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CHAPTER 6 Time Value of Money

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CHAPTER 6Time Value of Money

Future value

Present value

Annuities

Rates of return

Amortization

0

1

2

3

i%

CF0

CF1

CF2

CF3

- Show the timing of cash flows.
- Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period.

$100 lump sum due in 2 years

0

1

2

i%

100

3 year $100 ordinary annuity

0

1

2

3

i%

100

100

100

Uneven cash flow stream

0

1

2

3

i%

-50

100

75

50

0

1

2

3

10%

100

FV = ?

- Finding the FV of a cash flow or series of cash flows when compound interest is applied is called compounding.
- FV can be solved by using the arithmetic, financial calculator, and spreadsheet methods.

- After 1 year:
- FV1 = PV ( 1 + i ) = $100 (1.10) = $110.00

- After 2 years:
- FV2 = PV ( 1 + i )2 = $100 (1.10)2 =$121.00

- After 3 years:
- FV3 = PV ( 1 + i )3 = $100 (1.10)3 =$133.10

- After n years (general case):
- FVn = PV ( 1 + i )n

- Solves the general FV equation.
- Requires 4 inputs into calculator, and will solve for the fifth. (Set to P/YR = 1 and END mode.)

3

10

-100

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

133.10

- Finding the PV of a cash flow or series of cash flows when compound interest is applied is called discounting (the reverse of compounding).
- The PV shows the value of cash flows in terms of today’s purchasing power.

0

1

2

3

10%

PV = ?

100

- Solve the general FV equation for PV:
- PV = FVn / ( 1 + i )n
- PV = FV3 / ( 1 + i )3
= $100 / ( 1.10 )3

= $75.13

- Solves the general FV equation for PV.
- Exactly like solving for FV, except we have different input information and are solving for a different variable.

3

10

0

100

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

-75.13

- Solves the general FV equation for N.
- Same as previous problems, but now solving for N.

20

-1

0

2

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

3.8

Future Values – Example 3

- Suppose you had a relative deposit $10 at 5.5% interest 200 years ago. How much would the investment be worth today?
- 200 N
- 5.5 I/Y
- 10 PV
- CPT FV = -447,189.84

- What is the effect of compounding?
- Simple interest = 10 + 200(10)(.055) = 210.55
- Compounding added $446,979.29 to the value of the investment

Future Value as a General Growth Formula

- Suppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you currently sell 3 million widgets in one year, how many widgets do you expect to sell in 5 years?
- 5 N
- 15 I/Y
- 3,000,000 PV
- CPT FV = -6,034,072 units (remember the sign convention)

Ordinary Annuity

0

1

2

3

i%

PMT

PMT

PMT

Annuity Due

0

1

2

3

i%

PMT

PMT

PMT

- $100 payments occur at the end of each period, but there is no PV.

3

10

0

-100

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

331

- $100 payments still occur at the end of each period, but now there is no FV.

3

10

100

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

-248.69

- Now, $100 payments occur at the beginning of each period.
- Set calculator to “BEGIN” mode.

3

10

0

-100

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

364.10

- Again, $100 payments occur at the beginning of each period.
- Set calculator to “BEGIN” mode.

3

10

100

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

-273.55

Present Values – Example 2

- You want to begin saving for you daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today?
- N = 17
- I/Y = 8
- FV = 150,000
- CPT PV = -40,540.34 (remember the sign convention)

You have just received notification that you have won the $2 million first prize in the Millennium Lottery. However, the prize will be awarded on your 100th birthday (assuming you’re around to collect),

80 years from now. What is the present value of your windfall if the appropriate discount rate is 14 percent

PV = FV / (1 + r)t

PV = $2,000,000 / (1 + .14)80

PV = $56.06

Benjamin Franklin died on April 17, 1790. In his will, he gave 1,000 pounds sterling to Massachusetts and the city of Boston. He gave a like amount to Pennsylvania and the city of Philadelphia. The money was paid to Franklin when he held political office, but he believed that politicians should not be paid for their service(!).

Franklin originally specified that the money should be paid out 100 years after his death and used to train young people. Later, however, after some legal wrangling, it was agreed that the money would be paid out 200 years after Franklin’s death in 1990. By that time, the Pennsylvania bequest had grown to about $2 million; the Massachusetts bequest had grown to $4.5 million. The money was used to fund the Franklin Institutes in Boston and Philadelphia.

Assuming that 1,000 pounds sterling was equivalent to 1,000 dollars, what rate did the two states earn? (Note: the dollar didn’t become the official U.S. currency until 1792.)

Assuming that 1,000 pounds sterling was equivalent to 1,000 dollars, what rate did the two states earn?

