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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

10%

100

FV = ?

(121x1.1)

=133.10

- Finding the FV of a cash flow or series of cash flows is called compounding.

(100x1.1)1

=110

(110x1.1)1

=121

- 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

FVn= PV (1+I)n

FV3= PV (1+I)3

=100 (1+10%)3

=100 (1.10)3

=$133.10

FV3= PV (Fk% VFn)

= 100 (F10% VF3)

= 100 (1.331)

= $ 133.10

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.

- 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 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 = ?

82.64/(1.1)1

=75.13

100/(1.1)1

=90.91

90.91/(1.1)1

=82.64

100

- Solve the general FV equation for PV:
- PV = FVN / (1 + I)N
- PV = FV3 / (1 + I)3
= $100 / (1.10)3

= $75.13

FV = PV

(1+k)n

100 = PV

(1+10%)3

100 = PV

(1.1)3

75.13 = PV

PV= FV (Pk% VFn)

= 100 (P10% VF3)

= 100 (0.7513)

= 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.

25

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

0

1

2

3

FV = 125.97

PV=100

n = 3

i = ?

PV= 100

FV= 125.97

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

20

-200

0

400

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

3.8

?

?

?

FV = 400

PV=200

i = 20%

PV= 200

FV= 400

n = ?

?

a) Based on the following arrangements, what interest rates are charged by the lending institution to the borrower.

You borrow RM 500 and repay RM 551 in two years.

FV = PV ( F V F )

n=2 k =?

551= 500 ( F2 V Fk% )

551= ( F2 V Fk% )

1.10 = ( F2 V Fk% )

k= 5 %

from table A-1

PV = 500

FV = 551

n = 2

Pmt = 0

I/yr = ?

b) Borrow RM 100,000 and repay RM 146,932 in 5 years

146,932 = 100,000 ( F V F )

n = 5 k = ?

146,9325 = ( F5 V F k% )

100,000

1.46 = ( F5V Fk% )

k = 8 % (see table A-1)

c) Borrow RM 100,000 and repay RM 300,000 in 10 years

300,000 = 100,000 ( F V F )

n = 10 k = ?

300,000 = ( F10 V Fk% )

100,000

3 = ( F10 V Fk% )

k = 11.6%

PV = -100,000

FV = 146,932

n = 5

Pmt = 0

I/yr = ?

PV = -100,000

FV = 300,000

n = 10

Pmt = 0

I/yr = ?

How long does it take for the followings to happen

a) RM 856 to grow into RM1,122 at 7 %

FV = PV ( Fk% V Fn )

RM 1,122 = RM 856 ( F 7%V Fn=? )

RM 1,122 = ( F7% V Fn=? )

RM 856

1.31 = ( F7% V Fn=? )

4 = n (use table A1)

b) RM10,000 to grow into RM100,000 at 8 %

100,000 = 10,000 ( F V F )

100,000 = ( F8% V Fn=? )

10,000

10 = ( F8% V Fn=? )

n = 30 (use table A1)

PV = -856

FV = 1,122

k = 7%

Pmt = 0

n = ?

PV = -10,000

FV = 100,000

k = 8%

Pmt = 0

n = ?

c) RM10,000 to grow into RM200,000 at 8 %

200,000 = 10,000 ( F V F )

200,000 = ( F8% V Fn=? )

10,000

20 = ( F8% V Fn=? )

n = 38 (use table A1)

PV = -10,000

FV = 200,000

k = 7%

Pmt = 0

n = ?

1. En. Kamal is planning ahead for his son’s education. The boy is eight now and will start college in 12 years time. How much must he set aside now to have RM 100,000 when his son starts schooling. The interest rate is 8 %

k = 8% n= 12 years FV = RM 100,000 PV = ?

FV = PV

( 1 + k )n

PV= RM 100,000

(1 + 0.08)12

= RM 100,000

2.518

= RM 39,711

FV = 100,000

k = 8%

Pmt = 0

n = 12

PV = ?

