first principles of valuation

FIRST PRINCIPLES OF VALUATION

TIME VALUE OF MONEY (TVM)

MORE TVM

INVESTMENT RETURNS: IRR and NPV

TYPES OF LOANS

TIME VALUE OF MONEY:Six Functions of a Dollar

• Future Value of a One Time Deposit• Present Value of a Single Future Cash Flow• Future Value of an Annuity• Present Value of an Annuity• Sinking Fund Payment• Payment to Amortize Debt

TVM: Future Value of a One Time Deposit with Annual Compounding

What is the value in two years of $1,000 deposited today in a savingsaccount bearing 6% annual interest compounded annually?

Value at the end of year 1:

FV = $1,000 (1 + 0.06) = $1,060

Value at the end of year 2:

FV = $1,000 (1 + 0.06) (1 + 0.06)

= $1,000 + $60 + $60 + $3.60

= $1,123.60


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1 P/YR

1000 PV

6 I/YR

2 N

FV

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Annual compounding

$1,000 present value

6% annual interest

Two year term

Compute future value


F V P V i n ( )1

w h e r e

P V = t h e a m o u n t o f t o d a y ’ s d e p o s i t

i = t h e a n n u a l i n t e r e s t r a t e

n = t h e n u m b e r o f y e a r s i n t h e i n v e s t m e n t t e r m

F V = t h e f u t u r e v a l u e

TVM: Future Value of a One Time Deposit with Periodic Compounding

Suppose the deposit in the previous example was compoundedmonthly instead of annually. What is the future value?

Monthly interest rate = 0.06/12 = 0.005Number of compounding periods = 2 x 12 = 24

End of month Future Value

1 $1000.00 (1.005) = $1005.00 2 $1005.00 (1.005) = $1010.25 .12 $1056.40 (1.005) = $1061.68 .24 $1121.55 (1.005) = $1127.16


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12 P/YR

1000 PV

6 I/YR

2

FV

x P/YR

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Monthly compounding


6% annual interest

24 months



F V P Vi

kn k ( )1

w h e r eP V = t h e a m o u n t o f t o d a y ’ s d e p o s i t

i = t h e a n n u a l i n t e r e s t r a t e

n = t h e n u m b e r o f y e a r s i n t h e t e r m

k = t h e n u m b e r o f c o m p o u n d i n g p e r i o d s p e r y e a r

F V = t h e f u t u r e v a l u e

TVM: Quoted Rates vs Effective Annual Rates

Quoted interest rate (QR): the interest rate expressed interms of the interest paymentmade each period.

Effective Annual Rate (EAR): the interest rate expressed asif it were compoundedONCE per year.

Compounding can produce significant differences betweenquoted and effective annual rates.


6% annual interest, compounded annually, produced a futurevalue of $1060 at the end of year 1-- the quoted and effectiveinterest rates are the same.

6% annual interest, compounded monthly, produced a futurevalue of $1061.68 at the end of year 1 -- the effective interestrate is 6.168%.


I n g e n e r a l , t h e e f f e c t i v e a n n u a l r a t e ( E A R ) i s :

w h e r e Q R = t h e q u o t e d ( a n n u a l ) i n t e r e s t r a t e , a n d

k = t h e n u m b e r o f c o m p o u n d i n g p e r i o d sp e r y e a r .

E A RQ R

kk [ ]1 1


E x a m p l e : t h e e f f e c t i v e a n n u a l r a t e f o r a d e p o s i t t h a t r e c e i v e s6 % a n n u a l i n t e r e s t , c o m p o u n d e d m o n t h l y , i s :

= 1 . 0 0 5 1 2 - 1

= 1 . 0 6 1 6 7 8 - 1

= 0 . 0 6 1 6 7 8 o r 6 . 1 6 7 8 %

E A R [.

]10 0 6

1 211 2


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12 P/YR

I/YR

EFF %

DISP

6

4


Continuous compounding occurs when deposits receiveinstantaneous interest compounded throughout the year.

The effective interest rate with continuous compounding is:

EAR = eQR - 1, where QR is the quoted rate.