A.For Pennsylvania, the future value is $ 2 million and the present value is $ 1,000. There are 200 years involved, so we need to solve for r in the following:

$ 1,000 = $ 2 million/(1 + r)200

(1 + r)200 = 2,000.00

Solving for r, the Pennsylvania money grew at about 3.87% per year. The Massachusetts money did better; check that the rate of return in this case was 4.3%. Small differences can add up!

4

0

1

2

3

10%

100

300

300

-50

90.91

247.93

225.39

-34.15

530.08 = PV

- Input cash flows in the calculator’s “CFLO” register:
- CF0 = 0
- CF1 = 100
- CF2 = 300
- CF3 = 300
- CF4 = -50

- Enter I/YR = 10, press NPV button to get NPV = $530.09. (Here NPV = PV.)

- Solves the general FV equation for I.

3

-100

0

125.97

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

8

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) 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?

- If she begins saving today, and sticks to her plan, she will have $1,487,261.89 when she is 65.

45

12

0

-1095

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

1,487,262

- If a 40-year-old investor begins saving today, and sticks to the plan, he or she will have $146,000.59 at age 65. This is $1.3 million less than if starting at age 20.
- Lesson: It pays to start saving early.

25

12

0

-1095

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

146,001

- To find the required annual contribution, enter the number of years until retirement and the final goal of $1,487,261.89, and solve for PMT.

25

12

0

1,487,262

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

-11,154.42

0

1

2

3

10%

100

133.10

0

1

2

3

4

5

6

0

1

2

3

5%

100

134.01

- LARGER, as the more frequently compounding occurs, interest is earned on interest more often.

Annually: FV3 = $100(1.10)3 = $133.10

Semiannually: FV6 = $100(1.05)6 = $134.01

Decisions, Decisions

- Your broker calls you and tells you that he has this great investment opportunity. If you invest $100 today, you will receive $40 in one year and $75 in two years. If you require a 15% return on investments of this risk, should you take the investment?
- Use the CF keys to compute the value of the investment
- CF; CF0 = 0; C01 = 40; F01 = 1; C02 = 75; F02 = 1
- NPV; I = 15; CPT NPV = 91.49

- No – the broker is charging more than you would be willing to pay.

- Use the CF keys to compute the value of the investment

Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today?

PV = 333,333.33[1 – 1/1.0530] / .05 = 5,124,150.29

30 N; 5 I/Y; 333,333.33 PMT; CPT PV = 5,124,150.29

- Nominal rate (iNOM) – also called the quoted or state rate. An annual rate that ignores compounding effects.
- iNOM is stated in contracts. Periods must also be given, e.g. 8% Quarterly or 8% Daily interest.

- 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. m = 4 for quarterly and m = 12 for monthly compounding.

- Effective (or equivalent) annual rate (EAR = EFF%) – the annual rate of interest actually being earned, taking into account compounding.
- EFF% for 10% semiannual investment
EFF%= ( 1 + iNOM / m )m - 1

= ( 1 + 0.10 / 2 )2 – 1 = 10.25%

- An investor would be indifferent between an investment offering a 10.25% annual return and one offering a 10% annual return, compounded semiannually.

- EFF% for 10% semiannual investment

- An investment with monthly payments is different from one with quarterly payments. Must put each return on an EFF% basis to compare rates of return. Must use EFF% for comparisons. See following values of EFF% rates at various compounding levels.
EARANNUAL10.00%

EARQUARTERLY10.38%

EARMONTHLY10.47%

EARDAILY (365)10.52%

- Yes, but only if annual compounding is used, i.e., if m = 1.
- If m > 1, EFF% will always be greater than the nominal rate.

- iNOMwritten into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.
- iPERUsed in calculations and shown on time lines. If m = 1, iNOM = iPER = EAR.
- EARUsed to compare returns on investments with different payments per year. Used in calculations when annuity payments don’t match compounding periods.

1

2

3

0

1

2

3

4

5

6

5%

100

100

100

- Payments occur annually, but compounding occurs every 6 months.
- Cannot use normal annuity valuation techniques.

1

2

3

0

1

2

3

4

5

6

5%

100

100

100

110.25

121.55

331.80

FV3 = $100(1.05)4 + $100(1.05)2 + $100

FV3 = $331.80

- Find the EAR and treat as an annuity.
- EAR = ( 1 + 0.10 / 2 )2 – 1 = 10.25%.

3

10.25

0

-100

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

331.80

- Could solve by discounting each cash flow, or …
- Use the EAR and treat as an annuity to solve for PV.

3

10.25

100

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

-247.59

- Amortization tables are widely used for home mortgages, auto loans, business loans, retirement plans, etc.
- Financial calculators and spreadsheets are great for setting up amortization tables.
- EXAMPLE: Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.

