2_Time Value of Money (1)

1Dr Pankaj Varshney

LBSIM-PGDM(General) Batch 2014-16

Time Value of Money

Time Value of Money

• Cash flow occurring at the end of 2nd year is not equal

to the cash flow occurring now because of Time Value

of Money (TVM).

• Why TVM?

� Inflation - value of currency decreases over time.

� Preference for present consumption over future

consumption- to induce people to give up present

consumption, you have to offer them more in the

future.

� Uncertainty about the future - higher uncertainty (risk)

means less valuable the future cash flow.

• Thus, the cash flows occurring in different time periods

have to be made comparable. 2

2Dr Pankaj Varshney


Time Line

Cash Flows at-the-end of period

Cash Flows at-the-beginning of period

0 4321

1000 1000 1000 1000

Period 1 Period 2 Period 3 Period 4

• Cash Flows of Rs 1000/- each at Year-end for 4 years

• Cash Flows of Rs 1000/- each at Year-beginning for 4 years

0 4321

1000 1000 1000 1000

Period 1 Period 2 Period 3 Period 4

3

Time Value of Money

• Take today’s cash flows into the future : Future Value

• Bring future cash flows to today’s value: Present Value

4

0 4321

P FV

5

@ r%

0 4321

P FV

5

@ r%

3Dr Pankaj Varshney


Future Value

• Future Value of amount P , after ‘n’ years would be:

FVn = P*(1+r)n

5

Future Value Interest Factor

[FVIF (n,r)]

0 4321

P = 1,000 FV = 1,276.28

5

@ 5%1000*(1.05)5

0 4321

P FV

5

@ r%

Future Value

6

FVIF (5 yrs,5%)

• Values of FVIF for various combinations of ‘r’ and ‘n’ are

given in Future Value Interest Factor tables.

4Dr Pankaj Varshney


Future Value (Contd.)

If you invest Rs.80,000/- @ 14%p.a., how much would it amount to

in 5 years?

0 4321

80,000 1,54,033

5

7

� Future Value of Rs. 80,000/- @14% after 5 years would

be: 80,000*(1+0.14)5 or 80,000*FVIF (5 years,14%)

= 80,000*1.9254 = Rs 1,54,033/-

Future Value (Contd.)

8

5Dr Pankaj Varshney


Future Value

9

Compounding Rate (r)

Present Value (PVN)

Time Period (N)

Returns the Future Value

=FV(Interest Rate, Time,, Present

Value,0(or1))

Future value

10

5%

8%

10%

12%

15%

18%

900

1400

1900

2400

2900

3400

3900

4400

4900

5400

5900

1 2 3 4 5 6 7 8 9 10

Fu

ture

Va

lue

Years

Higher the interest rate, faster the savings grow

6Dr Pankaj Varshney


Compounding more than once a year

n*m

n

rFV = P*(1+ )

m

• Interest may be paid more than once a year.

• Future Value is :

where ‘m’ is no. of times interest is paid.

11

� If you invest Rs.80,000/- @ 14%p.a., how much would it amount to

in 5 years, if interest is compounded (a)semi-annually, (b) quarterly?

Frequency m Future Value

Semi-Annually 2 80000*(1+0.14/2)(5*2) = 80000*(1.07)(10)

= 80000*1.9672 = 1,57,372/-

Quarterly 4 80000*(1+0.14/4)(5*4) = 80000*(1.035)(20)

= 80000*1.9898 = 1,59,183/-

Compounding more than once a year

12

� If you invest Rs.80,000/- @ 14%p.a., how much would it amount

to in 5 years, if interest is continuously compounded?

