topic 2 1

Frino, Hill, Chen: Introduction to Corporate Finance, 4e © 2009 Pearson Australia

Chapter 2

A review of financial

mathematics


Overview

– In this lecture we will:

1.) Review the basic mathematics used in valuing assets


Overview– In this lecture we will:

2.) Examine the mathematics for calculating and comparing:

1. Financial arrangements consisting of different interest rates

2. The present value of different cash flow structures3. Accumulated future cash positions for different cash

flow structures


Key concepts

1) The aim of financial mathematics

2) Interest rate arrangements

3) Present value of a single lump sum

4) Present value and future value of multiple amounts

5) Annuities


Aim of financial mathematics

– The aim of financial mathematics is to convert single or multiple cash flows that will be received at different points in time to one number

– This number represents the value of all of an asset’s cash flows at a given point in time



– This number represents the value of all of an asset’s cash flows at a given point in time



WHY?

– This is used to:

1. Make a rational choice between different assets

2. Determine the maximum amount an investor is willing to pay for an asset – i.e. the intrinsic value of the asset


A $20 US gold coin


A $20 US gold coin

The US $20 gold coin has a STATED value: $20

But the INTRINSIC value is determined by the current market price for GOLD.


Time value of money

Example: Which of these assets would you rather own?

0 7

$100

2 3 4 5 61Year

Asset 1

0 7

$100

2 3 4 5 61Year

Asset 2


Time value of money (cont)

– The intrinsic value of Asset 2 is greater because of the time value of money

– If you intended to consume in – year 5, you would prefer Asset 2 because you

could reinvest the $100 received in year 3 and accumulate more than $100 by year 5

– year 2, you would still prefer Asset 2 – since you will have to borrow in year 2 you will be better off repaying the loan in year 3 rather than year 5


Interest rate arrangements

– The time value of money is often measured by using an interest rate, which

1. Compensates those who defer consumption until later and

2. Imposes a charge on those who wish to consume more now than their income allows

– We will use the symbol ‘ r ’ to denote the interest rate in the following calculations


Simple interest

– Interest is earned on the initial amount invested or ‘principal’

– If you invest an initial amount (PV, the present value) you would accumulate an amount (FV, the future value) equal to the principal plus interest:

– The interest is the product of PV and the simple interest rate, r

FV PV interest


Simple interest (cont)

– Thus:

where:FV = the accumulated (future) valuePV = the initial amount invested or borrowedr = the simple interest rate over the entire period

FV PV 1 r

)365/(1 dtmr



If simple interest is applied to periods of less than 1 year, the interest rate is:

where dtm = days to maturity of the loan

)365/(1 dtmr



A credit union pays 5% p.a. simple interest. If $1000 is invested today, how much will the account accrue in 4 years?



A credit union pays 5% p.a. simple interest. If $1000 is invested today, how much will the account accrue in 4 years?

_______________________________________________

FV = PV(1 + r)r = 4-year interest rate = 0.05 x 4 = 0.20

Hence:

FV= 1000 (1 + 0.20) = $1200



A credit union pays 5% p.a. simple interest. If $1000 is invested today, how much will the account accrue in 30 days?



A credit union pays 5% p.a. simple interest. If $1000 is invested today, how much will the account accrue in 30 days?

_______________________________________________

FV = PV(1 + r)r = 30-day interest rate = 0.05 x 30/365

= 0.05 x .082192 = .0041096

Hence:

FV= 1000 (1 + 0.00410906) = $1004.11


Compound interest

– The amount of interest accrued each period is added to the principal, and this new balance is used to calculate the interest amount for the next period

– Thus, interest is paid on interest that has accrued in previous periods


Compound interest (cont)

– The formula for the accrued value under compound interest is:

where:

FV = the accumulated amount in period n

PV = the initial amount invested

r = the interest rate per period

n = the number of periods

nrPVFV 1


The Seven Wonders of the World (New)

What are the Seven Wonders of the World?


