1. CHAPTER 5 Time Value of Money
Future value
Present value
Annuities
Rates of return
Amortization

2. Last week Objective of the firm Business forms Agency conflicts
Capital budgeting decision and capital structure decision

3. The plan of the lecture Time value of money concepts
present value (PV)
discount rate/interest rate (r)
Formulae for calculating PV of
perpetuity
annuity
Interest compounding
How to use a financial calculator

4. Financial choices with time
Which would you rather receive?
$1000 today
$1040 in one year
Both payments have no risk, that is,
there is 100% probability that you will be paid

5. Financial choices with time
Why is it hard to compare ?
$1000 today
$1040 in one year
This is not an “apples to apples” comparison. They have different units
$1000 today is different from $1000 in one year
Why?
A cash flow is time-dated money

6. Present value To have an “apple to apple” comparison, we
convert future payments to the present values
or convert present payments to the future values
This is like converting money in Canadian $ to money in US $.

7. Some terms
Finding the present value of some future cash flows is called discounting.
Finding the future value of some current cash flows is called compounding.

8. What is the future value (FV) of an initial $100 after 3 years, if i = 10%?
Finding the FV of a cash flow or series of cash flows is called compounding.
FV can be solved by using the arithmetic, financial calculator, and spreadsheet methods.
FV = ? 1 2 3
10%
100

9. Solving for FV: The arithmetic method
After 1 year:
FV1 = c ( 1 + i ) = $100 (1.10) = $110.00
After 2 years:
FV2 = c (1+i)(1+i) = $100 (1.10) =$121.00
After 3 years:
FV3 = c ( 1 + i )3 = $100 (1.10) =$133.10
After n years (general case):
FVn = C ( 1 + i )n

10. Set up the Texas instrument
2nd, “FORMAT”, set “DEC=9”, ENTER
2nd, “FORMAT”, move “↓” several times, make sure you see “AOS”, not “Chn”.
2nd, “P/Y”, set to “P/Y=1”
2nd, “BGN”, set to “END”
P/Y=periods per year,
END=cashflow happens end of periods

11. Solving for FV: The calculator method
Solves the general FV equation.
Requires 4 inputs into calculator, and it will solve for the fifth. 3 10
-100
INPUTS N
I/YR PV
PMT FV
OUTPUT
133.10

12. What is the present value (PV) of $100 received in 3 years, if i = 10%?
Finding the PV of a cash flow or series of cash flows is called discounting (the reverse of compounding).
The PV shows the value of cash flows in terms of today’s worth. 1 2 3
10%
PV = ?
100

13. Solving for PV: The arithmetic method
i: interest rate, or discount rate
PV = C / ( 1 + i )n
PV = C / ( 1 + i )3
= $100 / ( 1.10 )3
= $75.13

14. Solving for PV: The calculator method
Exactly like solving for FV, except we have different input information and are solving for a different variable. 3 10
100
INPUTS N
I/YR PV
PMT FV
OUTPUT
-75.13

15. Solving for N: If your investment earns interest of 20% per year, how long before your investments double? 20 -1 2
INPUTS N
I/YR PV
PMT FV
OUTPUT
3.8

16. Solving for i: What interest rate would cause $100 to grow to $125
Solving for i: What interest rate would cause $100 to grow to $ in 3 years? 3
-100
125.97
INPUTS N
I/YR PV
PMT FV
OUTPUT 8

17. Now let’s study some interesting patterns of cash flows…
Annuity
Perpetuity

18. ordinary annuity and annuity due
PMT 1 2 3 i%
PMT 1 2 3 i%
Annuity Due

19. Value an ordinary annuity
Here C is each cash payment
n is number of payments
If you’d like to know how to get the formula below (not required), see me after class.

20. Solving for FV: 3-year ordinary annuity of $100 at 10%
$100 payments occur at the end of each period. Note that PV is set to 0 when you try to get FV. 3 10
-100
INPUTS N
I/YR PV
PMT FV
OUTPUT
331

21. Solving for PV: 3-year ordinary annuity of $100 at 10%
$100 payments still occur at the end of each period. FV is now set to 0. 3 10
100
INPUTS N
I/YR PV
PMT FV
OUTPUT

22. Example
you win the $1million dollar lottery! but wait, you will actually get paid $50,000 per year for the next 20 years if the discount rate is a constant 7% and the first payment will be in one year, how much have you actually won?

23. Solving for FV: 3-year annuity due of $100 at 10%
$100 payments occur at the beginning of each period.
FVAdue= FVAord(1+i) = $331(1.10) = $
Alternatively, set calculator to “BEGIN” mode and solve for the FV of the annuity:
BEGIN 3 10
-100
INPUTS N
I/YR PV
PMT FV
OUTPUT
364.10

24. Solving for PV: 3-year annuity due of $100 at 10%
$100 payments occur at the beginning of each period.
PVAdue= PVAord(1+I) = $248.69(1.10) = $
Alternatively, set calculator to “BEGIN” mode and solve for the PV of the annuity:
BEGIN 3 10
100
INPUTS N
I/YR PV
PMT FV
OUTPUT

25. What is the present value of a 5-year $100 ordinary annuity at 10%?
Be sure your financial calculator is set back to END mode and solve for PV:
N = 5, I/YR = 10, PMT = 100, FV = 0.
PV = $379.08

26. What if it were a 10-year annuity? A 25-year annuity? A perpetuity?
N = 10, I/YR = 10, PMT = 100, FV = 0; solve for PV = $
25-year annuity
N = 25, I/YR = 10, PMT = 100, FV = 0; solve for PV = $
Perpetuity (N=infinite)
PV = PMT / i = $100/0.1 = $1,000.