� Future Value on continuous compounding would be:

80000*e0.14*5 = 80000*2.0138 = 1,61,100/-

• As ‘m’ approaches infinity, the term (1+r/m)n*m approaches er*n,

where ‘e’ is approx. 2.71828 and is defined as m

m

1e=limit(1+ )

m→∞

• Future Value on continuous compounding basis is:r n

nFV = P*e

7Dr Pankaj Varshney


Nominal vs. Effective Interest Rate

13

Frequency (m)

Nominal

Rate

Future

Value

Effective

Annual Rate

Annual 1 10% 1100.00 (1.10)-1 10.0000%

Semi-annual 2 10% 1102.50 (1+ 0.10/2)2 - 1 10.2500%

Quarterly 4 10% 1103.81 (1+ 0.10/4)4 - 1 10.3813%

Monthly 12 10% 1104.71 (1+ 0.10/12)12 - 1 10.4713%

Daily 365 10% 1105.15 (1+ 0.10/365)365 - 1 10.5156%

Continuous 10% 1105.17 exp (0.10) - 1 10.5171%

Daily Compounding is same as Continuous Compounding

mStated Annual Interest rate

EIR = 1+ - 1m

Present Value

• The process of calculating the present value of the future Cash

Flows is called discounting and the interest rate used for

discounting is called the discount rate.

14

8Dr Pankaj Varshney


Present Value of a Single Cash Flow

Present Value Interest Factor

[PVIF(n,r)]

15

• Values of PVIF for various combinations of “r” and “n” are given in

Present Value tables.

• From our understanding of Future Value, we know that

FVn = P (1+r)n

hence,

= FVn* PVIF (n, r)

n n

1P=FV *

(1+r)

PVIF

(5 yrs,5%)


What is the worth of Rs.10,000/- received at the end of 5 years from

now, if the discount rate is 6% p.a.?

510,000=P*(1.06)

16

�Present Value of Rs. 10,000 (Future Value) would be :

5

10,000PV= =10,000*0.74726 = Rs.7,472.60

(1.06)

0 321

10,000

??

4 5

9Dr Pankaj Varshney



17

Present Value

18

Discounting Rate (r)

Future Value (FVn)

Time Period n)

= ‘0’ End of period

= ‘1’ Start of period

Returns the Present Value

=PV(Interest Rate, Time,,

-Future Value, 0(or1))

10Dr Pankaj Varshney



19

5%

100

200

300

400

500

600

700

800

900

1000

1100

1 2 3 4 5 6 7 8 9 10

Pre

sen

t V

alu

e

Years

18%

Annuity

Annuity is a stream of ‘n’ equal cash flows (inflows or

outflows) at regular intervals for a fixed period of time.

� If each investment is made at the END of each period, the

annuity is called Regular Annuity or Annuity in arrears

� If each investment is made at the BEGINNING of each

period, the annuity is called Annuity Due.

A

0 21

A A

n3

A

20

A

0 21

A A

n-13

AA



Future Value of Annuity

n

RA

(1+r) -1FVA = A

r

21

• Future Value of Regular Annuity:

A

0 21

A A

n3

AA A

n-2 n-1

A(1+r)n-1

A(1+r)n-2

A(1+r)n-3

A(1+r)1

A(1+r)2

n-1 n-2 n-3 2 1

RAFVA = A(1+r) + A(1+r) + A(1+r) +........+ A(1+r) +A(1+r) + A

Future Value Interest Factor Annuity

[FVIFA(n,r)]

Future Value of an Annuity

22

n

AD

(1+r) -1FVA = A (1+r)

r

• Future Value of Annuity Due:

n n-1 n-2 2 1

ADFVA = A(1+r) + A(1+r) + A(1+r) +........+ A(1+r) + A(1+r)

0 21

A A

n3

A(1+r)n

A(1+r)n-1

A(1+r)n-2

A A

n-2 n-1

A(1+r)1

A(1+r)2

A



Future Value of an Annuity (Annuity Regular)

23

Future Value of an Annuity (Annuity Due)

24



Future Value of an Annuity

Sairam deposits Rs.50,000/- every year in a 5-year recurring deposit earning

interest @8%p.a. How much money would get accumulated in the recurring

deposit account, by the end of 5 years?

5

RA

(1.08) -1FVA = 50,000

0.08

25

Case-1 Regular Annuity:

Case-2 Annuity Due:

5

AD

(1.08) -1FVA = 50,000 (1.08)

0.08

= 50,000*6.3359= Rs.3,16,796/-

??0 21

50,000

43

50,000 50,000 50,000

5

50,000

0 21

50,000

43

50,000 50,00050,000

5

50,000

??

