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CHAPTER 2 Time Value of Money. Future value Present value Annuities Rates of return Amortization. 0. 1. 2. 3. 10%. 100. FV = ? (121x1.1) =133.10. What is the future value (FV) of an initial $100 after 3 years, if I/YR = 10%?.

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Chapter 2 time value of money

CHAPTER 2Time Value of Money

Future value

Present value

Annuities

Rates of return

Amortization


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

0

1

2

3

10%

100

FV = ?

(121x1.1)

=133.10

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 is called compounding.

(100x1.1)1

=110

(110x1.1)1

=121


Solving for fv the step by step and formula methods
Solving for FV:The step-by-step and formula 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


FVn= PV (1+I)n

FV3= PV (1+I)3

=100 (1+10%)3

=100 (1.10)3

=$133.10


Or using fvf table a1
or, using FVF table (A1)

FV3= PV (Fk% VFn)

= 100 (F10% VF3)

= 100 (1.331)

= $ 133.10


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.


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 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


Solving for pv the formula method
Solving for PV: I/YR = 10%?The formula method

  • 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 I/YR = 10%?

(1+k)n

100 = PV

(1+10%)3

100 = PV

(1.1)3

75.13 = PV


Or using table a 2
or, using table (A-2) I/YR = 10%?

PV= FV (Pk% VFn)

= 100 (P10% VF3)

= 100 (0.7513)

= 75.13


Solving for pv the calculator method
Solving for PV: I/YR = 10%?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.

25

10

0

100

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

-75.13


Solving for i what interest rate would cause 100 to grow to 125 97 in 3 years
Solving for I: I/YR = 10%?What interest rate would cause $100 to grow to $125.97 in 3 years?

  • 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 I/YR = 10%?

1

2

3

FV = 125.97

PV=100

n = 3

i = ?

PV= 100

FV= 125.97


Solving for n if sales grow at 20 per year how long before sales double
Solving for N: I/YR = 10%?If sales grow at 20% per year, how long before sales double?

  • 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


? I/YR = 10%?

?

?

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 are charged by the lending institution to the borrower.

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 = ?


Finding implied interest and number of factoring years
Finding Implied Interest And Number Of Factoring Years are charged by the lending institution to the borrower.

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 % are charged by the lending institution to the borrower.

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 = ?


Example of present value problem
Example of present value problem are charged by the lending institution to the borrower.

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 = ?


Or using present value factor table
or, using present value factor table are charged by the lending institution to the borrower.

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


Drawing time lines

FV 3 year $100 ordinary annuity @ 10% are charged by the lending institution to the borrower.

0

1

2

3

I%

100

100

(100x1.1)1

=110

100

(100x1.1)1

=110

(110x1.1)1

=121

Drawing time lines

Σ 100 + 110 + 121 = 331


Solving for fv 3 year ordinary annuity of 100 at 10
Solving for FV: are charged by the lending institution to the borrower.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


Problems of future value annuity for all future value annuity problems please refer to table a 3
Problems of future value annuity are charged by the lending institution to the borrower.(for all future value annuity problems, please refer to table A-3 )

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 = ?


Drawing time lines1

PV 3 year $100 ordinary annuity @ 10% account that pays 20% interest rate. How much will you have in your account at the end of the 25 year

0

1

2

3

I%

100

100

100/(1.1)1

=90.91

Drawing time lines

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


Solving for pv 3 year ordinary annuity of 100 at 10
Solving for PV: account that pays 20% interest rate. How much will you have in your account at the end of the 25 year 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


Example
Example: account that pays 20% interest rate. How much will you have in your account at the end of the 25 year

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


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

Ordinary Annuity account that pays 20% interest rate. How much will you have in your account at the end of the 25 year

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

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

100(1.1)1

110 (1.1)1

121 (1.1)1


Solving for fv 3 year annuity due of 100 at 10
Solving for FV: account that pays 20% interest rate. How much will you have in your account at the end of the 25 year 3-year annuity due of $100 at 10%

  • 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


Fv annuity due
FV Annuity due account that pays 20% interest rate. How much will you have in your account at the end of the 25 year

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


Present value annuity due
Present value annuity due account that pays 20% interest rate. How much will you have in your account at the end of the 25 year

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%


Pv annuity due
PV Annuity due year one and

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


Solving for pv 3 year annuity due of 100 at 10
Solving for PV: year one and 3-year annuity due of $100 at 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


What is the present value of a 5 year 100 ordinary annuity at 10
What is the present value of a 5-year $100 ordinary annuity at 10%?

  • 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


Present value annuity
Present value annuity at 10%?

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


What if it were a 10 year annuity a 25 year annuity a perpetuity
What if it were a 10-year annuity? A 25-year annuity? A perpetuity?

  • 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.


A perpetuity

perpetuity?

100

1

100

2

100

A perpetuity

PV =Σ ?

Pmt = 100

i = 10

PV = ?

Pmt = 100

i 0.10

= 1,000


0 perpetuity?

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

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: 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: perpetuity?

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


Will the perpetuity?PV of a lump sum be larger or smaller if discounted more often, holding the stated I% constant

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 perpetuity?

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 perpetuity?

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


The power of compound interest
The Power of Compound Interest perpetuity?

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 perpetuity?

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 = ?


Solving for fv if she begins saving today how much will she have when she is 65

45 perpetuity?

12

0

-1095

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

1,487,262

Solving for FV:If she begins saving today, how much will she have when she is 65?

  • If she sticks to her plan, she will have $1,487,261.89 when she is 65.


Solving for fv if you don t start saving until you are 40 years old how much will you have at 65
Solving for FV: perpetuity?If you don’t start saving until you are 40 years old, how much will you have at 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 perpetuity?

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


Solving for pmt how much must the 40 year old deposit annually to catch the 20 year old
Solving for PMT: perpetuity?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


40 perpetuity?

41

42

43

44

64

65

?

?

?

n= 65-40

= 25

i / yr = 12%

PV = 0

Pmt = ?

FV = 1,487,261.89


The effective annual rate ear
The Effective Annual rate(EAR) perpetuity?

Example

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

CompoundingFinal balance

Annual 112.00

Semiannual 112.36

Quarterly 112.55

Monthly 112.68


When is each rate used
When is each rate used? perpetuity?

  • INOM written into contracts, quoted by banks and brokers. Not used in calculations or shown on timelines.

  • EAR Used 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?


Effective annual rate ear
Effective Annual Rate (EAR) perpetuity?

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


Loan amortization
Loan amortization perpetuity?

  • 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?


? perpetuity?

?

?

?

?

?

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 = ?


Step 1 find the required annual payment
Step 1: perpetuity?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.

180

1

-200,000

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

2,400


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