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Time Value of Money

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Financial Management by Paresh Shah Biztantra NextBack

Time Value of Money

Prof. (Dr.) Paresh ShahFCMA., Ph.D. (Finance),

F.D.P. (IIM-Ahd.)

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Introduction to Financial

Mathematics

Financial mathematics deals with the problem of

investing money, or capital. If a company (or an

individual investor) puts some capital into an

investment, it (or he/she) will want a financial return

for it.

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Time lines show timing of cash flows.

CF0 CF1 CF3CF2

0 1 2 3i%

Tick marks at ends of periods, so Time 0is today; Time 1 is the end of Period 1;or the beginning of Period 2.

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Time line for a Rs.100 lump sumdue at the end of Year 2.

100

0 1 2 Yeari%

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Notation Subscripts

In both compounding and discounting, subscripts

are used to show the period of time to which a

sum of money relates.

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Interest

Interest is the amount of money, which a capital

investment earns when it is invested for a certain

length of time.

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Simple Interest

Simple interest is the amount of interest, which is earned inan equal amount every year (or month), and which is a givenproportion of the (original investment or principal).

If a sum of money is invested for a period of time than theamount of simple interest, which accrues is equal to thenumber of accounting periods x the interest rate X theamount invested, i.e.

S n= P + nr P

Where P = Original sum invested;

r = Interest rate (Expressed as a proportion, e.g.

10% = 0.1)

n = Number of periods (Normally years);

S = Sum after n periods, consisting of the originalcapital plus interest earned

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Compound Interest

Interest is normally calculated by means of

compounding. If a sum of money, the principal, is

invested at a fixed rate of interest and the interest is

added to the principal and no withdrawals are made,

then the amount invested grows at an increasing rate

as time progresses, because interest earned in earlier

periods itself also earns interest later on.

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Whats the FV of an initial Rs.100after 3 years if i = 10%?

FV = ?

0 1 2 3

10%

Finding FVs (moving to the right

on a time line) is called compounding.

100

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After 1 year: FV1 = PV + INT1 = PV + PV (i)= PV(1 + i)

= Rs.100(1.10)

= Rs.110.00.

After 2 years: FV2 = PV(1 + i)2

= Rs.100(1.10)2= Rs.121.00.

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After 3 years: FV3 = PV(1 + i)3

= Rs.100(1.10)3

= Rs.133.10.

In general,

FVn = PV(1 + i)n.

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Will the FV of a lump sum be larger orsmaller if we compound more often, holding

the stated I% constant? Why?

LARGER! If compounding is more

frequent than once a year--for

example, semiannually, quarterly,

or daily--interest is earned on interestmore often.

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0 1 2 3

10%

0 1 2 3

5%

4 5 6

134.01

100 133.10

1 2 30

100

Annually: FV3= Rs.100(1.10)3= Rs.133.10.

Semiannually: FV6= Rs.100(1.05)6= Rs.134.01

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We will deal with 3 differentrates:

iNom = nominal, or stated, orquoted, rate per year.

iPer = periodic rate.

EIR = .

effective annualrate

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iNomis stated in contracts. Periods per year

(m) must also be given.

Examples:

8%; Quarterly

8%, Daily interest (365 days)

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Periodic rate= iPer= iNom/m, where m is numberof compounding periods per year. m = 4 for

quarterly, 12 for monthly, and 360 or 365 for

daily compounding.

Examples:

8% quarterly: iPer= 8%/4 = 2%.

8% daily (365): iPer

= 8%/365 = 0.021918%.

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The annual rate which causes PV to grow tothe same FV as under multi-periodcompounding.

Example: EFF% for 10%, semiannual:FV = (1 + iNom/m)

m

= (1.05)2 = 1.1025.

EFF%= 10.25%because(1.1025)1= 1.1025.

Any PV would grow to same FV at 10.25%

annually or 10% semiannually.

Effective Annual Rate (EAR =

EFF%):

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An investment with monthly payments

is different from one with quarterly

payments. Must put on EIR% basis to

compare rates of return. Use EIR% only

for comparisons.

Banks say interest paid daily. Same

as compounded daily.

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How do we find EFF% for a nominal rate of 10%,

compounded semiannually?

Or use a financial calculator.

EIR% = - 1(1 + )iNomm

m

= - 1.0(1 + )0.102

2

= (1.05)2- 1.0= 0.1025 = 10.25%.

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EAR = EFF% of 10%

EARAnnual = 10%.

EARQ = (1 + 0.10/4)4- 1 = 10.38%.

EARM = (1 + 0.10/12)12- 1 = 10.47%.

EARD(360) = (1 + 0.10/360)360- 1 = 10.52%.

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FV of Rs.100 after 3 years under 10%

semiannual compounding? Quarterly?

= Rs.100(1.05)6 = Rs.134.01.

FV3Q = Rs.100(1.025)12= Rs.134.49.

FV = PV 1 .+i

mn

Nom

mn

FV = Rs.1001 +0.10

2

3S

2x3

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Can the effective rate ever be equal

to the nominal rate?

Yes, but only if annual compounding isused, i.e., if m = 1.

If m > 1, EFF% will always be greater than

the nominal rate.

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When is each rate used?

iNom: Written into contracts, quoted

by banks and brokers. Notused in calculations or shown

on time lines.

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Changes in the Rate of Interest

If the rate of interest changes during the period of aninvestment, the compounding formula would be amended

slightly as follows

Sn = P(1+ r1)x (1+r2)(n-x)

Where r1= Initial rate of interest

x = Number of years in which the interest rate r1

applies

r2= Next rate of interest

n

x =Balancing number of years in which theinterest rate r2applies

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Increasing the Sum Invested

Each Year

There is an alternative formula for calculating thevalue of an investment at the end of n years, which

you may prefer to use, or which you may be

required to use in an examination.

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Discounting

The discounting is the reverse of compounding.

Its major application in business is in the evaluation ofthe capital expenditure projects, to decide whether theyoffer a satisfactory return to the investor.

This technique of discounting is known as DiscountedCash Flow, (DCF), and this will be described later.

Wh t th PV f R 100 d i 3

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10%

Whats the PV of Rs.100 due in 3 years

if i = 10%?

Finding PVs is discounting, and its

the reverse of compounding.

100

0 1 2 3

PV = ?

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Solve FVn= PV(1 + i )nfor PV:

PV =FV

1+ i = FV

1

1+ i

n

n n

n

PV = 100 11.10= 100 0.7513 = Rs.75.13.

3

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Present Values

The term present valueis often used in

discounting cash flow. It simply means the amount

of money, which must be invested now for n years

at an interest rate of r %, to earn a future sum of

money at the end of year n.

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What is the PV of this uneven cash

flow stream?

0

100

1

300

2

300

310%

-50

4

90.91

247.93

225.39-34.15

530.08 = PV

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Annuities

An annuity is a stream of equal annual cashflows.

A more general definition of an annuity would be a

constant sum of money each year for a given number

of years.

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Ordinary Annuity

PMT PMTPMT

0 1 2 3

i%

PMT PMT

0 1 2 3i%

PMT

Annuity Due

Whats the difference between an

ordinary annuity and an annuity due?

PV

FV

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Whats the FV of a 3-year ordinary annuity of

Rs.100 at 10%?

100 100100

0 1 2 310%

110

121FV = 331

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Whats the PV of this ordinary annuity?

100 100100

0 1 2 310%

90.91

82.6475.13

248.69 = PV

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Cash Flows of Perpetuity

The present value of an annuity for every year in

perpetuity is PV = a / r where r is the cost of

capital as proportion.

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