Chapter 6 time value of money
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CHAPTER 6 Time Value of Money. Future value Present value Annuities Rates of return Amortization. Time lines. 0. 1. 2. 3. i%. CF 0. CF 1. CF 2. CF 3. Show the timing of cash flows.

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CHAPTER 6 Time Value of Money

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Chapter 6 time value of money

CHAPTER 6Time Value of Money

Future value

Present value

Annuities

Rates of return

Amortization


Time lines

Time lines

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.


Drawing time lines 100 lump sum due in 2 years 3 year 100 ordinary annuity

$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

Drawing time lines:$100 lump sum due in 2 years;3-year $100 ordinary annuity


Drawing time lines uneven cash flow stream cf 0 50 cf 1 100 cf 2 75 and cf 3 50

Uneven cash flow stream

0

1

2

3

i%

-50

100

75

50

Drawing time lines:Uneven cash flow stream; CF0 = -$50, CF1 = $100, CF2 = $75, and CF3 = $50


What is the future value fv of an initial 100 after 3 years if i yr 10

0

1

2

3

10%

100

FV = ?

What is the future value (FV) of an initial $100 after 3 years, if I/YR = 10%?

  • 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.


Solving for fv the arithmetic method

Solving for FV:The arithmetic method

  • 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


Solving for fv the calculator method

Solving for FV:The calculator method

  • 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


What is the present value pv of 100 due in 3 years if i yr 10

What is the present value (PV) of $100 due in 3 years, if I/YR = 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


Solving for pv the arithmetic method

Solving for PV:The arithmetic method

  • Solve the general FV equation for PV:

    • PV = FVn / ( 1 + i )n

    • PV = FV3 / ( 1 + i )3

      = $100 / ( 1.10 )3

      = $75.13


Solving for pv the calculator method

Solving for PV:The calculator method

  • 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


Solving for n if sales grow at 20 per year how long before sales double

Solving for N:If sales grow at 20% per year, how long before sales double?

  • 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


Chapter 6 time value of money

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


Chapter 6 time value of money

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)


What is the difference between an ordinary annuity and an annuity due

Ordinary Annuity

0

1

2

3

i%

PMT

PMT

PMT

Annuity Due

0

1

2

3

i%

PMT

PMT

PMT

What is the difference between an ordinary annuity and an annuity due?


Solving for fv 3 year ordinary annuity of 100 at 10

Solving for FV:3-year ordinary annuity of $100 at 10%

  • $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


Solving for pv 3 year ordinary annuity of 100 at 10

Solving for PV:3-year ordinary annuity of $100 at 10%

  • $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


Solving for fv 3 year annuity due of 100 at 10

Solving for FV:3-year annuity due of $100 at 10%

  • 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


Solving for pv 3 year annuity due of 100 at 10

Solving for PV:3 year annuity due of $100 at 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


Chapter 6 time value of money

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)


Chapter 6 time value of money

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


Chapter 6 time value of money

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.)


Chapter 6 time value of money

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!


What is the pv of this uneven cash flow stream

4

0

1

2

3

10%

100

300

300

-50

90.91

247.93

225.39

-34.15

530.08 = PV

What is the PV of this uneven cash flow stream?


Solving for pv uneven cash flow stream

Solving for PV:Uneven cash flow stream

  • 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.)


Solving for i what interest rate would cause 100 to grow to 125 97 in 3 years

Solving for I:What interest rate would cause $100 to grow to $125.97 in 3 years?

  • Solves the general FV equation for I.

3

-100

0

125.97

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

8


The power of compound interest

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) 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?


Solving for fv savings problem

Solving for FV:Savings problem

  • 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


Solving for fv savings problem if you wait until you are 40 years old to start

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 $146,000.59 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


Solving for pmt how much must the 40 year old deposit annually to catch the 20 year old

Solving for PMT:How much must the 40-year old deposit annually to catch the 20-year old?

  • 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


Chapter 6 time value of money

0

1

2

3

10%

100

133.10

0

1

2

3

4

5

6

0

1

2

3

5%

100

134.01

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.

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

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


Chapter 6 time value of money

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.


Chapter 6 time value of money

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


Classifications of interest rates

Classifications of interest rates

  • 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.


Classifications of interest rates1

Classifications of interest rates

  • 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.


Why is it important to consider effective rates of return

Why is it important to consider effective rates of return?

  • 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%


Can the effective rate ever be equal to the nominal rate

Can the effective rate ever be equal to the nominal rate?

  • 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.


When is each rate used

When is each rate used?

  • 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.


What is the fv of 100 after 3 years under 10 semiannual compounding quarterly compounding

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


What s the fv of a 3 year 100 annuity if the quoted interest rate is 10 compounded semiannually

1

2

3

0

1

2

3

4

5

6

5%

100

100

100

What’s the FV of a 3-year $100 annuity, if the quoted interest rate is 10%, compounded semiannually?

  • Payments occur annually, but compounding occurs every 6 months.

  • Cannot use normal annuity valuation techniques.


Method 1 compound each cash flow

1

2

3

0

1

2

3

4

5

6

5%

100

100

100

110.25

121.55

331.80

Method 1:Compound each cash flow

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

FV3 = $331.80


Method 2 financial calculator

Method 2:Financial calculator

  • 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


Find the pv of this 3 year ordinary annuity

Find the PV of this 3-year ordinary annuity.

  • 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


Loan amortization

Loan amortization

  • 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.


Step 1 find the required annual payment

Step 1:Find the required annual payment

  • 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


Step 2 find the interest paid in year 1

Step 2:Find the interest paid in Year 1

  • 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


Step 3 find the principal repaid in year 1

Step 3:Find the principal repaid in Year 1

  • 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


Step 4 find the ending balance after year 1

Step 4:Find the ending balance after Year 1

  • 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


Constructing an amortization table repeat steps 1 4 until end of loan

Constructing an amortization table:Repeat steps 1 – 4 until end of loan

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


Illustrating an amortized payment where does the money go

Illustrating an amortized payment:Where does the money go?

$

  • Constant payments.

  • Declining interest payments.

  • Declining balance.

402.11

Interest

302.11

Principal Payments

0

1

2

3


Partial amortization

Partial amortization

  • 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.


Calculating annual loan payments

Calculating annual loan payments

  • 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


Determining the balloon payment

Determining the balloon payment

  • 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


Chapter 6 time value of money

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.)


Chapter 6 time value of money

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.


Chapter 6 time value of money

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.


Chapter 6 time value of money

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.


Chapter 6 time value of money

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.


Chapter 6 time value of money

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?


Chapter 6 time value of money

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%


Chapter 6 time value of money

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.


Chapter 6 time value of money

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.


Chapter 6 time value of money

2. Greatest Present Wealth

Find PV of note, and compare

with its $850 cost:

PV=$1,000/(1.00019178)456

=$916.27.


Chapter 6 time value of money

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.


Chapter 6 time value of money

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.


Chapter 6 time value of money

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%.


Chapter 6 time value of money

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.


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