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CHAPTER 2 Time Value of Money

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CHAPTER 2Time 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

0

2

3

1

I%

-50

100

75

50

Uneven cash flow stream (mixed stream)

Today

Future

We know that receiving $1 today is worth more than $1 in the future. This is due to opportunity costs.

The opportunity cost of receiving $1 in the future is the interestwe could have earned if we had received the $1 sooner.

Today

Future

?

Today

Future

?

- Translate $1 today into its equivalent in the future(compounding).
- Translate $1 in the future into its equivalent today(discounting).

Compound Interest and Future Value

- 1) Future Value - single sumsIf you deposit $100 in an account earning 6%, how much would you have in the account after 1 year?

PV = -100 FV = 106

Mathematical Solution:

FV = PV (FVIF i, n )

FV = 100 (FVIF .06, 1 ) (use FVIF table, or)

FV = PV (1 + i)n

FV = 100 (1.06)1 = $106

- 2) Future Value - single sums
- If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years?

Mathematical Solution:

FV = PV (FVIF i, n )

FV = 100 (FVIF .06, 5 ) (use FVIF table, or)

FV = PV (1 + i)n

FV = 100 (1.06)5 = $133.82

- 3) Future Value - single sums
- If you deposit $100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years?

Mathematical Solution:

FV = PV (FVIF i, n )

FV = 100 (FVIF .015, 20 ) (can’t use FVIF table)

FV = PV (1 + i/m) m x n

FV = 100 (1.015)20 = $134.68

- 4) Future Value - single sums
- If you deposit $100 in an account earning 6% with monthly compounding, how much would you have in the account after 5 years?

Mathematical Solution:

FV = PV (FVIF i, n )

FV = 100 (FVIF .005, 60 ) (can’t use FVIF table)

FV = PV (1 + i/m) m x n

FV = 100 (1.005)60 = $134.89

- 5) Future Value - continuous compounding
- What is the FV of $1,000 earning 8% with continuous compounding, after 100 years?

Mathematical Solution:

FV = PV (e in)

FV = 1000 (e .08x100) = 1000 (e 8)

FV = $2,980,957.99

0

1

2

3

10%

100

FV = ?

- Finding the FV of a cash flow or series of cash flows is called compounding.
- FV can be solved by using the step-by-step (formula/table), financial calculator, and spreadsheet methods (excel).

- 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

Present Value

- 1) Present Value - single sums
- If you receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%?

Mathematical Solution:

PV = FV (PVIF i, n )

PV = 100 (PVIF .06, 1 ) (use PVIF table, or)

PV = FV / (1 + i)n

PV = 100 / (1.06)1 = $94.34

- 2) Present Value - single sums
- If you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%?

Mathematical Solution:

PV = FV (PVIF i, n )

PV = 100 (PVIF .06, 5 ) (use PVIF table, or)

PV = FV / (1 + i)n

PV = 100 / (1.06)5 = $74.73

- 3) Present Value - single sums
- What is the PV of $1,000 to be received 15 years from now if your opportunity cost is 7%?

Mathematical Solution:

PV = FV (PVIF i, n )

PV = 100 (PVIF .07, 15 ) (use PVIF table, or)

PV = FV / (1 + i)n

PV = 100 / (1.07)15 = $362.45

- 4) Present Value - single sums
- If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?

Mathematical Solution:

PV = FV (PVIF i, n )

5,000 = 11,933 (PVIF ?, 5 )

PV = FV / (1 + i)n

5,000 = 11,933 / (1+ i)5

.419 = ((1/ (1+i)5)

2.3866 = (1+i)5

(2.3866)1/5 = (1+i) i = .19

- 5) Present Value - single sums
- Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500?

Mathematical Solution:

PV = FV / (1 + i)n

100 = 500 / (1+ .008)N

5 = (1.008)N

ln 5 = ln (1.008)N

ln 5 = N ln (1.008)

1.60944 = .007968 N N = 202 months

- 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 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 I.
- Hard to solve without a financial calculator or spreadsheet.

3

-100

0

125.97

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

8

- Solves the general FV equation for N.
- Hard to solve without a financial calculator or spreadsheet.

20

-1

0

2

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

3.8

- In every single sum present value and future value problem, there are four variables:
FV, PV, i and n.

- When doing problems, you will be given three variables and you will solve for the fourth variable.
- Keeping this in mind makes solving time value problems much easier!

0

1

2

3

4

- Annuity: a sequence of equal cash flows, occurring at the end of each period.

- If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond.
- If you borrow money to buy a house or a car, you will pay a stream of equal payments.

- 1) Future Value – annuity
- If you invest $1,000 each year at 8%, how much would you have after 3 years?