The Seven Wonders of the World

1) Colosseum in Rome

2) India's Taj Mahal

3) The Great Wall of China

4) Jordan's ancient city of Petra

5) The Inca ruins of Machu Picchu in Peru

6) The ancient Maya city of Chichén Itzá in Mexico

7) Christ Redeemer, Rio de Janeiro Brazil


Wonders of the World

What is the Eighth Wonder of the World?


Wonders of the World

COMPOUND INTEREST


SAYS WHO???!!!

SAYS THIS GUY!!


Compound interest


A bank pays 5% p.a. interest compounded annually. If $1000 is invested today, how much will the account accrue in 4 years?

_______________________________________________

FV = PV(1 + r)n

r = 4-year interest rate = 0.05 x 4 = 0.20

Hence:

FV= 1000 (1 + 0.05)4 = $1216



A bank pays 5% p.a. interest compounded monthly. If $1000 is invested today, how much will the account accrue in 4 years?

_______________________________________________

FV = PV(1 + r/12)n x 12

Hence:

FV= 1000 (1 + 0.05/12)4x12 = $????


Question 1

After two years a $10,000 investment earning 8% p.a. compounded six monthly will accumulate to:

FV = PV(1 + r)n


Comparing different financing arrangements

– Compound interest rates are quoted as:

–Nominal interest rate

– Quoted annual interest rate that is adjusted to match the frequency of payments or compounding by taking a proportion of the quoted nominal rate to obtain the actual interest rate per period.

– e.g. 10% p.a. compounded semi-annually = 5% per half-year


Comparing different financing arrangements

–Effective interest rate

–Accounts for the true amount of interest that is earned on both reinvested interest and principal earned over a year.


Effective interest rate– The effective rate is an annual rate that takes into

account the effect of compounding:

where:

rnom = the nominal ratem = the number of compounding periods underlying the nominal

rate

– The effective rate will be greater than the nominal rate for compounding periods of less than 1 year

Effective rate 1rnom

m

m

1


Example 2.3What is the effective interest rate on the outstanding balance on a credit card if the nominal rate quoted on the card is 15.75% per annum, compounding daily?

______________________________________________

Comparing interest rates

Effective rate 1 rnom

m

m

1

10.1575

365

365

1

0.17054, or 17.05%


Which is cheaper?

a. a loan that charges 14% annual interest with monthly compounding

b. a loan with a 14.75% interest rate with annual compounding?

Would you prefer a loan with a 14.5% interest rate with semi-annual compounding?

_______________________________________________

Comparing interest rates (cont)


Which is cheaper?

a. a loan that charges 14% annual interest with monthly compounding

b. a loan with a 14.75% interest rate with annual compounding?

Would you prefer a loan with a 14.5% interest rate with semi-annual compounding?

_______________________________________________


Nominal Rate Effective Rate

14.75% p.a. with annual compounding14% p.a. with monthly compounding

14.5% p.a. with semi-annual compounding

14.75%

14.93%

15.03%

11214.01 12

12145.01 2


– We can compare simple interest with compound interest by converting the compound interest into an equivalent simple interest rate

where:

m = compounding interval

t = years over which the amount is invested

m = interest rate with compounding frequency m

Simple interest ratet 1 compound interest ratem mt 1



A bank pays interest at 5% p.a. compounded annually. If $1000 is invested for five years, what is the equivalent simple interest rate that the amount will earn?

______________________________________________

Effective simple interest rate

1 compound interest ratem mt 1

t

1 0.05 5 1

t0.056, or 5.6%



Discrete vs. Continuous Intervals

– Discrete compounding means we can count the number of compounding periods per year– e.g., once a year, twice a year, quarterly, monthly,

or daily

– Continuous compounding results when there is an infinite number of compounding periods


Continuous compounding

– Continuous compounding is the theoretical case where interest is calculated at every single point in time

– Another way of thinking about this is that the compounding period is infinitely small

– Although interest is not literally calculated in this way, this concept is used extensively in many finance applications and security valuation models


Continuous compounding (cont)

– To determine the accumulated value (FV) of an amount invested at a continuously compounding rate of interest:

where:PV = the cash flow invested or borrowedr = the continuously compounded rate of returnt = the time over which the cash flow is invested/borrowede = the base of natural logarithms (a constant),

equal to 2.718

rtPVeFV


Example 2.7On 8 January 2008, Lend Lease Corporation (LLC) opened for trading at a price of $13.75 and closed at a price of $13.90. If you bought LLC shares at the opening price and sold them at the end of the day, what is the continuously compounded rate of return you would have earned for the day?

