1. Present Value

2. Time Value of Money
Financial managers compare the marginal benefits and marginal cost of investment projects.
Projects usually have a long-term horizon: timing of benefits and costs matters.
Time-value of money concerned with adjusting for different timing of benefits and costs.

3. Future Value depends on:
The value of a lump sum or stream of cash payments at a future point in time:
FVn = PV x (1+r)n
Future Value depends on:
Interest rate
Number of periods
Compounding interval
Formula can be rearranged to compute required return, if price and dividend known:
Equity Valuation
As will be discussed in chapter 5, the required return on common stock is based on its beta, derived from the CAPM
Valuing CS is the most difficult, both practically & theoretically
Preferred stock valuation is much easier (the easiest of all)
Whenever investors feel the expected return, rˆ, is not equal to the required return, r, prices will react:
If exp return declines or reqd return rises, stock price will fall
If exp return rises or reqd return declines, stock price will rise
Asset prices can change for reasons besides their own risk
Changes in asset’s liquidity, tax status can change price
Changes in market risk premium can change all asset values
Most dramatic change in market risk: Russian default Fall 98
Caused required return on all risky assets to rise, price to fall

4. Future Value of $200 (4 Years, 7% Interest )
FV4 = $262.16
FV3 = $245.01
FV2 = $228.98
FV1 = $214
PV = $200
End of Year
Compounding: the process of earning interest in each successive year

5. Compounding Earns 7% interest on initial $200 FV1 = $200+$14 = $214
Year 1:
FV1 = $214
Earns 7% interest on initial $200
FV1 = $200+$14 = $214
Year 2:
FV2 = $228.98
Earn $14 interest again on $200 principal
Earns $0.98 on previous year’s interest of $14: $14 x 7% = $0.98
FV2 = $214+$14+$0.98 = $228.98
Year 3:
FV3 = $245.01
Earn $14 interest again on $200 principal
Earns $2.03 on previous years’ interest of $28.98: $28.98 x 7% = $2.03
FV3 = $ $14+$2.03 = $245.01
Earn $14 interest again on $200 principal
Earns $3.15 on previous years’ interest of $45.01: $45.01 x 7% = $3.15
FV4 = $ $14+$3.15 = $262.16
Year 4:
FV4 = $262.16

6. The Power of Compound Interest
20%
Future Value of One Dollar ($)
15%
10% 5% 0%
Periods

7. Present Value
Today's value of a lump sum or stream of cash payments received at a future point in time:
Formula can be rearranged to compute required return, if price and dividend known:
Equity Valuation
As will be discussed in chapter 5, the required return on common stock is based on its beta, derived from the CAPM
Valuing CS is the most difficult, both practically & theoretically
Preferred stock valuation is much easier (the easiest of all)
Whenever investors feel the expected return, rˆ, is not equal to the required return, r, prices will react:
If exp return declines or reqd return rises, stock price will fall
If exp return rises or reqd return declines, stock price will rise
Asset prices can change for reasons besides their own risk
Changes in asset’s liquidity, tax status can change price
Changes in market risk premium can change all asset values
Most dramatic change in market risk: Russian default Fall 98
Caused required return on all risky assets to rise, price to fall

8. Present Value of $200 (4 Years, 7% Interest )
Discounting
FV4 = $200
PV = $152.58
FV3 = $200
PV = $163.26
PV = $186.92
FV1 = $200
FV2 = $200
PV = $174.69
End of Year
Discounting: the process of converting a future cash flow into a present value

9. The Power of High Discount Rates
1.00 0%
0.75
Present Value of One Dollar ($)
0.5 5%
10%
0.25
15%
20%
Periods

10. Future Value and Present Value of an Ordinary Annuity
Compounding
Future Value
$1, $1, $1, $1, $1,000
End of Year
Present Value
Discounting

11. Future Value of Ordinary Annuity (End of 5 Years, 5.5% Interest Rate)
$1,238.82
$1,174.24
$1,113.02
$1,055.00
$1,000.00
$1, $1, $1, $1, $1,000
End of Year
How is annuity due different ?

12. Future Value of Annuity Due (End of 5 Years, 5.5% Interest Rate)
$1,306.96
$1,238.82
$1,174.24
$1,113.02
$1,055.00
$1, $1, $1, $1, $1,000
End of Year
Annuity due: payments occur at the beginning of each period

13. Present Value of Ordinary Annuity (5 Years, 5.5% Interest Rate)
$1, $1, $1, $1, $1,000
End of Year
$947.87
$898.45
$851.61
$807.22
$765.13

14. Present Value of Annuity Due (5 Years, 5.5% Interest Rate)
$1, $1, $1, $1, $1,000
End of Year
$947.87
$898.45
$851.61
$807.22

15. FV and PV of Mixed Stream (5 Years, 4% Interest Rate)
Compounding
- $12,166.5
FV $6,413.8
$3,509.6
$5,624.3
$4,326.4
$3,120.0
-$10, $3, $5, $4, $3, $2,000.0
End of Year
$2,884.6
PV $5,271.7
$4,622.8
$3,556.0
$2,564.4
$1,643.9
Discounting

16. Present Value of Perpetuity ($1,000 Payment, 7% Interest Rate)
Stream of equal annual cash flows that lasts “forever”
What if the payments grow at 2% per year?

17. Present Value of Growing Perpetuity
CF1 = $1,000
r = 7% per year
g = 2% per year
$1, $1,000(1+0.02)1 $1,000(1+0.02)2 $1,000(1+0.02)3 …
$1, $1, $1, $1,061.2

18. Compounding Intervals
m compounding periods
The more frequent the compounding period, the larger the FV!

19. Compounding More Frequently Than Annually
FV at end of 2 years of $125,000 deposited at 5% interest
For semiannual compounding, m equals 2:
For quarterly compounding, m equals 4:

20. Continuous Compounding
In extreme case, interest compounded continuously:
FVn = PV x (e r x n)
FV at end of 2 years of $125,000 at 5 % annual interest, compounded continuously:

21. The Stated Rate versus the Effective Rate
Stated rate: the contractual annual rate charged by lender or promised by borrower
Effective rate: the annual rate actually paid or earned

22. The Stated Rate versus the Effective Rate
FV of $100 at end of 1 year, invested at 5% stated annual interest, compounded:
Annually: FV = $100 (1.05)1 = $105
Semiannually: FV = $100 (1.025)2 = $105.06
Quarterly: FV = $100 (1.0125)4 = $105.09
Stated rate of 5% does not change. What about the effective rate?

23. Effective Rates: Always Greater Than or Equal to Stated Rates
For annual compounding, effective = stated
For semiannual compounding
For quarterly compounding

24. Much Of Finance Involves Finding Future And (Especially) Present Values
Central to all financial valuation techniques
Techniques used by investors and firms alike