- All input information is already given, just remember that the FV = 0 because the reason for amortizing the loan and making payments is to retire the loan.

3

10

-1000

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

402.11

- The borrower will owe interest upon the initial balance at the end of the first year. Interest to be paid in the first year can be found by multiplying the beginning balance by the interest rate.
INTt = Beg balt (i)

INT1 = $1,000 (0.10) = $100

- If a payment of $402.11 was made at the end of the first year and $100 was paid toward interest, the remaining value must represent the amount of principal repaid.
PRIN= PMT – INT

= $402.11 - $100 = $302.11

- To find the balance at the end of the period, subtract the amount paid toward principal from the beginning balance.
END BAL= BEG BAL – PRIN

= $1,000 - $302.11

= $697.89

- Interest paid declines with each payment as the balance declines. What are the tax implications of this?

$

- Constant payments.
- Declining interest payments.
- Declining balance.

402.11

Interest

302.11

Principal Payments

0

1

2

3

- Bank agrees to lend a home buyer $220,000 to buy a $250,000 home, requiring a $30,000 down payment.
- The home buyer only has $7,500 in cash, so the seller agrees to take a note with the following terms:
- Face value = $22,500
- 7.5% nominal interest rate
- Payments made at the end of the year, based upon a 20-year amortization schedule.
- Loan matures at the end of the 10th year.

- Based upon the loan information, the home buyer must make annual payments of $2,207.07 on the loan.

20

7.5

-22500

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

2207.07

- Using an amortization table (spreadsheet or calculator), it can be found that at the end of the 10th year, the remaining balance on the loan will be $15,149.54.
- Therefore,
- Balloon payment = $15,149.54
- Final payment = $17,356.61

On January 1 you deposit $100 in an account that pays a nominal interest rate of 10%, with daily compounding (365 days).

How much will you have on October 1, or after 9 months (273 days)? (Days given.)

iPer= 10.0% / 365

= 0.027397% per day.

0

1

2

273

0.027397%

...

FV = ?

-100

273

(

)

FV

=

$100

1.00027397

273

(

)

=

$100

1.07765

=

$107.77.

Note: % in calculator, decimal in equation.

iPer=iNom/m

=10.0/365

=0.027397% per day.

INPUTS

273-100 0

107.77

N

I/YR

PV

PMT

FV

OUTPUT

Enter i in one step.

Leave data in calculator.

Now suppose you leave your money in the bank for 21 months, which is 1.75 years or 273 + 365 = 638 days.

How much will be in your account at maturity?

Answer:Override N = 273 with N = 638.

FV = $119.10.

iPer = 0.027397% per day.

0

365

638 days

...

...

-100

FV = 119.10

FV=$100(1 + .10/365)638

=$100(1.00027397)638

=$100(1.1910)

=$119.10.

You are offered a note that pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank that pays a 7.0% nominal rate, with 365 daily compounding, which is a daily rate of 0.019178% and an EAR of 7.25%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless.

Should you buy it?

iPer =0.019178% per day.

0

365

456 days

...

...

-850

1,000

3 Ways to Solve:

1. Greatest future wealth: FV

2. Greatest wealth today: PV

3. Highest rate of return: Highest EFF%

1. Greatest Future Wealth

Find FV of $850 left in bank for

15 months and compare with

note’s FV = $1,000.

FVBank= $850(1.00019178)456

= $927.67 in bank.

Buy the note: $1,000 > $927.67.

Calculator Solution to FV:

iPer=iNom/m

=7.0/365

=0.019178% per day.

INPUTS

456-850 0

927.67

N

I/YR

PV

PMT

FV

OUTPUT

Enter iPer in one step.

2. Greatest Present Wealth

Find PV of note, and compare

with its $850 cost:

PV=$1,000/(1.00019178)456

=$916.27.

7/365 =

INPUTS

456 .019178 0 1000

-916.27

N

I/YR

PV

PMT

FV

OUTPUT

PV of note is greater than its $850 cost, so buy the note. Raises your wealth.

3. Rate of Return

Find the EFF% on note and compare with 7.25% bank pays, which is your opportunity cost of capital:

FVn= PV(1 + i)n

$1,000 = $850(1 + i)456

Now we must solve for i.

456-850 0 1000

0.035646%

per day

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

Convert % to decimal:

Decimal = 0.035646/100 = 0.00035646.

EAR = EFF%= (1.00035646)365 – 1

= 13.89%.

Using interest conversion:

P/YR = 365.

NOM% = 0.035646(365) = 13.01.

EFF% = 13.89.

Since 13.89% > 7.25% opportunity cost,

buy the note.