______________________________________________

FV PVe rt

13.90 13.75ert

r ln 13.90 13.75 0.0109 1.09%



Topic Two – Class two

Present value

and

Future value

– Receiving $1 today is worth more than $1 in the future

– The opportunity cost of $1 in the future is the interest we could have earned on $1 if received earlier

Today Future

Time value of money


–Compounding–Translating $1 today into its equivalent

future value

–Discounting–Translating a future $1 into its

equivalent present value today

Compounding & discounting


– For one period:FV1 = PV ( 1 + r ) = PV ( 1 + r ) 1

– For two periods:FV2 = PV ( 1 + r ) ( 1 + r ) = PV ( 1 + r ) 2

– For n periods:FVn = PV ( 1 + r ) n

Future value


– The future value, FVt of a single amount invested today at r % for n periods is:

– The expression (1 + r)n is the future value interest factor (FVIF).

nr 1 V FV P

Future value

Future value of a single amount

Example 1

You invest $100 in a savings account that earns 10% interest per annum (compounded) for three years.

After one year: $100 (1+0.1) = $110

After two years: $100 (1+0.1)(1+0.1) = $121

After three years: $100 (1+0.1)(1+0.1)(1+ 0.1) = $133.10


Future value of multiple cash flows

– The general formula for future values of a multiple cash flow stream is:

where:FV = future value of a multiple stream of cash flowsXt = cash flow received in period tr = the compound interest rate on an alternative

comparable investmentt = the number of periods before Xt is received

n

t

tt rXFV

1

1

Future value of multiple UnevenCash Flows

– You deposit $1 000 now, $1 500 in one year, $2000 in two years and $2 500 in three years in an account paying 10% interest per annum. How much do you have in the account at the end of the third year?

– calculate the future value of each cash flow first and then total them.

Solution 0 1 2 3

$1,000 $1,500 $2,000 $2,500

$2,200

$1,815

$1,331

$7,846

$1 000 (1.10)3 = $1 331

$1 500 (1.10)2 = $1 815$2 000 (1.10)1 = $2 200

$2 500 1.00 = $2 500

Total = $7 846

End End End


The present value of a single cash flow

– The present worth of a payment to be received in the future, taking into account the time value of money

where:FV = the future cash flow to be receivedPV = the present value of the future cash flowr = the compound interest rate on an alternative

comparable investment

n = the number of periods before FV is received

PV FV

1 r n


Example 1

You are offered an asset that pays $1500 in five years. Currently, an investment opportunity of similar risk is available to you, paying interest at 5% p.a. compounded annually. What is the (present) value of the asset?

______________________________________________

Present value of a single amount (cont)

PV FV 1 r n

1500 1 0.05 5

$1175


0 1 2 3

$751.32

$826.45

$909.09$1,000

Example 2

If you will receive $1 000 in three years’ time. What is its PV if your opportunity cost/discount rate/interest rate is 10%?Can do it the long way, period by period

Discount one year: $1000 (1+0.10)-1 = $909.09Discount two years: $1000 (1+0.10)-2 = $826.45Discount three years: $1000 (1+0.10)-3 = $751.32

Straight Application of Formula

If you will receive $1,000 in three years’ time. what is its PV if your opportunity cost is 10%?

n

n-

r 1

FV r 1V PV

F

-30.1 1 1000 PV


Present Value

Example 3

If you will receive $1 000 in three years’ time. What is its PV if your opportunity cost/discount rate/interest rate is 10%?

Can do it the long way, period by period

Discount one year: $1000 (1+0.10)-1 = ?