= 50,000*5.8666= Rs.2,93,330/-

Present Value of an Annuity

RA n

1 1PVA =A -

r r(1+r)

26

• Present Value of Regular Annuity:

Present Value Interest

Factor Annuity

[PVIFA (n,r)]

RA 1 2 3 n-2 n-1 n

A A A A A APVA = + + +.........+ + +

(1+r) (1+r) (1+r) (1+r) (1+r) (1+r)

31 n-2 n-10 1 nn-2 n-1

A

2

A A

3

A

A/(1+r)1

A A

A/(1+r)3

A/(1+r)2

A/(1+r)n-2

A/(1+r)n-1

A/(1+r)n

2 n




27

• Present Value of Annuity Due:

AD n

1 1PVA =A - (1+r)

r r(1+r)

AD 1 2 3 n-2 n-1

A A A A APVA = A+ + + +.........+ +

(1+r) (1+r) (1+r) (1+r) (1+r)

nn-2 n-1

A

0 21

A A

3

A/(1+r)1

A A

A/(1+r)3

A/(1+r)2

A/(1+r)n-2

A/(1+r)n-1

1 3 4 n-1 n2

A

Present Value of an Annuity (Annuity Regular)

28



Present Value of an Annuity (Annuity Due)

29


30

=PV(Interest Rate, Time,

-Annuity,,0(or1))

Discounting Rate (r)

Time Period n)

-Annuity



Returns the Present Value of

the Annuity




31

Aditya is planning to buy a Single premium pension plan which

would give him an annual pension of Rs 50,000/- for the next 30

years. What should be the maximum premium that he should pay

now for the pension plan, assuming interest @9%?

30

1 1=50000 -

0.09 0.09(1.09)

=50000*10.2737=Rs.5,13,683/-

• Single premium (to be paid now) = Present value of the annuities

to be received over the life of the pension plan.

Equated Monthly or Yearly Installments

32

Year

Opening

Balance

(1)

Annual

Instalment

(2)

Interest

(3) = (1)*8%

Principal

Repayment

(4) = (2)-(3)

Closing

Balance

(5) = (1)-(4)

1 250,000 62,614 20,000 42,614 207,386

2 207,386 62,614 16,591 46,023 161,363

3 161,363 62,614 12,909 49,705 111,658

4 111,658 62,614 8,933 53,682 57,976

5 57,976 62,614 4,637 57,977 0

Suppose you take a loan of Rs 2,50,000/- @8% pa to be repaid in 5

yearly equal installments. Find the amount of each installment?

RA n

1 1PVA =A -

r r(1+r)

5

1 12,50,000 = A -

0.08 0.08(1.08)

2,50,000A = = 62,614/-

3.9927

3,13,0703,13,0703,13,0703,13,070 63,07063,07063,07063,070 2,50,0002,50,0002,50,0002,50,000



Installments

33

Discounting Rate

Loan Amount

Time Period



Returns the Amount of

Installment

=PMT(Interest Rate, Time,

-Loan Amount,,Type)

Saving for College Education

Sunil wants to send his daughter to a 4-year college, 18 years

from now. Tuition fees is Rs. 50,000 per year now which is

expected to rise @ 5% pa over the next 18 years. If Sunil’s

saving can earn @ 8% pa, (a) how much he should invest

(lumpsum) now to meet the expenditure, or (b) how much he

should invest each year for the same.



Saving for College Education

Refinancing a Housing Loan

Mudit had taken a 30-year loan for Rs. 2,00,000/-, 3 years

ago @ 9%pa. The interest rate has fallen now to 7.50%pa.

He is thinking of refinancing the loan. Cost of refinancing is

2.50% of the loan. Assuming the discount rate as 6%, should

the loan be refinanced?



Refinancing Housing Loan

Refinancing Housing Loan



Growing Annuity

Growing Annuity is a stream of ‘n’ cash flows growing @ ‘g’, paid

at regular intervals.