Example: the EAR for a deposit that receives 6% annualinterest, compounded continuously, is

EAR = e0.06 - 1 = 0.061837 or 6.1837%


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Subtract 1

Multiply by 100 to get

the percent.

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0.06 e X

- 1 =* 100 =


The following table provides the effective annual rates (EARs)for a deposit that receives 6% annual interest for variouscompounding frequencies:

Compounding Frequency EARAnnual compounding (k = 1) 6.0000%Semi-annual compounding (k = 2) 6.0900%Quarterly compounding (k = 4) 6.1364%Monthly compounding (k = 12) 6.1678%Weekly compounding (k = 52) 6.1800%Daily compounding (k = 365) 6.1831%Continuous compounding (k = large) 6.1837%

TVM: Present Value of a Future CFAnnual Discounting

An investment promises $1000 cash in two years. A potentialinvestor believes the investment should return 8% annually.How much would this investor be willing to pay today for thisopportunity?

$1000 = PV (1 + 0.08)2

= PV (1.1664), so

PV = $1000/1.1664

= $857.34At a required return of 10%, the present value is $826.45.

TVM: Present Value of a Future CF Annual Discounting: HP 10B Keystrokes

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FV

N

I/YR

PV

1

1000

2

8

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One payment per year

$1,000 future value

CF in two years

8% annual interest

Compute present value

TVM: Present Value of a Future CFAnnual Discounting

T h e p re s e n t v a lu e (P V ) o f a s in g le fu tu re c a s h f lo w (C F ) i s :

w h e re F V = th e s in g le fu tu re c a s h f lo w ,

d = th e a n n u a l d is c o u n t r a te , a n d

n = th e n u m b e r o f y e a r s in th e in v e s tm e n t te rm .

P V F Vd n

1

1( )

TVM: Annuities

Annuity: a finite sequence of equal paymentsmade across equally spaced timeintervals.

Ordinary Annuity: the payments occur at the endof the period.

Examples?

Annuity Due: the payments occur at the beginning of the period.

Examples?

TVM: Future Value of an Annual Ordinary Annuity

How much will $1,200, deposited at the end of each year,accumulate to in three years if the deposits earn 6% annualinterest (compounded annually).

Year Future Value + Contribution = Total FV

1 $0 + $1,200 = $1,200.00

2 $1,200x1.06 + $1,200 = $2,472.00

3 $2,472x1.06 + $1,200 = $3,820.32


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$1,200 payment/year

6% annual interest

For 3 years


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P/YR

PMT

I/YR

N

FV

1

1200

6

3


F V = P M T ( 1 + i ) n - 1 + P M T ( 1 + i ) n - 2 + . . . . . . . . +

P M T ( 1 + i ) 2 + P M T ( 1 + i ) + P M T

= P M T [ ( 1 + i ) n - 1 + . . . . + ( 1 + i ) 2 + ( 1 + i ) + 1 ]

= P M T i P M Ti

it

n n t n

( ){ ( ) }

11 1

1

TVM: Future Value of a Periodic Ordinary Annuity

Instead of depositing $1,200 at the end of the year for three years, computethe future value of the annuity if the deposits are $100 at the end of eachmonth for 36 months and earn 6% annual interest, compounded monthly.

Month Future Value + Contribution = Total FV

1 $ 0 + $100 = $ 100.00

2 $100 x (1.005) + $100 = $ 200.50

3 $200.5 x (1.005) + $100 = $ 301.50 . .36 $3,814.54 x (1.005) + $100 = $3,933.61


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PMT

I/YR

12

100

3

6

FV

x P/YR

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Monthly payments

$100 payment/mo.

36 months (3 years)

6% annual interest



The total future value is simply the sum of 36 future values:

FV = $100(1.005)35 + $100(1.005)34 + .... + $100

= $100[(1.005)35 + (1.005)34 + .... + 1]

= $100[1.1907 + 1.1848 + .... + 1]

= $100[39.33610]

= $3,933.61


In general,

FV = PMT

where PMT = the equal periodic payment made at theend of the period;

i/k = the periodic interest rate;nk = the number of periods in the annuity;FV = the future value.