Discount two years: $1000 (1+0.10)-2 = ?

Discount three years: $1000 (1+0.10)-3 = ?


Present Value

Example 3

If you will receive $1 000 in three years’ time. What is its PV if your opportunity cost/discount rate/interest rate is 10%?

Can do it the long way, period by period

Discount one year: $1000 (1+0.10)-1 = $909.09

Discount two years: $1000 (1+0.10)-2 = $826.45

Discount three years: $1000 (1+0.10)-3 = $751.32


Present Value

$100 to be received at the end of 3 years is worth how much today,

assuming a discount rate of

1. 10 per cent

2. 100 per cent

3. 0 per cent?

n

n

rFVPV

rFVPV

)1/(

1

or


nrFVPV 1

PV 100 1 010 13

3. $75.

PV 100 1 1 50

3$12.

PV 100 1 0 00

3$100.

Present Value

Example

Your rich grandmother promises* to give you $10,000 in 10 years’ time. If interest rates are 12% per annum, how much is that gift worth today?

*Don’t believe what grandma tells you!! Make her put it in writing!!

n

n-

r 1

FV r 1 V PV

F


Example

Your rich grandmother promises to give you $10000 in 10 years’ time. If interest rates are 12% per annum, how much is that gift worth today?

220 $3

0.3220 000 $10

0.12 1 000 $10 PV 10

Example

Your poor grandmother has a debt she may repay by paying $5000 NOW, or $10,000 in FOUR YEARS TIME.

If the interest rate is 14% compounded monthly, would you advise her to pay the debt NOW or IN FOUR YEARS?

We need to compare the present value of $10,000 to be paid in 4 years time with $5,000 now. The interest rate per compounding period is 0.14/12 = 0.01167, and the number of compounding periods in four years is 4 12 = 48.


Example

Your poor grandmother has a debt she may repay by paying $5000 NOW, or $10,000 in FOUR YEARS TIME.

If the interest rate is 14% compounded monthly, would you advise her to pay the debt NOW or IN FOUR YEARS?

64.730,5$01167.1

000,10

1 48

nr

PVPV


Present value of multiple amounts

– The present value of a stream of cash flows can be determined using the following equation:

where:PV = present value of future multiple cash flows

Xt = cash flow received in period tr = the compound interest rate on an alternative comparable

investment

t = the number of periods before Xt is received

PV X t

1 r tt1

n

X t 1 r t

t1

n

Present value of multiple Uneven cash flows

– You deposit $1 500 in one year, $2000 in two years and $2 500 in three years in an account paying 10% interest per annum. What is the present value of these cash flows?

– calculating the present value of each cash flow first and then total them.

Solution

0 1 2 3

$1,500 $2,000 $2,500

$1 500 (1.10)-1 = $1 364

$2 000 (1.10)-2 = $1 653

$2 500 (1.10)-3 = $1 878

Total = $4 895

$1 364

$1 653

$1 878

$4 895

EndEndEnd


Example 2.9You are offered an investment that promises:

$1000 in year 1, $2000 in year 2, $3000 in year 3 $500 in year 4.

If an investment opportunity of similar risk pays 10% p.a. compounded annually, what is the maximum amount that you would pay for this investment?

______________________________________________

Step 1: Construct a timeline

500

0

1000

2 3 41

Years

Cash flows 2000 3000?

Present value of multiple amounts (cont)


Example 2.9 (cont) You are offered an investment that promises:

$1000 in year 1, $2000 in year 2, $3000 in year 3 $500 in year 4.

If an investment opportunity of similar risk pays 10% p.a. compounded annually, what is the maximum amount that you would pay for this investment ______________________________________________

Step 2: Determine the present value

410.1

500

10.1

1000

210.1

2000

310.1

3000

0

Present value

$5157



Example 2.9 (cont)If we invest $5157 at 10% interest we can create the equivalent cash flow

stream, as follows:

At an interest rate of 10%, $5157 is equivalent to the cash flow stream described in the example

YEAR 1 2 3 4

Investment at beginning $5157 $4673 $3140 $454

Interest at 10% $516 $467 $314 $45

Balance $5673 $5140 $3454 $500

Less withdrawal $1000 $2000 $3000 $500

Balance at end of year $4673 $3140 $454 $0



Solving for the Rate of Return

Example – You currently have $100 available for investment for a

21- year period.