PV = FV ( P V F )

K% n

PV = RM 100,000 ( P V F )

8 % 12

PV = RM 100,000 ( 0.3971 ) from Table A-2

PV = RM 39,710 ~ RM 39,71

FV 3 year $100 ordinary annuity @ 10%

0

1

2

3

I%

100

100

(100x1.1)1

=110

100

(100x1.1)1

=110

(110x1.1)1

=121

Σ 100 + 110 + 121 = 331

- $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

1. Assume that you have deposited RM100 each year in a bank account that pays 15% interest rate. How much will you have in your account at the end of the fifth year?

F VF = P m t ( F V F A )

k% n

k = 15% n = 5

F VF A = 100 ( F 15%V F A 5 )

= 100 (6.7424 ) (use table A-3)

F V A = 674.24

PV = 0

k = 15%

Pmt = -100

n = 5

FV = ?

2. Assume you have deposited RM200 each year in a bank account that pays 20% interest rate. How much will you have in your account after 25 years.

F VF = P m t ( F V F A )

k% n

k = 20% n =2 5

F VF A = 200 ( F 20% V F A25 )

= 200 (471.98 ) (use table A-3)

F V A = 94,396

PV = 0

k = 20%

Pmt = -200

n = 25

FV = ?

3. Assume you have deposited RM1000 each year in a bank account that pays 20% interest rate. How much will you have in your account at the end of the 25 year

F VF = P m t ( F V F A )

k % n

k = 20% n =2 5

F VF A = 1000( F 20%V F A 25 )

= 1000 (471.98 )

F V A = 471,980

PV = 0

k = 20%

Pmt = -1,000

n = 25

FV = ?

PV 3 year $100 ordinary annuity @ 10%

0

1

2

3

I%

100

100

100/(1.1)1

=90.91

100/(1.1)1

=90.91

90.91/(1.1)1

=82.64

82.64/(1.1)1

=75.13

100

100/(1.1)1

=90.91

90.91/(1.1)1

=82.64

Σ 90.91 + 82.64 + 75.13 = 248.68

- $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

We would like to know what is the present value of four payments in the amount of RM1,000 each to be received in the next four years. The current market interest rate is 10%

P V F = P m t ( P V F A )Pmt = RM1,000, k%= 10, n = 4

k % n

PVF = 1000 (P10%VFA4) (please refer to table A-4 for this value)

PVF = 1000 (3.1699)

PVF = 3,169.9

Ordinary Annuity

1

2

3

0

i%

PMT

PMT

PMT

Annuity Due

1

2

3

0

i%

PMT

100

PMT

100

100(1.1)

=110

PMT

100

100(1.1)1

=110

110 (1.1)1

=121

100(1.1)1

110 (1.1)1

121 (1.1)1

- 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

3 year annuity due of $100 at 10%,

n= 3, Pmt=100, k=10%, FV=?

FVA = Pmt (F10% VFA3)

= 100 (3.31) table A-3

FVA = 331 (1 + 10%)

FVAdue = 331 (1.10)

= 364.10

The process of discounting the future payment back to the present time in the case of Present value annuity due is different than the normal present value annuity. The figure below illustrate the difference

Present value annuity due

P m t 1 P m t 2 P m t 3

PVA 1/(1 + k)1 1/ (1 + k)2

Present value annuity 0 P m t 1 P m t 2 P m t 3

PVA 1/(1 + k)1 1/(1 + k)2 1/(1 + k)3

Because the first payment took place at the beginning of year one and not the end of year one, that payment will be discounted one less period. Hence that factor has to be taken into consideration when we compute the present value annuity due amount. Illustrated below, we have added (1+k) factor to the original formula use to compute an ordinary present value annuity amount

P V A due = P m t ( P V F A ) ( 1 + k ) Annuity due factor

N K%

3 year annuity due of $100 at 10%

n= 3, Pmt= 100, k=10%, PV= ?

