04/19/23 Richard MacMinn 1

The Time Value of Money

04/19/23 Richard MacMinn 2

Objectives

To stress that the time value of money is an integral part of financial decision making.

To discuss compounding techniquesTo discuss discounting techniques

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Future Value

Suppose you place $100 in a savings account that pays 7.2% interest, compounded annually. How does the account grow?

fv1 = 100 (1.072)

fv2 = 100 (1.072)2

fvT = 100 (1.072)T

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Future Value

The future value of a one dollar stream of savings for n years, i.e., one dollar saved at dates t = 1, 2, . . . , T, is

2 1

1

0

1 (1 ) (1 ) (1 )

(1 )

TT

Tt

t

fv r r r

r

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Future Value

If a dollar is compounded m times per period then the future value may be expressed as

, 1mT

m Tr

fvm

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Future Value

As m approaches infinity, the factor

where e = 2.71828Given continuous compounding, the future

value of a dollar n periods hence is

1m rr

m e

r TTfv e

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Present Value

A dollar today is worth more than a dollar tomorrow!

Present value is the inverse of future value.

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Present Value

The present value of a dollar received in one year is

The present value of a dollar received in T years is

11 r

1

1 Tr

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Present Value

The present value of a stream of one dollar returns received at dates t = 1, 2, . . . , T is

2 3

1

1 1 1 1(1 ) (1 ) (1 ) (1 )

1

(1 )

T

T

tt

pvr r r r

r

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Annuity

An annuity is a stream of equal dollar returns for a specified number of periods, e.g., pension funds, insurance contracts, sinking funds.

The value of a T year annuity that pays one dollar per year is

1

11

1 (1 )

(1 )

T T

tt

rrr

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Annuity

Loans paid off in equal installments over a fixed period are annuities, or equivalently, amortized loans.

The value L of the of the amortized loan is 1

1(1 )mTr

mrm

L p

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Perpetuity

A perpetuity is an annuity payable indefinitely.

The present value of an indefinitely long one dollar stream is

1

1 1

(1 )tt rr

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Growing Perpetuity

A growing perpetuity is a perpetuity with a payment that grows at a constant rate g.

The payment stream may be specified as 1, (1 + g), (1 + g)2, (1 + g)3, . . . , and its value as

1

1

(1 ) 1

(1 )

t

tt

gr gr

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Annuity Again

The present value of an annuity is

This is the difference between a perpetuity that begins paying at t = 1 and one that begins paying at t = T + 1.

1

1 1 1

(1 ) (1 )

T

t Tt rr r r

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Bond Value

A bond that pays a level interest stream and then the principal at maturity is a combination of an annuity and a single payment.

Bonds also typically pay interest semiannually and the bond value B is

1 (1 ) (1 )

crmTm

t mTr rt m m

F FB

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Links

QUICKEN FINANCIAL PLANNER Extensive planning software http://www.quicken.com/

RETIRE SECUREPrice Waterhouse Cooper's retirement aid http://www.pwcglobal.com/

S&P PERSONAL WEALTH Standard & Poor's asset allocation modelhttp://www.personalwealth.com/

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Links

T. ROWE PRICE Good, simple ''what if'' program http://www.troweprice.com

VANGUARD Solid online retirement planning http://www.vanguard.com