At what interest rate must you invest this amount in order for it to be worth $500 at maturity?



– Given any three factors in the present value or future value equation, the fourth factor can be solved.

r can be solved by one of two ways:

1. take the nth root of both sides of the equation2. use a financial calculator;



Example 1 Continued

– take the nth root of both sides of the equation

100 x(1+r)21 = 500

(1+r)21 = 500/100 = 5

(1+r)21/1 x 1/21 = 5 1/21 = 1.0797

r = 7.97%



Example – You currently have $5000 available for investment for a

30- year period.

At what interest rate must you invest this amount in order for it to be worth $15,000 at maturity?


Solving for the Rate of ReturnExample 1 Continued

– take the nth root of both sides of the equation

5000 x(1+r)30 = 15,000

(1+r)30 = 15,000/5000 = 3

(1+r)30/1 x 1/30 = 3 1/30 = 1.037299

r = 3.730%



Practice example – You currently have $25,000 available for investment for a

15 - year period.

At what interest rate must you invest this amount in order for it to be worth $150,000 at maturity?


Annuities

–Annuities are a special case of multiple cash flow streams, where the cash flows are of equal size and occur at regular time intervals


Annuities

– Examples of annuities include:

1. A constant retirement payment made to retirees on a monthly basis throughout the remainder of their lives

2. Rental payments

3. Interest payments on a bond


Annuities (cont)

– Types of annuity:1. Ordinary annuity

1. Cash flows occur at the end of each period and, hence, the first cash flow occurs at the end of the first year

2. Annuity due 1. Cash flows occur at the beginning of each period and,

hence, the first cash flow occurs immediately

3. Deferred annuity 1. The first cash flow in an annuity is delayed by x periods

Types of annuity:


Annuities


Future value of an annuity

– A four-year ordinary annuity paying $100 cash flows would look like this

– The future value of the four payments of this annuity can be calculated by compounding each cash flow forward to the fourth year and adding them together


Future value of an annuity (cont)

– To determine the accumulated value of an ordinary annuity of $A :

where:FV = the accumulated or future value of the annuityA = the cash flow received/paid under the annuityn = the number of cash flows that form the annuityr = the compound interest rate per period

Note: 1) There is no cash flow at time 02) There is a cash flow at time n

FV A1 r n 1

r


Example 2.12 At the end of each year, you place $500 in an account that

earns 5% interest p.a. compounded annually. How much will be in the account at the end of five years?

______________________________________________

FV A1 r n 1

r

5001 .05 5 1

.05

$2763

Future value of an annuity (cont)


0

What are we solving for?


Annuity – Future Value

A university student is planning to invest the sum of $200 per month for the next three years in order to accumulate sufficient funds to pay for a trip overseas once she has graduated. Current rates of return are 6 per cent per annum, compounding monthly.

How much will the student have available when she graduates?


36 years 12 3

0.005or 0.50% p.a.12

6.00%

$200

n

i

A


$7867.22

1]200[39.336

]0.005

1(1.005)200[

]0.005

10.005) (1 200[

:therefore

36

36

PV


Present value of an annuity

– The present value of an annuity can also be found by discounting the individual cash flows

– The present values of the four components of this annuity can be calculated by discounting each cash flow to the present and adding them together


Example 2.13 If you receive $100 at the end of each year for four years, what is

the value of these cash flows if the current rate of interest is 5% p.a. compounded annually?