A(1+g)2

0 21

A A(1+g)

n3

A(1+g)n-1

39

• Growing Regular Annuity:

• Growing Annuity Due:

n-10 21 3

A(1+g)2A A(1+g) A(1+g)n-1A(1+g)3

n

Present Value of a Growing Annuity

40

• Present Value of Growing Regular Annuity :n-10 21 3

A(1+g)2A A(1+g) A(1+g)n-2

1

2

A (1 + g )

(1 + r)

2

3

A (1 + g )

(1 + r )

n -2

n -1

A (1 + g )

(1 + r)

1

A

(1 + r)

A(1+g)n-1

n

n -1

n

A (1 + g )

(1 + r) 2 n-1

RA 1 2 3 n

A A(1+g) A(1+g) A(1+g)PVGA = + + +......+

(1+r) (1+r) (1+r) (1+r)n

RA

A 1+gPVGA = 1-

r-g 1+r

For g ≠ r




41

• Present Value of Growing Annuity Due:

n-10 21 3

A(1+g)2A A(1+g) A(1+g)n-1

2

2

A(1+g)

(1+r)

3

3

A(1+g)

(1+r)

n-1

n-1

A(1+g)

(1+r)

1

A(1+g)

(1+r)

A(1+g)3

n

1 2 n-1

AD 0 1 2 n-1

A A(1+g) A(1+g) A(1+g)PVGA = + + +......+

(1+r) (1+r) (1+r) (1+r)

n

AD

A 1+gPVGA = 1- (1+r)

r-g 1+r

For g ≠ r

42


Growing Regular Annuity:

If Growing Annuity Due:

5

RA

10,000 1.10PVGA = 1- = Rs.48,042/-

0.08 - 0.10 1.08

The annual (year-end) lease payment of a building increase by 10%

for the next 5 years. If 8% is the appropriate discount rate and first

years’ payment is Rs. 10,000/-, what is the maximum amount that an

investor pay to be the recipient of these lease payments?

5

AD

10,000 1.10PVGA = 1- (1.08) = Rs.51,886/-

0.08 - 0.10 1..08



Future Value of a Growing Annuity

n

nA 1+g= 1- *(1+r)

r-g 1+r

43

n

RA RAFVGA = PVGA *(1+r)

• Future Value of Growing Regular Annuity :n-10 21 3

A(1+g)2A A(1+g) A(1+g)n-2

1

2

A(1+g)

(1+r)

2

3

A(1+g)

(1+r)

n-2

n-1

A(1+g)

(1+r)

1

A

(1+r)

A(1+g)n-1

n

n-1

n

A(1+g)

(1+r)


n

n

AD

A (1+r) 1+gFVGA = 1- *(1+r)

r-g 1+r

44

n

AD ADFVGA = PVGA *(1+r)

• Future Value of Growing Annuity Due:

n-10 21 3

A(1+g)2A A(1+g) A(1+g)n-1

2

2

A(1+g)

(1+r)

3

3

A(1+g)

(1+r)

n-1

n-1

A(1+g)

(1+r)

1

A(1+g)

(1+r)

A(1+g)3

n




0 21

10,000

303

35 36 37 38 65

10,000(1.05) 10,000(1.05)2 10,000(1.05)29

n 30FVGA = PVGA*(1+r) = 1,50,464*(1.10) = Rs. 2.625 Mn

3010,000 1.05

PVGA = 1- = Rs. 1,50,464/-0.10 - 0.05 1.10

45

Mr. Sairam is 35 years old now and wants to save each year until he

is 65. If he saves Rs 10,000/- every year and the savings grow@ 5%

pa (after the first year),how much will he have saved by age 65 if

the interest rate is 10% pa?

Step-1 : Calculate PVGA:

Step-2 : Calculate FVGA using PVGA:

Perpetuity

Perpetuity is a stream of equal cash flows at regular intervals which lasts forever.

1 2 3

A A APVP= + + +......