Calculator Solution:

P/Y = 1I = 8N = 3

PMT = -1,000

FV = $3,246.40

Mathematical Solution:

FV = PMT (FVIFA i, n)

FV = 1,000 (FVIFA .08, 3) (use FVIFA table, or)

FV = PMT (1 + i)n - 1

i

FV = 1,000 (1.08)3 - 1 = $3246.40

.08

- 1) Present Value - annuity
- What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

Calculator Solution:

P/Y = 1I = 8N = 3

PMT = -1,000

PV = $2,577.10

Mathematical Solution:

PV = PMT (PVIFA i, n)

PV = 1,000 (PVIFA .08, 3) (use PVIFA table, or)

1

PV = PMT 1 - (1 + i)n

i

1

PV = 1000 1 - (1.08 )3 = $2,577.10

.08

Ordinary Annuity (end)

0

1

2

3

i%

PMT

PMT

PMT

Annuity Due (beginning)

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.
- FVAdue= FVAord(1+I) = $331(1.10) = $364.10.
- Alternatively, set calculator to “BEGIN” mode and solve for the FV of the annuity:

BEGIN

3

10

0

-100

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

364.10

- Again, $100 payments occur at the beginning of each period.
- PVAdue= PVAord(1+I) = $248.69(1.10) = $273.55.
- Alternatively, set calculator to “BEGIN” mode and solve for the PV of the annuity:

BEGIN

3

10

100

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

-273.55

- 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

- 10-year annuity
- N = 10, I/YR = 10, PMT = 100, FV = 0; solve for PV = $614.46.

- 25-year annuity
- N = 25, I/YR = 10, PMT = 100, FV = 0; solve for PV = $907.70.

- Perpetuity
- PV = PMT / I = $100/0.1 = $1,000.

- Perpetuities
- Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity.
- You can think of a perpetuity as an annuity that goes on forever.

- When we find the PV of an annuity, we think of the following relationship:

PV = PMT (PVIFA i, n )

1

n

1 -

(1 + i)

i

Mathematically,

(PVIFA i, n ) =

We said that a perpetuity is an annuity where n = infinity. What happens to this formula when n gets very, very large?

1

n

1 -

(1 + i)

i

1

i

When n gets very large,

this becomes zero.

So we’re left with PVIFA =

PMT

PV =

i

Present Value of a Perpetuity

- So, the PV of a perpetuity is very simple to find:

PMT $10,000

i .08

= $125,000

PV = =

What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment?

A 20-year-old student wants to save $3 a day for her retirement. Every day she places $3 in a drawer. At the end of the year, she invests the accumulated savings ($1,095) in a brokerage account with an expected annual return of 12%.

How much money will she have when she is 65 years old?

45

12

0

-1095

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

1,487,262

- If she sticks to her plan, she will have $1,487,261.89 when she is 65.
- N = 45, I/YR = 12, PMT = -1095, PV = 0; solve for FV = $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

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 = $530.087. (Here NPV = PV.)

-10,000 2,000 4,000 6,000 7,000

0 1 2 3 4

- Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.

-10,000 2,000 4,000 6,000 7,000

0 1 2 3 4

period CF PV (CF)

0-10,000 -10,000.00

1 2,000 1,818.18

2 4,000 3,305.79

3 6,000 4,507.89

4 7,000 4,781.09

PV of Cash Flow Stream: $ 4,412.95

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 (n=1): FV3 = $100(1.10)3 = $133.10

Semiannually (nxm=1x2=2): FV6 = $100(1.05)6 = $134.01

- 1) 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.

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

- 3) Effective (or equivalent) annual rate (EAR = EFF%) – the annual rate of interest actually being earned, accounting for compounding.
- EFF% for 10% semiannual investment
EFF%= ( 1 + INOM / M )M - 1

= ( 1 + 0.10 / 2 )2 – 1 = 10.25%

- Should be indifferent between receiving 10.25% annual interest and receiving 10% interest, compounded semiannually.

- EFF% for 10% semiannual investment

Which is the better loan:

- 8% compounded annually, or
- 7.85% compounded quarterly?
- We can’t compare these nominal (quoted) interest rates, because they don’t include the same number of compounding periods per year!
We need to calculate the APY.

APY = ( 1 + ) m - 1

quoted rate

m

APY = ( 1 + ) 4 - 1

APY = .0808, or 8.08%

.0785

4

- Find the APY for the quarterly loan:
- The quarterly loan is more expensive than the 8% loan with annual compounding!

- Investments with different compounding intervals provide different effective returns.
- To compare investments with different compounding intervals, you must look at their effective returns (EFF% or EAR).
- See how the effective return varies between investments with the same nominal rate, but different compounding intervals.
EARANNUAL10.00%

EARQUARTERLY10.38%

EARMONTHLY10.47%

EARDAILY (365)10.52%

- 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.

- 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.

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

0

1

2

3

4

5

6

5%

100

100

100

110.25

121.55

FV3 = 331.80

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

FV3 = $331.80

0

1

2

3

4

5

6

5%

100

100

100

- 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