5_fmkts_tvm_pgdm

8/10/2019 5_FMkts_TVM_PGDM

1/33


2/33

Premise A $ today is not equal in value to a $

tomorrow or a $ yesterday (incomparable).

Time and $ value both are bidimensional.

Value of money changes through time (T)by a factor which represents rate ofinterest (r).

If we estimate a composite value of CashFlows (C) at a common point in time for allCs which actually occurred at differentpoints in time, those Cs values can be

compared.


3/33

The One-Period Case

If you were to invest $10,000 at 5-percentinterest for one year, your investment wouldgrow to

$500 would be interest ($10,000 .05)$10,000 is the principal repayment ($10,000 1)

$10,500 is the total due. It can be calculated as:

$10,500 = $10,000(1.05)

The total amount due at the end of theinvestment is call the Future Value(FV).


4/33

Future Value

In the one-period case, the formula for FVcan be written as:

FV=

C0

(1 +r)

Where C0is cash flow today (time zero), and

ris the appropriate interest rate.


5/33

Present Value

If you were to be promised $10,000 due in oneyear when interest rates are 5-percent, yourinvestment would be in todays dollars worth

05.1

000,10$81.523,9$

The amount that a borrower would need to setaside today to be able to meet the promised

payment of $10,000 in one year is called thePresent Value(PV).

Note that $10,000 = $9,523.81(1.05).


6/33

Present Value

In the one-period case, the formula for PVcan be written as:

r

CPV

1

1

Where C1is cash flow at date 1, and

r is the appropriate interest rate.


7/33

Net Present Value

The Net Present Value (NPV) of aninvestment is the present value of the

expected cash flows, less the cost of theinvestment.

Suppose an investment that promises to

pay $10,000 in one year is offered for salefor $9,500. Your interest rate is 5%.Should you buy?


8/33

Net Present Value

81.23$

81.523,9$500,9$

05.1000,10$500,9$

NPV

NPV

NPV

The present value of the cash inflow is greaterthan the cost. In other words, the Net PresentValue is positive, so the investment should bepurchased.

In the one-period case, the formula for NPVcanbe written as:

NPV=Cost+ PV


9/33

The Multi period Case

The general formula for the future value ofan investment over many periods can bewritten as:

FV=

C0

(1 +r)T

Where

C0is cash flow at date 0,

r is the appropriate interest rate, and

Tis the number of periods over which the cashis invested.


10/33

Future Value

Suppose a stock currently pays adividend of $1.10, which is expected togrow at 40% per year for the next five

years.

What will the dividend be in five years?

FV= C0(1 + r)T

$5.92 = $1.10(1.40)5


11/33

Future Value and Compounding

0 1 2 3 4 5

10.1$

3)4 0.1(1 0.1$

02.3$

)4 0.1(1 0.1$

54.1$

2)40.1(10.1$

16.2$

5)40.1(10.1$

92.5$

4)40.1(10.1$

23.4$


12/33

Present Value and Discounting

How much would an investor have to setaside today in order to have $20,000 fiveyears from now if the current rate is 15%?

0 1 2 3 4 5

$20,000PV

5)15.1(

000,20$53.943,9$


13/33

How Long is the Wait?

If we deposit $5,000 today in an account paying10%, how long does it take to grow to $10,000?

TrCF V

)1(0

T

)1 0.1(0 0 0,5$0 0 0,1 0$

2000,5$

000,10$)10.1( T

)2ln()1 0.1ln( T

years27.7

0953.0

6931.0

)10.1ln(

)2ln(T


14/33

Assume the total cost of a college education will be$50,000 when your child enters college in 12 years.You have $5,000 to invest today. What rate ofinterest must you earn on your investment to coverthe cost of your childs education?

What Rate Is Enough?

TrCF V )1(0

1 2)1(0 0 0,5$0 0 0,5 0$ r

10000,5$

000,50$)1( 12 r 1 211 0)1( r

2 1 1 5.12 1 1 5.111 0

1 21

r

About 21.15%.


15/33

Multiple Cash Flows

Consider an investment that pays $200one year from now, with cash flowsincreasing by $200 per year through year

4. If the interest rate is 12%, what is thepresent value of this stream of cash flows?

If the issuer offers this investment for

$1,500, should you purchase it?