( )11

i

kt

nknk t

TVM: Future Value of a Periodic Annuity Due

What is the future value of the three year, 6% annual interestrate, $100 monthly annuity if the cash flows occur at thebeginning, rather than the end, of the month? Interest iscompounded monthly.

FV = $100(1.005)36 + $100(1.005)35 +...+ $100(1.005)

= $100[1.1967 + 1.1907 + .... + 1.005]

= $100[39.5328]

= $3,953.28


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BEG/END

FV

12

6

100

3 x P/YR

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12 payments per year

6% annual interest

$100 payment/mo.

Annuity due (BEGIN)

36 monthly payments



In general,

FV = PM T

w here

PM T = the equal paym ent that occurs atthe beginning of the period;

i/k = the periodic in terest rate; and

nk = the num ber of periods in the annuity.

( )11

1

i

kt

nknk t

TVM: Present Value of an Annual Ordinary Annuity

What is the present value of receiving $1,200 at the end the

year for three years if future cash flows are discounted at 8%

annually?

Year Present Value of Receipt Total PV

1 $1,200 x (1/1.08) = $1,111.11 $1,111.11

2 $1,200 x (1/1.082) = $1,028.81 $2,139.92

3 $1,200 x (1/1.083) = $ 952.60 $3,092.52


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PMT

N

I/YR

PV

1

1200

3

8

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1 payment per year

$1,200 payment/year

3 year term

8% annual interest



The total present value is the sum of the present values:

PV = $1,200 (1/1.08) + $1,200 (1/1.082) + $1,200 (1/1.083)

= $1,200 [(1/1.08) + (1/1.082) + (1/1.083)]

= $1,200[0.9259 + 0.8573 + 0.7938]

= $1,200[2.5771]

= $3,092.52


The PV of an annual ordinary annuity is:

PV = PMT + ..... + PMT

= PMT

=PMT

dd

n11

1

{

( )}

where d is the annual discount rate.

1

1 d

1

1( ) d n

1

11 ( ) d tt

n

TVM: Present Value of a Periodic Ordinary Annuity

Compute the present value of an annuity that pays $100 at theend of each month for three years if future cash flows arediscounted monthly at an annual rate of 8%.

PV = $100 + ..... + $100

= $100[0.9934 + 0.9868 + ..... + 0.7872]

= $100[31.9118]

= $3,191.18

1

10812

1(.

)

1

10812

36(.

)


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P/YR

PMT

I/YR

PV

12

100

3

8

x P/YR

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Monthly payments

$100 payment/mo.

For 36 months

8% annual interest



In general, the PV of a periodic ordinary annuity is:

PV = PMT

where PMT = the periodic payment;

d/k = the periodic discount rate;

nk = the number of discounting periods inthe annuity term.

1

11 ( ) d

ktt

nk

TVM: Present Value of a Periodic Annuity Due

Recompute the present value of the $100 monthly annuity if thecash flows are received at the beginning of the month and arediscounted monthly at an annual rate of 8% .

PV = $100 + $100 + ... + $100

= $100 [ 1 + 0.9934 + ..... + 0.7925 ]

= $100 [32.1246]

= $3,212.46

1

108

121(

.)

1

10812

35(.

)


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12 P/YR

100 PMT

8 I/YR

3

BEG/END

PV

x P/YR

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Monthly payments

$100 monthly payment

8% annual interest36 months

Annuity due (BEGIN)



In general, the present value of a periodic annuity due is:

PV = PMT

where d/k = the periodic discount rate,

nk = the number of discounting periods, and

PMT = the equal periodic payment.

1

1 11 ( ) d

ktt

nk

TVM: Annual Sinking Fund Payment for Ordinary Annuities

How much has be set aside at the end of each year in anaccount paying 6% annual interest (compounded annually) toaccumulate to a future value of $5,000 in 3 years?

$5,000 = PMT (1.06)2 + PMT (1.06) + PMT

= PMT [ 1.1236 + 1.06 + 1]

= PMT [ 3.1836 ]

Solve for the annual sinking fund payment:

PMT = $5,000/3.1836 = $1,570.55

TVM: Annual Sinking Fund Payment for Ordinary Annuities

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6 I/YR

5000 FV

3 N

PMT

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6% annual interest

$5,000 future value

In three years

Compute the annual

payment.

