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Chapter 3 The Time Value of Money

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2005, Pearson Prentice Hall

Compounding and Discounting Single Sums

Today

Future

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

The opportunity cost of receiving $1 in the future is theinterestwe 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

Calculator Solution:

P/Y = 1I = 6

N = 1 PV = -100

FV = $106

PV = -100 FV =

0 1

Calculator Solution:

P/Y = 1I = 6

N = 1 PV = -100

FV = $106

PV = -100 FV = 106

0 1

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

PV = -100 FV = 106

0 1

Calculator Solution:

P/Y = 1I = 6

N = 5 PV = -100

FV = $133.82

PV = -100 FV = 133.82

0 5

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

PV = -100 FV = 133.82

0 5

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

Calculator Solution:

P/Y = 4I = 6

N = 20 PV = -100

FV = $134.68

PV = -100 FV = 134.68

0 20

Future Value - single sumsIf 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

PV = -100 FV = 134.68

0 20

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

Calculator Solution:

P/Y = 12I = 6

N = 60 PV = -100

FV = $134.89

PV = -100 FV = 134.89

0 60

Future Value - single sumsIf 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

PV = -100 FV = 134.89

0 60

Mathematical Solution:

FV = PV (e in)

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

FV = $2,980,957.99

PV = -1000 FV = $2.98m

0 100

Present Value

PV = -94.34 FV = 100

0 1

Calculator Solution:

P/Y = 1I = 6

N = 1 FV = 100

PV = -94.34

PV = -94.34 FV = 100

0 1

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

PV = -74.73 FV = 100

0 5

Calculator Solution:

P/Y = 1I = 6

N = 5 FV = 100

PV = -74.73

PV = -74.73 FV = 100

0 5

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

PV = -362.45 FV = 1000

0 15

Calculator Solution:

P/Y = 1I = 7

N = 15 FV = 1,000

PV = -362.45

PV = -362.45 FV = 1000

0 15

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

PV = -5000 FV = 11,933

0 5

Calculator Solution:

P/Y = 1N = 5

PV = -5,000 FV = 11,933

I = 19%

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

PV = -100 FV = 500

0 ?

Present Value - single sumsSuppose 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?

Calculator Solution:

- P/Y = 12FV = 500
- I = 9.6PV = -100
- N = 202 months

Present Value - single sumsSuppose 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

- 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

Compounding and Discounting

Cash Flow Streams

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.

- 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.

0 1 2 3

Calculator Solution:

P/Y = 1I = 8N = 3

PMT = -1,000

FV = $3,246.40

10001000 1000

0 1 2 3

Calculator Solution:

P/Y = 1I = 8N = 3

PMT = -1,000

FV = $3,246.40

10001000 1000

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

0 1 2 3

Calculator Solution:

P/Y = 1I = 8N = 3

PMT = -1,000

PV = $2,577.10

10001000 1000

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