Present value of an annuity (cont)

PV1 $100 x (1 + i)-1 $100 / 1.05 $95.24

PV2 $100 x (1 + i)-2 $100 / 1.1025 $90.70

PV3 $100 x (1 + i)-3 $100 / 1.1576 $86.39

PV4 $100 x (1 + i)-4 $100 / 1.2155 $82.27

Total $354.60


Present value of an annuity (cont)

– Alternatively, to determine the present value of an ordinary annuity of $A:

where:PV = the present value of the annuityA = the cash flow received/paid under the annuityn = the number of cash flows that form the annuityr = the compound interest rate per period

r

rAPV

n11


Annuity due

– To date we have assumed that payments occur at the end of each period throughout the annuity

– ordinary annuity

– Often payments are made at the beginning of each period (e.g. rental agreements)

– annuity due


Annuity due

where:PV = the present value of the annuityA = the cash flow received/paid under the annuityn = the number of cash flows that form the annuityr = the compound interest rate per period

PV A A1 1 r n 1

r


What is the present value of a $1000 annuity that pays five regular payments when the interest rate is 10% p.a. with the first payment due immediately?

______________________________________________

Present value of an annuity due


This is equivalent to an amount of $1000 now plus an ordinary annuity of four payments of $1000

Present value of an annuity due

PV 1000 10001 1 .10 5 1

.10

$4170


Present value of annuity with different rates of return

– The present value of a stream of cash flows can be determined using the following equation:

where:PV = present value of future multiple cash flowsr1 = rate for first cash flowr2 = rate for second cash flown = the number of periods

) ^-n1.0r )^-n(r PMT(1 )^-nr (1 PMT PV 121



– Example: – The present value of a stream of two annual cash flows of $1000

beginning in one years time, where the interest rate is 8% p.a. for the first year and 10% p.a. for the second year is:

– First cash flow 1000 (1.08)^-1 = 925.93– Second cash flow 1000(1.10)^-1 (1.08)^-1 = 841.75– – PV = 925.93 + 841.75 = 1767.68

–

^-n^-n^-n )r(1)r PMT(1 )r PMT(1 PV 121



– Example: – The present value of a stream of four annual cash flows

of $100 beginning in one years time, where the interest rate is 5% p.a. for the first year, 8% p.a. for the second year. 9% p.a. for the third year and 10% p.a. for the fourth year is:

where:PV = present value of future multiple cash flowsr1 = rate for first cash flowr2 = rate for second cash flowr3 = rate for third cash flowr4 = rate for fourth cash flown = the number of periods


)^-nr)^-n)(1r)^-n(1r)^-n(1rPMT(1

)^-n)r)^-n(1r)^-n(1rPMT(1

^-n))^-n(1.0rr PMT(1

)^-nr PMT(1 PV

4321

321

21

1



Deferred annuities

–A deferred annuity is an ordinary annuity that does not begin in one period’s time, but at a later date


Deferred annuities (cont)

– To determine the present value of a deferred annuity (where x is the number of periods before the first cash flow occurs):

– The term in large brackets is the present value of an ordinary annuity of n payments

– This second term gives the present value of the single lump sum calculated from the first term, discounted back x - 1 periods

PV A 1 1 r n

r

1 r x 1


What is the present value of an asset that commences paying cash flows of $2 million in two years’ time for four years, when the interest rate is 5%?

______________________________________________

*An ordinary annuity would occur in one year’s time, so the above series of cash flows is the equivalent of an ordinary annuity deferred for one year.

$2m

0 2 3 41Year

$2m $2m $2m

5



Example 2.16 (cont) What is the present value of an asset that commences paying

cash flows of $2 million in two years’ time for four years, when the interest rate is 5%?

______________________________________________

Step 1:

Present value of $2 million payments at start of year 2 (= end of year 1)

millionmPV 09.7$

05.

05.112$

4

2




cash flows of $2 million in two years’ time for four years, when the interest rate is 5%?

______________________________________________

Step 2:

Calculate the present value of this lump sum cash flow occurring at the end of period 1 back to time 0

m

r

FVx 75.6$

05.1

09.7

1PV 1



Another example What is the present value of an asset that commences paying

cash flows of $4 million in two years’ time for five years, when the interest rate is 8%?

______________________________________________

Step 1:

Present value of $2 million payments at start of year 2 (= end of year 1)

840,970,15$

08.