PVA = Pmt ( PV10%FA3)

= 100 ( 2.486 ) table A-4

= 248.6

PVAdue = 248.6 ( 1.10 )

= 273.46

- 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

Pmt =100, n =5, i =10%, FV =0, PV =?

PVA = 100 ( PVk% FAn )

= 100 ( PV10% FA5 )

= 100 ( 3.7908 ) Table A-4

PVA= 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.

∞

100

1

100

2

100

∞

PV =Σ ?

Pmt = 100

i = 10

PV = ?

Pmt = 100

i 0.10

= 1,000

0

Yr= 1

Yr= 2

Yr= 3

10%

100

133.10

0

Yr= 1

Yr= 2

Yr= 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= 3, PV= -100, I/YR= 10, PMT=0, FV =133.10

Semiannually: N= 2x3, PV= -100, I/YR= 10/2, PMT=0, FV =134.01

Quaterly:

N= 4x3, PV= 100, I/yr= 10/4, FV= ?

= 12 = 2.5 134.49

Monthly:

N= 12x3, PV= 100, I/yr= 10/12, FV= ?

= 36 = 0.833 134.80

Annually

FV= 100, I/yr= 10%, n= 3, PV= ?

75.13

Semiannually

FV= 100, I/yr= 10%/2, n= 3x2, PV= ?

= 5= 6 74.62

Quarterly

FV= 100, I/yr= 10%/4, n= 3x4, PV= ?

= 2.5= 12 74.36

Monthly

FV= 100, I/yr= 10%/12, n= 3x12, PV= ?

= 0.833 = 36 74.18

Yr 1

Yr 1

Yr 1

3

100

0

1

2

“

“

100

(1+i )n

Yr 1

Yr2

Yr 3

0

1

2

3

4

5

6

100

“

“

“

“

“

100

(1+i )n

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?

20

21

22

23

64

65

1,095

1,095

1,095

1,095

1,095

1,095

n= 65-20

= 45

i / yr = 12%

PV = 0

Pmt = 1,095

FV = ?

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.

- 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

40

41

42

43

44

64

65

1,095

1,095

1,095

n= 65-40

= 25

i / yr = 12%

PV = 0

Pmt = ?

FV = 146,000.59

- 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

40

41

42

43

44

64

65

?

?

?

n= 65-40

= 25

i / yr = 12%

PV = 0

Pmt = ?

FV = 1,487,261.89

Example

Borrow: RM 100, Payback time: 1yr @ 12%

CompoundingFinal balance

Annual112.00

Semiannual112.36

Quarterly112.55

Monthly112.68

- INOMwritten into contracts, quoted by banks and brokers. Not used in calculations or shown on timelines.
- EARUsed to compare returns on investments with different compounding periods.
- i.e. A credit card holder must pay 18% monthly compounding interest on the amount outstanding. What is the EAR of the credit card?

EAR = ( 1 + I nom )m – 1.0

m

= (1 + 0.18/12 )12 – 1.0

= 0.1956 ≈ 19.56%

ie: A credit card holder must pay 18% monthly compounding interest on the amount outstanding. What is the EAR of the credit card?

I nom = 18% annually

m = Compounding period

- Amortization tables are widely used for home mortgages.
Ie. A home owner borrows, RM 200,000 on a mortgage loan. The loan is to be repaid in 180 ( 12months x 15 years) equal payments at the end of each month. The bank charges 12% on the loan per year ( 1% monthly). What is the monthly repayment?

?

?

?

?

?

?

200,000

Pmt 1

Pmt 2

Pmt 3

Pmt 178

Pmt 179

Pmt 180

n= 12 months x 15

= 180

i / yr = 12% / 12

= 1%

PV = 200,000

FV = 0

Pmt = ?

- 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.

180

1

-200,000

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

2,400