Corporate Finance
Time Value of Money

Session One
Decision Dilemma—Take a Lump Sum
or Annual Installments

 A suburban Chicago couple won the
Lottery.

 They had to choos...
Time Value of Money

“The idea that money available
at the present time is worth
more than the same amount in
the future d...
Future Values and Compound Interest
Future Value – Amount to which an investment
will grow after earning interest.
Compoun...
Future Values
You have $100 invested in a bank account. Suppose banks are
currently paying an interest rate of 6 percent p...
Future Values
Example - Simple Interest
Interest earned at a rate of 6% for five years on a
principal balance of $100.
Tod...
Future Values
Example - Compound Interest
Interest earned at a rate of 6% for five years on the
previous year’s balance.
I...
Future Values
Example - Compound Interest
Interest earned at a rate of 6% for five years on the
previous year’s balance.
T...
Practice Problem: Warren Buffett’s Berkshire
Hathaway
• Went public in 1965: $18 per share
• Worth today (August 22, 2003)...
Market Value
• Assume that the company’s stock will continue to
appreciate at an annual rate of 24.58% for the next 27
yea...
Manhattan Island Sale
Peter Minuit bought Manhattan Island for $24 in 1629.
Was this a good deal? Suppose the interest rat...
Manhattan Island Sale
Peter Minuit bought Manhattan Island for $24 in 1629.
Was this a good deal?
To answer, determine $24...
Present Values
P resen t V a lu e = P V

PV =

F u tu re V alu e after t p erio d s
(1 + r)

t
Present Values
Example
You just bought a new computer for $3,000. The payment
terms are 2 years same as cash. If you can e...
PV of Multiple Cash Flows
Example
Your auto dealer gives you the choice to pay $15,500 cash
now, or make three payments: $...
Present Values
$8,000
$4,000

$ 4,000

Present Value
Year 0

0

$8,000

4000/1.08

= $3,703.70

4000/1.082

= $3,429.36

T...
PV of Multiple Cash Flows
• PVs can be added together to evaluate
multiple cash flows.

PV

C1
(1 r )

C2
1

(1 r )

2

.....
Perpetuities & Annuities
Perpetuity
A stream of level cash payments that
never ends.

Annuity
Equally spaced level stream ...
Perpetuities
PV of Perpetuity Formula

PV
C = cash payment
r = interest rate

C
r
Perpetuities
Example - Perpetuity
In order to create an endowment, which pays
$100,000 per year, forever, how much money m...
Perpetuities
Example - continued
If the first perpetuity payment will not be received
until three years from today, how mu...
Annuities
PV of Annuity Formula

PV

C

1
r

1
r (1 r )

t

C = cash payment
r = interest rate
t = Number of years cash pa...
Annuities
PV Annuity Factor (PVAF) - The present value of
$1 a year for each of t years.

PVAF

1
r

1
r (1 r )

t
Annuities
Example - Annuity
You are purchasing a car. You are scheduled to make
3 annual installments of $4,000 per year. ...
Annuities
Applications
• Value of payments
• Implied interest rate for an annuity
• Calculation of periodic payments
– Mor...
Annuities
Example - Future Value of annual payments
You plan to save $4,000 every year for 20 years and
then retire. Given...
Effective Interest Rates
Effective Annual Interest Rate - Interest rate
that is annualized using compound
interest.

Annua...
Effective Interest Rates
example
Given a monthly rate of 1%, what is the Effective
Annual Rate(EAR)? What is the Annual Pe...
Inflation
Inflation - Rate at which prices as a whole are
increasing.
Nominal Interest Rate - Rate at which money
invested...
Inflation
1

1 + n o m in a l in te re st ra te
re a l in te re st ra te =
1 + in fla tio n ra te

approximation formula

...
Inflation
Example
If the interest rate on one year govt. bonds is 6.0%
and the inflation rate is 2.0%, what is the real
in...
The End
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Time value of money

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The idea that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. This core principle of finance holds that, provided money can earn interest, any amount of money is worth more the sooner it is received.