TVM: Periodic Sinking Fund Payment for Ordinary Annuities

In general, the periodic sinking fund payment (PMT) forordinary annuities is:

PMT = FV

where i/k = the periodic interest rate,

nk = the number of periodic sinking fund payments,

FV= the desired future value.

1

11

( )

i

kt

nk nk t


Example: compute the quarterly payment, made at the end ofeach quarter, necessary to accumulate to $25,000 in six years ifthe payments earn 8% annual interest compounded quarterly.

PMT = $25,000

= $821.78

1

1008

41

24 24

(.

)

t

t


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P/YR

25000 FV

6

8 I/YR

PMT

x P/YR

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Quarterly payments

$25,000 future value

24 payments (6 years)

8% annual interest

Compute quarterly

payment

TVM: Periodic Sinking Fund Payment for an Annuity Due

Example: compute the quarterly payment, made at thebeginning of the period, necessary to accumulate to $25,000 insix years if the payments earn 8% annual interest compoundedquarterly.

PMT = $25,000

= $805.66 ($16.11 less than the ordinary annuity pmt)

1

1008

41

24 24 1

(.

)

t

t

TVM: Periodic Sinking Fund Payment for an Annuity Due

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4 P/YR

25,000 FV

6

8 I/YR

BEG/END

PMT

x P/YR

Clear registersQuarterly payments$25,000 future value24 quarterly payments8% annual interestAnnuity due (BEGIN)Compute quarterly

payment

TVM: Annual Payment to Amortize Debt for Ordinary Annuities

Compute the annual payment necessary to fully amortize(completely repay) an 8%, 3 year, $5000 loan if the paymentsare made at the end of the year.

$5,000 = PMT/1.08 + PMT/1.082 + PMT/1.083

= PMT [ 1/1.08 + 1/1.082 + 1/1.083 ]

= PMT [ 0.9259 + 0.8573 + 0.7938 ]

= PMT [ 2.5771 ]

So, PMT = $5,000/2.5771 = $1,940.17

TVM: Annual Payment to Amortize Debt for Ordinary Annuities

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5,000 PV

3 N

8 I/YR

PMT

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Annual payments


3 year term

8% annual interest

Compute annual payment

TVM: Periodic Payment to Amortize Debt for Ordinary Annuities

In general, the periodic payment (PMT) made at the end of theperiod necessary to amortize (completely repay) a loan is:

PMT = PV

wherePV = the loan amount, or present value,

d/k = the periodic discount (or interest) rate, and

nk = the number of periods in the loan term.

11

11( ) d

ktt

nk


Example: compute the constant monthly payment necessary tofully amortize a $100,000, 7.5% annual interest rate, 30 yearmortgage.

PMT = $100,000

= $ 699.21

11

10075

121

360

(.

)

tt


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PV

I/YR

PMT

12

100,000

7.5

30 x P/YR

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Monthly payments

$100,000 loan (PV)

7.5% annual interest

360 monthly pmts

Compute monthly payment

MORE TIME VALUE OF MONEY

• Perpetuities

• Growing Perpetuities

• The PV of Uneven Cash Flows

• The PV of Grouped Cash Flows

MORE TIME VALUE OF MONEY:Perpetuities

PERPETUITY: an infinite sequence of equal periodic cash flows (also called CONSOLS).

The present value of a perpetuity with periodic cash flow CFand periodic discount rate r is:

PV =

=

CF

r

CF

r

CF

r( ) ( ) ( ).......

1 1 12 3

CF

r

MORE TIME VALUE OF MONEY:Perpetuities

T h e p re s e n t v a lu e o f a p e rp e tu i ty th a t p ro v id e s $ 5 0 ,0 0 0 p e ry e a r ( fo re v e r ) d is c o u n te d a t 1 0 % a n n u a l ly is :

P V = $ 5 0 ,

.$ 5 0 0 ,

0 0 0

0 1 00 0 0

MORE TIME VALUE OF MONEY:Growing Perpetuities

The present value of a periodic perpetuity with end of periodcash flows

discounted at rate r (for r > g) is:

PV =

=

CF CF gjj ( )1

CF g

r

CF g

r

CF g

r

( )

( )

( )

( )

( )

( ).....

