a set of future cash flows into a present value.

To compute present value of a single cash flow, we need: Future value of the cash flow (FV) Interest rate (r) and Time Period (n)

PV0 = FVn / (1 + r)n

PVIF (r,n) Examples

Assume that you need to have exactly $4,000 saved 10 years from now. How much must you deposit today in an account that pays 6% interest, compounded annually, so that you reach your goal of $4,000?

0 5 5 10

$4,000$4,000

6%

PVPV00

Present Value (Graphic)Present Value (Graphic)

PV0 = FV / (1+r)10

= $4,000 / (1.06)10

= $2,233.58

Present Value (Formula)Present Value (Formula)

0 5 5 10

$4,000$4,000

6%

PVPV00

Joann needs to know how large of a deposit to make today so that the money will grow to $2,500 in 5 years. Assume today’s deposit will grow at a compound rate of 4% annually.

Present Value ExamplePresent Value Example

0 1 2 3 4 55

$2,500$2,500PVPV00

4%

Calculation based on general formula: PVPV00 = FVFVnn / (1+r)n

PVPV00 = $2,500/(1.04)$2,500/(1.04)55

= $2,054.81

Calculator keystrokes: 1.04 2nd yx 5 = 2nd 1/x X 2500 =

Present Value SolutionPresent Value Solution

General Formula:

FVn = PVPV00(1 + [r/m])mn

n: Number of Years

m: Compounding Periods per Year

r: Annual Interest Rate

FVn,m: FV at the end of Year n

PVPV00: PV of the Cash Flow today

Frequency of CompoundingFrequency of Compounding

Frequency of Compounding Example

Suppose you deposit $1,000 in an account that pays 12% interest, compounded quarterly. How much will be in the account after eight years if there are no withdrawals?

PV = $1,000

r = 12%/4 = 3% per quarter

n = 8 x 4 = 32 quarters

Solution based on formula:

FV= PV (1 + r)n

= 1,000(1.03)32

= 2,575.10

Calculator Keystrokes:

1.03 2nd yx 32 X 1000 =

Present Value versus Future Value

Present value factors are reciprocals of future value factors

Interest rates and future value are positively related Interest rates and present value are negatively related Time period and future value are positively related Time period and present value are negatively related

Determining the Interest Rate (r)

At what rate of interest should we invest our money today to get a desired amount of money after a certain number of years?

Essentially, we are trying to determine the interest rate given present value (PV), future value (FV), and time period (n)

Examples The rate which money can be doubled/tripled

Determining the Time Period (n)

For how long should we invest money today to get a desired amount of money in the future at a given rate of interest

Determining the time period (n) for which a current amount (PV) needs to be invested to get a certain future value (FV) given a rate of interest (r).

Examples The time period needed to double/triple our current

investment

Future Value of Multiple Uneven Cash Flows

Compute the future value of each single cash flow using future value formula and add them up over all the cash flows

Example

Present Value of Multiple Uneven Cash Flows

Compute the present value of each single cash flow using present value formula and add them over all the cash flows

Examples

Annuities

A series of level/even/equal sized cash flows that occur at the end of each time period for a fixed time period

Examples of Annuities: Car Loans House Mortgages Insurance Policies Some Lotteries Retirement Money

Present Value of an Annuity– Examples

Computing Cash Flow per period in annuity– Examples

Perpetuities

A series of level/even/equal sized cash flows that occur at the end of each period for an infinite time period

Examples of Perpetuities: Consoles issued by British Government Preferred Stock

Present Value of a Perpetuity

Effective Annual Rate

Compounding other than annual