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Time value of money

  1. 1. Corporate Finance Time Value of Money Session One
  2. 2. Decision Dilemma—Take a Lump Sum or Annual Installments  A suburban Chicago couple won the Lottery.  They had to choose between a single lump sum $104 million, or $198 million paid out over 25 years (or $7.92 million per year).  The winning couple opted for the lump sum.  Did they make the right choice? What basis do we make such an economic comparison?  To make such comparisons (the lottery decision problem), we must be able to compare the value of money at different point in time.
  3. 3. Time Value of Money “The idea that money available at the present time is worth more than the same amount in the future due to its potential earning capacity.”
  4. 4. Future Values and Compound Interest Future Value – Amount to which an investment will grow after earning interest. Compound Interest – Interest earned on interest. Simple Interest – Interest earned only on original investment; no interest is earned on interest.
  5. 5. Future Values You have $100 invested in a bank account. Suppose banks are currently paying an interest rate of 6 percent per year on deposits. So after a year, your account will earn interest of $6: FV = PV (1 + r) ͭ FV = 100 (1+06)¹ Value of investment after 1 year = $106
  6. 6. Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Today 1 Interest Earned Value 100 6 106 Value at the end of Year 5 = $130 Future Years 2 3 6 112 6 118 4 6 124 5 6 130
  7. 7. Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Interest Earned Per Year =Prior Year Balance x .06
  8. 8. Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Today 1 Interest Earned Value 100 6 106 Future Years 2 3 6.36 112.36 Value at the end of Year 5 = $133.82 6.74 119.10 4 7.15 126.25 5 7.57 133.82
  9. 9. Practice Problem: Warren Buffett’s Berkshire Hathaway • Went public in 1965: $18 per share • Worth today (August 22, 2003): $76,200 • Annual compound growth: 24.58% • Current market value: $100.36 Billion • If he lives till 100 (current age: 73 years as of 2003), his company’s total market value will be ?
  10. 10. Market Value • Assume that the company’s stock will continue to appreciate at an annual rate of 24.58% for the next 27 years. F $100.36(1 0.2458) $37.902 trillions 27
  11. 11. Manhattan Island Sale Peter Minuit bought Manhattan Island for $24 in 1629. Was this a good deal? Suppose the interest rate is 8%.
  12. 12. Manhattan Island Sale Peter Minuit bought Manhattan Island for $24 in 1629. Was this a good deal? To answer, determine $24 is worth in the year 2003, compounded at 8%. FV $ 24 (1 . 08 ) 374 $ 75 . 979 trillion FYI - The value of Manhattan Island land is well below this figure.
  13. 13. Present Values P resen t V a lu e = P V PV = F u tu re V alu e after t p erio d s (1 + r) t
  14. 14. Present Values Example You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years? PV 3000 ( 1 . 08 ) 2 $ 2 ,572
  15. 15. PV of Multiple Cash Flows Example Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer? Im ediate m PV 1 PV 2 Total PV paym ent 4 , 000 (1 1 3 , 703 . 70 2 3 , 429 . 36 . 08 ) 4 , 000 (1 . 08 ) 8,000.00 $15,133.06
  16. 16. Present Values $8,000 $4,000 $ 4,000 Present Value Year 0 0 $8,000 4000/1.08 = $3,703.70 4000/1.082 = $3,429.36 Total = $15,133.06 1 2 Year
  17. 17. PV of Multiple Cash Flows • PVs can be added together to evaluate multiple cash flows. PV C1 (1 r ) C2 1 (1 r ) 2 ....
  18. 18. Perpetuities & Annuities Perpetuity A stream of level cash payments that never ends. Annuity Equally spaced level stream of cash flows for a limited period of time.
  19. 19. Perpetuities PV of Perpetuity Formula PV C = cash payment r = interest rate C r
  20. 20. Perpetuities Example - Perpetuity In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today if the rate of interest is 10%? PV 100 ,000 .1 0 $ 1, 0 0 0 , 0 0 0
  21. 21. Perpetuities Example - continued If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today? PV 1,000 ,000 ( 1 .1 0 ) 3 $ 7 5 1, 3 1 5
  22. 22. Annuities PV of Annuity Formula PV C 1 r 1 r (1 r ) t C = cash payment r = interest rate t = Number of years cash payment is received
  23. 23. Annuities PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years. PVAF 1 r 1 r (1 r ) t
  24. 24. Annuities Example - Annuity You are purchasing a car. You are scheduled to make 3 annual installments of $4,000 per year. Given a rate of interest of 10%, what is the price you are paying for the car (i.e. what is the PV)? 1 .1 0 PV 4 ,0 0 0 PV $ 9 , 9 4 7 .4 1 1 .1 0 ( 1 .1 0 ) 3
  25. 25. Annuities Applications • Value of payments • Implied interest rate for an annuity • Calculation of periodic payments – Mortgage payment – Annual income from an investment payout – Future Value of annual payments t FV C PVAF (1 r )
  26. 26. Annuities Example - Future Value of annual payments You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account? 1 .1 0 FV 4 ,0 0 0 FV $ 2 2 9 ,1 0 0 1 .1 0 ( 1 .1 0 ) 20 ( 1 .1 0 ) 20
  27. 27. Effective Interest Rates Effective Annual Interest Rate - Interest rate that is annualized using compound interest. Annual Percentage Rate - Interest rate that is annualized using simple interest.
  28. 28. Effective Interest Rates example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)? EAR = (1 + .01) 12 -1 = r EAR = (1 + .01) 12 - 1 = .1268 or 12.68% APR = .01 x 12 = .12 or 12.00%
  29. 29. Inflation Inflation - Rate at which prices as a whole are increasing. Nominal Interest Rate - Rate at which money invested grows. Real Interest Rate - Rate at which the purchasing power of an investment increases.
  30. 30. Inflation 1 1 + n o m in a l in te re st ra te re a l in te re st ra te = 1 + in fla tio n ra te approximation formula R eal in t. rate n o m in al in t. rate - in flatio n rate
  31. 31. Inflation Example If the interest rate on one year govt. bonds is 6.0% and the inflation rate is 2.0%, what is the real interest rate? 1 1+.06 real int erestrat e = 1+.02 1 real int erestrat e = 1.039 real int erestrat e= .039or 3.9% Approximat ion = .06 - .02 = .04 or 4.0%
  32. 32. The End

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