(1+r) (1+r) (1+r)∞

A

0 21

A A

∞3

46

• Present Value of a Perpetuity:

APVP=

r



Perpetuity

You want to endow an annual MBA graduation party at your alma amter. The event would cost Rs.50,000/- each year forever. If the business school earns @ 8%p.a. on its investments and the first party is in one year’s time, how much will you need to donate to endow the party?

50,000PVGP= =Rs.6,25,000/-

0.08

1 2 3

50,000 50,000 50,000PVGP= + + +......

(1.08) (1.08) (1.08)∞

47

Growing Perpetuity

Growing Perpetuity is a stream of cash flows at regular intervals and grows at a constant rate forever.

2 3

1 2 3 4

A A(1+g) A(1+g) A(1+g)PVGP= + + + ......

(1+r) (1+r) (1+r) (1+r)∞

A(1+g)2

0 21

A A(1+g)

∞3

A(1+g)3

4

48

• Present Value of a Growing Perpetuity:

APVGP=

r-g



Growing Perpetuity

But then you are informed that the cost of the party would increase by 4% per year, (after the first year).How much will you now need to donate to endow the party?

50,000PV= =Rs.12,50,000/-

0.08 - 0.04

2

1 2 3

50,000 50,000(1.04) 50,000(1.04)PV= + + +......

(1.08) (1.08) (1.08)∞

You need to double the amount of your gift !!!

49

Summary

50

Present Value Future Value

Single Cash Flow

Annuity

(Regular Annuity)

Annuity

(Annuity Due)

Growing Annuity

(Regular Annuity)

Growing Annuity

(Annuity Due)

Perpetuity

Growing Perpetuity

n

1 1A -

r r(1+r)

n

1C

(1+r)

nA 1+g

1-r-g 1+r

A

r

A

r - g

n

1 1A - (1+r)

r r(1+r)

n(1+r) -1

Ar

n(1+r) -1 A (1+r)

r

n

nA 1+g1- (1+r)

r-g 1+r

nA 1+g

1- (1+r)r-g 1+r

n

nA(1+r) 1+g1- (1+r)

r-g 1+r

nC(1+r)



Where to Invest ?

Atul want to invest Rs. 10 Lac for a period of 10 years. He can invest in Government bonds which mature in 6 years and earn interest @ 8% pa. The expected fixed deposit rate for 6 years hence, given by a local bank is 3.5% pa, with half yearly compounding. Meanwhile, Yep Bank has offered an investment proposal offering 6.5% pa with quarterly compounding for 10 years. Which proposal is better for Atul?

Option 1: Invest in Government Bonds @ 8% pa for 6 years & @ 3.5%

pa (half yearly compounding) in bank fixed deposit for next 4 years

thereafter.

51

• 10,00,000 * (1.08)6 = Rs. 15,86,874.32

• 15,86,874.32 * (1.0175)8 = Rs. 15,86,874.32 *1.14888 = Rs. 18,23,131/-

Option 2: Invest in Yep Bank 10-year Fixed deposit @ 6.5% pa

(quarterly compounding) for 10 years.• 10,00,000 * (1.01625)40 = 10,00,000* 1.90556 = Rs. 19,05,560/-

Better to deposit with Yep Bank.

The MBA Decision

52



Valuation of Securities

54

Valuing a Zero-coupon Bond

Consider a Zero-coupon Bond with face value of Rs. 5,000/- payable

at the end of 3 years. What should be the price of the bond today,

if the required rate of return is 5%?

n

n

FValue of ZCB =

(1+r)

0 3

5,000V = = Rs.4,319.19

(1.05)

0 321

5,000

??



55

Valuing a Coupon Bond

n0 1 2 3 n n

FA A A AV = + + +.....+ +

(1+r) (1+r) (1+r) (1+r) (1+r)

NTPC issues 14% bonds of Rs.10,000/- face value, redeemable after 5 years. Assuming the required rate of return is 10%, what should be the price of the bond today?

