1. Introduction to Present Value
Text: Chapter 4

2. How much are you willing to pay for a project with payoff of $40,000 in a year?

3. Present Value
If the project is riskless, then use return of government bond as a guide, say 8%, how much are you willing to pay?
$40,000/1.08 = 37,037
[ $40,000 * (1/1.08) = $40,000 * 0.926]
Present value of this project is $37,037
Future value of this project is $40,000
Discount rate
Discount factors

4. Present Value PV = C1 * [1 / (1+r)] = C1 * discount factor
where r is discount rate, opportunity cost of capital
[1/1+r] is discount factor

5. Future Value
Future Value = PV * (1+r)

6. The First Rule in Finance
One dollar today is worth more than one dollar tomorrow

7. Present Value and Future Value

8. Present Value What if the project is risky?
Suppose the return is as risky as the stock index, which offers 12% in the long run, what is the PV of $40,000 next year?
$40,000 / (1.12) = $ < $37037
(8% discount rate)

9. The Second Rule in Finance
A safe dollar is worth more
than a risky one!

10. Introduction to Net Present Value
If the riskless project is sold at $37,000, is this a good deal?
Net Present Value = C0 + C1* (1/1+r)
NPV = = 37
Accept any project with positive net present value!

11. NPV > 0 <==> rate of return > opportunity cost of capital
NPV and Cost of Capital
In the previous example, the rate of return
[40,000 / 37,000] - 1 = 8.1% > 8% (discount rate)
NPV > 0 <==> rate of return > opportunity cost of capital

12. Opportunity Cost of Capital
Example
You may invest $100,000 today. Depending on the state of the economy, you may get one of three possible payoffs:

13. Opportunity Cost of Capital
Example - continued
The stock is expected to have a 15% rate of return.
Discounting the expected payoff at the expected return leads to the PV of the project
Initial Investment: $100,000

14. Opportunity Cost of Capital
Example - continued
Notice that you come to the same conclusion if you compare the expected project return with the cost of capital.
But ….Cost of capital: 15%

15. No money machine exists for a lasting time.
A simple question …..
Should DF2 always be less than DF1?
[ie, 1/(1+r2)2 > 1/(1+r1) ]?
If not, say DF2 = .8, and DF1=.7, then we can
1. Borrow $.8 and return $1 in period 2
2. Lend $.7 and get $1 back at period 1, then hold it to period 2
We end with $.1 without investing anything!
But if everybody does this, then
r2  (DF2 ) and r1  (DF1 )
until no arbitrage conditions exists any more
No money machine exists for a lasting time.

16. The Calculation of Present Value

17. Present Values
PVs can be added together to evaluate multiple cash flows.

18. Present Value of Multiple Cash Flow
$200
$100
Present Value
Year 0
100/1.07
200/1.0772
Total
= $93.46
= $172.42
= $265.88
Year

19. Short Cuts
Sometimes there are shortcuts that make it very easy to calculate the present value of an asset that pays off in different periods. These tolls allow us to cut through the calculations quickly.

20. Perpetuity
A Perpetuity is a constant stream of CF without end.
PVt = Ct+1 / r
…forever...
| | | |
(r = 10%)
$100 $100 $ forever…
PV0 = $100 / 0.1 = $1000

21. Growing Perpetuity
a stream of cash flows that grows at a constant rate forever.
Simplification: PVt = Ct+1 / (r - g)
…forever...
| | | | (r = 10%)
$ $102 $ … (g = 2%)
PV0 = $100 / ( ) = $1250

22. Annuity
Annuity - An asset that pays a fixed sum each year for a specified number of years.
| | | | (r = 10%)
$ $100 $100

23. Short Cuts
Annuity - An asset that pays a fixed sum each year for a specified number of years.
Asset
Year of Payment
…..t t + 1
Present Value
Perpetuity (first payment in year 1)
Perpetuity (first payment in year t + 1)
Annuity from year 1 to year t

24. Short Cuts
Annuity - An asset that pays a fixed sum each year for a specified number of years.

25. Annuity Short Cut
Example
You agree to lease a car for 4 years at $300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease?

26. Annuity Short Cut Example - continued
You agree to lease a car for 4 years at $300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease?

27. Compound Interest i ii iii iv v Periods Interest Value Annually
per per APR after compounded
year period (i x ii) one year interest rate
% % % = = = = =

28. Compound Interest

29. Effective Annual Interest Rates
Compounding periods (m)
How often is interest computed
Stated (nominal) Annual Percentage Rate (r)
What the bank usually quotes
Effective Annual Interest Rate (EAR):
EAR = (1 + r / m) m - 1
As m approaches infinity, (1 + r / m) m > er
EAR of continuous compounding = er - 1

30. 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.

31. Inflation
Example
If the interest rate on one year govt. bonds is 5.9% and the inflation rate is 3.3%, what is the real interest rate?
Savings
Bond