27. What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%?
$297.22
$323.97
$ $ $ $100

28. What is the PV of this uneven cash flow stream?
100 1
300 2 3
10%
-50 4
90.91
247.93
225.39
-34.15
= PV

29. Solving for PV: Uneven cash flow stream
Input cash flows in the calculator’s “CF” register:
CF0 = 0
CF1 = 100
CF2 = 300
CF3 = 300
CF4 = -50
Enter I/YR = 10, press NPV button to get NPV = $ (Here NPV = PV.)

30. Detailed steps (Texas Instrument calculator)
To clear historical data:
CF, 2nd ,CE/C
To get PV:
CF , ↓,100 , Enter , ↓,↓ ,300 , Enter, ↓,2,
Enter, ↓, 50, +/-,Enter, ↓,NPV,10,Enter, ↓,CPT
“NPV= ”

31. The Power of Compound Interest
A 20-year-old student wants to start saving for retirement. She plans to save $3 a day. Every day, she puts $3 in her drawer. At the end of the year, she invests the accumulated savings ($1,095=$3*365) in an online stock account. The stock account has an expected annual return of 12%.
How much money will she have when she is 65 years old?

32. Solving for FV: Savings problem
If she begins saving today, and sticks to her plan, she will have $1,487, when she is 65. 45 12
-1095
INPUTS N
I/YR PV
PMT FV
OUTPUT
1,487,262

33. Solving for FV: Savings problem, if you wait until you are 40 years old to start
If a 40-year-old investor begins saving today, and sticks to the plan, he or she will have $146, at age 65. This is $1.3 million less than if starting at age 20.
Lesson: It pays to start saving early. 25 12
-1095
INPUTS N
I/YR PV
PMT FV
OUTPUT
146,001

34. Will the FV of a lump sum be larger or smaller if compounded more often, holding the stated i% constant?
LARGER, as the more frequently compounding occurs, interest is earned on interest more often. 1 2 3
10%
100
133.10
Annually: FV3 = $100(1.10)3 = $133.10 1 2 3 5% 4 5 6
134.01
100
Semiannually: FV6 = $100(1.05)6 = $134.01

35. What is the FV of $100 after 3 years under 10% semiannual compounding
What is the FV of $100 after 3 years under 10% semiannual compounding? Quarterly compounding?

36. Classifications of interest rates
1. Nominal rate (iNOM) – also called the APR, quoted rate, or stated rate. An annual rate that ignores compounding effects. Periods must also be given, e.g. 8% Quarterly.
2. Periodic rate (iPER) – amount of interest charged each period, e.g. monthly or quarterly.
iPER = iNOM / m, where m is the number of compounding periods per year. e.g., m = 12 for monthly compounding.

37. Classifications of interest rates
3. Effective (or equivalent) annual rate (EAR, also called EFF, APY) : the annual rate of interest actually being earned, taking into account compounding.
If the interest rate is compounded m times in a year, the effective annual interest rate is

38. Example, EAR for 10% semiannual investment
An investor would be indifferent between an investment offering a 10.25% annual return, and one offering a 10% return compounded semiannually.

39. EAR on a Financial Calculator
Texas Instruments BAII Plus
keys:
description:
[2nd] [ICONV]
Opens interest rate conversion menu
[↑] [C/Y=] 2 [ENTER]
Sets 2 payments per year
[↓][NOM=] 10 [ENTER]
Sets 10 APR.
[↓] [EFF=] [CPT]
10.25

40. Why is it important to consider effective rates of return?
An investment with monthly payments is different from one with quarterly payments.
Must use EAR for comparisons.
If iNOM=10%, then EAR for different compounding frequency:
Annual %
Quarterly %
Monthly %
Daily %

41. If interest is compounded more than once a year
EAR (EFF, APY) will be greater than the nominal rate (APR).

44. What’s the FV of a 3-year $100 annuity, if the quoted interest rate is 10%, compounded semiannually?1
100 2 3 5% 4 5 6
Payments occur annually, but compounding occurs every 6 months.
Cannot use normal annuity valuation techniques.

45. Method 1: Compound each cash flow1
100 2 3 5% 4 5 6
110.25
121.55
331.80
FV3 = $100(1.05)4 + $100(1.05)2 + $100
FV3 = $331.80

46. Method 2: Financial calculator
Find the EAR and treat as an annuity.
EAR = ( / 2 )2 – 1 = 10.25%. 3
10.25
-100
INPUTS N
I/YR PV
PMT FV
OUTPUT
331.80

47. When is periodic rate used?
iPER is often useful if cash flows occur several times in a year.