1. Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

2. Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg
Notation:
CF => Cash Flow
CF0 => Cash flow now
CF1 => Cash flow one period ahead
CFt => Cash flow t period ahead
PV => Present Value
FV => Future Value
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

3. Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg
Simple Interest
FV = PV + Interest
FV = PV*(1 + i)
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

4. Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg
Value of Investing $1
Continuing in this manner you will find that the following amounts will be earned (10% interest rate):
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

5. Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg
Value of $5 Invested
More generally, with an investment of $5 at 10% we obtain
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

6. Generalizing the method
Generalizing the method requires some definitions. Let
i be the interest rate
n be the life of the lump sum investment
PV be the present value
FV be the future value
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

7. Future Value of a Lump Sum
FV with growths from 0% to +6%
1,000
1,500
2,000
2,500
3,000
3,500 2 4 6 8 10 12 14 16 18 20
Years
Future Value of $1000 6% 4% 2% 0%
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

8. Present Value of a Lump Sum
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

9. Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg
Lump Sums Formulae
You have solved a present value and a future value of a lump sum. There remains two other variables that may be solved for
interest, i
number of periods, n
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

10. Solving Lump Sum Cash Flow for Interest Rate
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

11. Solving Lump Sum Cash Flow for Number of Periods
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

12. The Frequency of Compounding
When there are n macroperiods
(1) FV = PV*(1+r)m*n
Or (2) FV = PV (1+i/m)m*n
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

13. The Frequency of Compounding
When we are presented with an APR and m compounding periods.
EAR = (1 + APR/m)m
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

14. Effective Annual Rates of an APR of 18%
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

15. The Frequency of Compounding
Note that as the frequency of compounding increases, so does the annual effective rate
What occurs as the frequency of compounding rises to infinity?
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

16. Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg
Multiple Cash Flows
Value a promise for $100 one year from today, and $200 two years from today. Given 10% annual rate.
Time line : CF $ $ $200 Time
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

17. Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg
Multiple Cash Flows
Generalizing the method. Let
i be the interest rate
t time periods counter
T time period of the last cash flow
CFt be cash flow at time t
PV be the present value
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

18. Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg
Multiple Cash Flows
Present value of multiple cash flows
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

19. Net Present Value (NPV)
NPV = - PV(All outflows) + PV(All inflows)
If NPV > 0 (inflows exceed outflows) -- Accept the project
If NPV < (inflows are less than outflows) -- Reject the project
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

20. Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg
Perpetuity
A stream of cash flows the last forever.
A constant cash flow:
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

21. Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg
Perpetuity
A g – constant growth cash flow, growth after the first period and g < i:
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

22. Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg
Annuities
a sequence of equally spaced identical (or constantly growing) cash flows
regular annuity with its first cash flow one period from now
annuity due with its first cash flow today
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

23. Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg
Annuities
Four period annuity replication with two perpetuity.
$ $ $ $ $ $ $
$ $ $
$ $ $ $ =
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

24. Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg
Annuities
Annuity Formula Notation
PV the present value of the annuity
I interest rate to be earned over the life of the annuity
n the number of payments
pmt the periodic payment
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

25. PV Annuity Formula: Payment
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

26. Annuity Formula: PV Annuity Due
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

27. Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg
Growing Annuities
Annuity cash flows that grow at a constant rate (g) after the first cash flow:
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg

28. PV Annuity Formula: Number of Payments
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Fin307 -Review Time Value of Money | Dr. Menahem Rosenberg