0 1 54

A

2

A A

3

A/(1+r)1

A+FA

A/(1+r)3

A/(1+r)2

A/(1+r)4

A+F/(1+r)5

∑n

t nt=1

Coupon Interest Face Value of BondValue of Coupon Bond = +

(1+r) (1+r)

56

Valuing a Coupon Bond

∑5

t 5t=1

1,400 10,000= +

(1.10) (1.10)

Face Value = Rs.10,000 ; Coupon rate = 14% pa;

Tenure = 5 Years; Required rate of return = 10%;

Value of Bond= ??

∑5

t0 t 5

t=1

A FV = +

(1+r) (1+r)

=1,400*PVIFA(5yrs, 10%) + 10,000*PVIF(5 yrs, 10%)

=1,400*3.79079 + 10,000*0.62092

= 5,307.10 + 6,209.21 = Rs.11,516.31



57

Valuing Equity : Dividend Discount Model (DDM)

• Value of an equity share is the present value of the stream of expected future dividends discounted at an appropriate discount rate.

∞

∞

31 2

1 2 3

DD D DValue = + + +..........+

(1+r) (1+r) (1+r) (1+r)

∞

∑ t

tt=1

DValue =

(1+r)General Form of DDM

Value of a stock = PV(expected future dividends)

58

DDM

ABC Co. is expected to pay a dividend of Rs.3/- forever.

What should be the Value of the equity share, if an

investor has a required rate of return as 13%?

1D 3.00

Value = = = Rs.23/-r 0.13



59

DDM - Constant Growth

• If the dividends are expected to grow at a constant rate ‘g’, and r > g, then,

Assumptions:

• D1 > 0

• Dividends grow at a constant growth rate “g” =ROE*b

• Dividend Payout ratio (1-b) is constant

∞

1 2 3

1 1 1 10 1 2 3 4

D D (1+g) D (1+g) D (1+g)V = + + + +..........+

(1+r) (1+r) (1+r) (1+r)

10

DV =

(r - g)

60

DDM - Constant Growth

1D 3.00

Value = = = Rs.50/-(r - g) (0.13 - 0.07)

ABC Co. is expected to pay a dividend of Rs.3/- which is

expected to grow @ 7% forever. What should be the Value

of the equity share, if an investor has a required rate of

return as 13%?



61

DDM - Multiple Growth Rate

∑n

t-1 t n n+10 nt n

t=1 n

D (1+g ) V DV = + where V =

(1+r) (1+r) r - g

nt-1 t n+1

0 t nt=1 n

D (1+g ) D1V = + ×

(1+r) (1+r) r - g

∑

• A firm may pass through different growth phases and hence

dividends may also grow at different rates.

∞

1 2

1 1 1 2 2 2 20 1 2 3 4

D D (1+g ) D (1+g ) D (1+g )V = + + + +..........+

(1+r) (1+r) (1+r) (1+r)

62

DDM - Multiple Growth Rate -Illustration

D0 = 3.50; g1-3=15% g 4-6=12% g7+=8% ; r = 12% ; Value = ??

D1 = 4.03; D2 = 4.63; D3 = 5.32; D4 = 5.96; D5 = 6.68; D6 = 7.48; D7 = 8.08

0 1 2 3 4 5 6 6

4.03 4.63 5.32 5.96 6.68 7.48 8.08 1V = + + + + + + ×

(1.12) (1.12) (1.12) (1.12) (1.12) (1.12) (0.12 - 0.08) (1.12)

0P = 124.78 Rs. 125/-≃



63

Value of a Business

= Rs.23,92,380/-

Case-1 Growing Regular Annuity:

Case-2 Growing Annuity Due:

50

RA

3,00,000 1.025PVGA = 1-

0.15 - 0.025 1.15

50

RA

3,00,000 1.025PVGA = 1- (1.15)

0.15 - 0.025 1.15

= Rs.27,51,245/-

Indicoffee, a popular coffee shop located in a busy shopping mall, is

expected to generate net cash flows of Rs 3 Lacs a year. If the

cash flows increase @ 2.5% pa for the next 50 years, what is the

worth of the coffee shop? (assume discount rate of 15%)

2_Time Value of Money (1)

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