1

1

1

1

1

1

2 3

32

CF g

r g

( )1

MORE TIME VALUE OF MONEY:Growing Perpetuities

T h e p r e s e n t v a l u e o f a $ 5 0 , 0 0 0 a n n u a l p e r p e t u i t y t h a t i n c r e a s e s4 % p e r y e a r a n d i s d i s c o u n t e d a t 1 0 % a n n u a l l y i s :

P V $ 5 0 , ( . )

.

$ 5 0 , ( . )

.. . . .

0 0 0 1 0 4

1 1

0 0 0 1 0 4

1 1

2

2

P V $ 5 2 ,

.

$ 5 4 ,

.. . . . .

0 0 0

1 1

0 8 0

1 1 2

P V

$ 5 2 ,

. .

$ 5 2 ,

.$ 8 6 6 , .

0 0 0

0 1 0 0 4

0 0 0

0 0 66 6 6 6 7

MORE TIME VALUE OF MONEY:The PV of Uneven Cash Flows

P V =

T h e p r e s e n t v a l u e o f a s e q u e n c e o f d i f f e r e n t p e r i o d i c c a s hf l o w s ( C F 1 , C F 2 , C F 3 , . . . . . ) , d i s c o u n t e d a t t h e p e r i o d i c d i s c o u n tr a t e r , i s :

C F

r

C F

r

C F

r1 2

23

31 1 1( ) ( ) ( ). . . . . .

MORE TIME VALUE OF MONEY:The PV of Uneven Cash Flows

W h a t i s t h e p r e s e n t v a l u e o f r e c e i v i n g $ 5 0 0 o n e y e a r f r o mt o d a y , $ 1 , 5 0 0 t h r e e y e a r s f r o m t o d a y , a n d $ 2 , 5 0 0 f o u r y e a r sf r o m t o d a y i f f u t u r e r e c i e p t s a r e d i s c o u n t e d a n n u a l l y a t 1 0 % ?

P V =

= $ 4 5 4 . 5 5 + $ 1 , 1 2 6 . 9 7 + $ 1 , 7 0 7 . 5 3

= $ 3 , 2 8 9 . 0 5

$ 5 0 0

.

$ 1 ,

.

$ 2 ,

.1 1

5 0 0

1 1

5 0 0

1 13 4

MORE TVMThe PV of Uneven Cash Flows

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CFj

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NPV

CFj

CFj

CFj

CFj

1

0

500

0

1500

2500

10

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Initial CF = 0

1st CF = $500

2nd CF = $0

3rd CF = $1,500

4th CF = $2,500

Discount rate = 10%

Compute (net) present value

MORE TIME VALUE OF MONEY:The PV of Grouped Cash Flows

W h a t i s t h e p r e s e n t v a l u e o f a n i n v e s t m e n t t h a t i s e x p e c t e d t or e t u r n $ 1 0 , 0 0 0 p e r y e a r a t t h e e n d o f t h e n e x t t h r e e y e a r s ,$ 1 5 , 0 0 0 a t t h e e n d o f y e a r s 4 a n d 5 , a n d t h e n $ 1 0 0 , 0 0 0 a t t h ee n d o f y e a r 6 i f e x p e c t e d c a s h f l o w s a r e d i s c o u n t e d a n n u a l l y a t1 2 % ?

P V =

= $ 9 2 , 7 2 5 . 6 0 .

$ 1 0 ,.

$ 1 5 ,.

$ 1 0 0 ,

.0 0 0

1

1 1 20 0 0

1

1 1 2

0 0 0

1 1 21

3

4

5

6tt

tt

MORE TIME VALUE OF MONEY:The PV of Grouped Cash Flows

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P/YR1

0

10000

3

15000

2

100000

12

CFj

CFj

CFj

CFj

I/YR

NPV

N j

N j

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Initial CF = $0

1st grouped CF = $10,000

CF occurs 3 times

2nd grouped CF = $15,000

CF occurs 2 times

3rd CF = $100,000 (1 time)

Annual discount rate = 12%

Compute (net) present value

first principles of valuation

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