16/33

Multiple Cash Flows

0 1 2 3 4

200 400 600 800178.57

318.88

427.07

508.41

1,432.93

Present Value (1432.93)


17/33

Compounding Periods

Compounding an investment mtimes ayear for Tyears provides for future value ofwealth:

Tm

m

rCFV

10


18/33

Compounding Periods

If you invest $50 for 3 years at 12%compounded semi-annually, yourinvestment will grow to

93.70$)06.1(50$

2

12.150$ 6

32

FV


19/33

Effective Annual Rates ofInterest

A reasonable question to ask in the aboveexample is what is the effective annual

rate of interest on that investment?

The Effective Annual Rate (EAR) of interest is theannual rate that would give us the same end-of-investment wealth after 3 years:

93.70$)06.1(50$)2

12.1(50$ 632 FV

9 3.7 0$)1(5 0$ 3 E A R


20/33

Effective Annual Rates ofInterest

So, investing at 12.36% compoundedannually is the same as investing at 12%compounded semi-annually.

9 3.7 0$)1(5 0$ 3 E A RF V

50$

93.70$

)1( 3

EAR

1236.150$

93.70$ 31

EAR


21/33

Continuous Compounding

The general formula for the future value of aninvestment compounded continuously overmany periods can be written as:

FV= C0erT

WhereC0is cash flow at date 0,

r is the stated annual interest rate,

Tis the number of years, andeis a transcendental number approximately

equal to 2.718. exis a key on yourcalculator.


22/33

Simplifications

Perpetuity A constant stream of cash flows that lasts forever

Growing perpetuity

A stream of cash flows that grows at a constant rate

forever

Annuity

A stream of constant cash flows that lasts for a fixednumber of periods

Growing annuity

A stream of cash flows that grows at a constant ratefor a fixed number of periods


23/33

Perpetuity

A constant stream of cash flows that lasts forever

0

1

C

2

C

3

C

32 )1()1()1( r

C

r

C

r

C

PV

r

CPV


24/33

Perpetuity: Example

What is the value of a British consol thatpromises to pay 15 every year forever?

The interest rate is 10-percent.

0

1

15

2

15

3

15

150

10.

15PV


25/33

Growing Perpetuity

A growing stream of cash flows that lasts forever

0

1

C

2

C(1+g)

3

C (1+g)2

3

2

2 )1(

)1(

)1(

)1(

)1( r

gC

r

gC

r

C

PV

gr

CPV


26/33

Growing Perpetuity: Example

The expected dividend next year is $1.30, anddividends are expected to grow at 5% forever.

If the discount rate is 10%, what is the value of

this promised dividend stream?

0

1

$1.30

2

$1.30(1.05)

3

$1.30 (1.05)2

00.26$05.10.

30.1$

PV


27/33

AnnuityA constant stream of cash flows with a fixed

maturity

0 1

C

2

C

3

C

T

r

C

r

C

r

C

r

CPV

)1()1()1()1( 32

Trr

CPV

)1(

11

T

C

A it E l


28/33

What is the present value of a four-year annuity of$100 per year that makes its first payment two years from

today if the discount rate is 9%?

22.297$09.1

97.327$

0 PV

0 1 2 3 4 5

$100 $100 $100 $100$323.97$297.22

97.323$)09.1(

100$

)09.1(

100$

)09.1(

100$

)09.1(

100$

)09.1(

100$4321

4

1

1 t

tPV

Annuity: Example

G i A it


29/33

Growing Annuity

A growing stream of cash flows with a fixedmaturity

0 1

C

T

T

r

gC

r

gC

r

CPV

)1(

)1(

)1(

)1(

)1(

1

2

T

r

g

gr

CPV

)1(

11

2

C(1+g)

3

C (1+g)2

T

C(1+g)T-1


30/33

Growing Annuity: Example

You are evaluating an income generating property. Net rent isreceived at the end of each year. The first year's rent is

expected to be $8,500, and rent is expected to increase 7%

each year. What is the present value of the estimated income

stream over the first 5 years if the discount rate is 12%?

0 1 2 3 4 5

5 0 0,8$

)0 7.1(5 0 0,8$

2)0 7.1(5 0 0,8$

095,9$ 6 5.7 3 1,9$

3)0 7.1(5 0 0,8$

8 7.4 1 2,1 0$

4)0 7.1(5 0 0,8$

7 7.1 4 1,1 1$

$34,706.26

Future and Present value of an


31/33

Future and Present value of anAnnuity Due

Present value of Cash Flows occurring at the beginning of the period

What would be the Future and the Present value of Cash Flowsoccurring at the beginning of the period as detailed above?


32/33

Future value of Annuity Due

FV of Annuity Due = C*FVIFA*(1+i)


33/33

Present value of Annuity Due

PV of Annuity Due = C*PVIFA*(1+i)

5_fmkts_tvm_pgdm

Documents