08.114$

5

2

mPV




cash flows of $4 million in two years’ time for five years, when the interest rate is 8%?

______________________________________________

Step 2:

Calculate the present value of this lump sum cash flow occurring at the end of period 1 back to time 0

mr

FV

x788.14$

08.1

971.15

1PV

1



Equivalent annuity

–It is sometimes necessary to compare cash flow streams in which cash flows occur at different time intervals

–This can be done by converting one or more cash flows to an equivalent annuity


Equivalent annuity

–This is done by:

1. Calculating the present value of a cash flow stream

2. Determining an annuity whose present value is equal to this cash flow stream


Equivalent annuities

– Consider two machines that differ in their cash flow requirements – which of these machines would a company prefer to operate continuously ?

– Convert Machine 1 cost to an equivalent annual cost, and choose the machine with the lower annual cost


Example 2.17 What is the equivalent annual cost of operating a machine that

requires an outlay of $7m every 5 years, with the first payment made at the end of the first year of operation? The opportunity cost of capital is 5%.

_________________________________________________________

Equivalent annuities (cont)

0 2 3 41Year

$7m

5

$x$x $x $x $x


We want to know the value of x in the following, if x’s present value is equal to the below:


0 2 3 41Year

$7m

5

$x$x $x $x $x


Example 2.17 (cont) What is the equivalent annual cost of operating a machine that requires an

outlay of $7m every 5 years, with the first payment made at the end of the first year of operation? The opportunity cost of capital is 5%.

_________________________________________________________

Step 1:

Calculate the present value of the cash payments made in operating Machine 1


PV FV 1 r n 7 1.05 $6.67m


Step 2:

Determine the ordinary annuity over 5 years with a present value of $6.67m.


5%at years 5for annuity offactor PV *

541.1*329.4/67.6

67.6$05.

05.11 5

mm

mAPV


Equivalent Annuities

– In the earlier example, the cost of operating Machine 1 ($7m every five years) has now been converted to an equivalent annual cost of $1.54m

– Machine 1 is clearly preferred because it has a lower equivalent annual operating cost than Machine 2


Which machine

should Mary recommend?


Can I afford a home in Vietnam?


Mortgage Finance

– Calculating the installment on a mortgage loan

])1(1

[

ii

amountLoantInstallmen n


Mortgage Finance (cont.)

– Calculating the installment on a mortgage loan (cont.)

– A company is seeking a fully-amortised commercial mortgage loan of $650,000 from its bank. The conditions attached to the loan include an interest rate of 8 per cent per annum, payable over five years by equal end-of-quarter installments. The company treasurer needs to ascertain the quarterly installment amount.



– Calculating the installment on a mortgage loan (cont.)

tinstallmenquarterly 751.87 $39

]0.02

0.02)(11[

000 $650

2045

0.024

0.08

000 $650

20

R

n

i

A


– After 2.5 years you want to pay off the entire balance of the mortgage. How much would that balance be?

loan of termremaining *

39751.87 PV 0.02)20.(1 - 1 *10



– After 2.5 years you want to pay off the entire balance of the mortgage. How much would that balance be?

55.074,357

98259.8 39751.87 PV


Net Present Value

The net present value (NPV) of a capital projectis calculated by subtracting the present value

of future cash outflows from the presentvalue of future cash inflows.

The net cash flows for all years are discountedusing the firm’s blended cost of capital.

NPV Example

Assume that the Whitewater AdventureCompany is considering a computer

upgrade of $100,000.

This project should result in net cash inflowsof $31,000 per year for the next four years.

The blended cost of capital is equal to 14%.

Therefore, the discount rate used is 14%.


NPV Example

$31,000$(100,000) $31,000 $31,000 $31,000 $31,000

Years in the Life of the Project0 1 32 4 5

$31,000 × 3.433 = $106,423

$ 106,423$ 6,423

NPV = $106,423 – $100,000

NPV = $6,423


NPV Example

Assume that Whitewater’s computerupgrade will require $12,000 in

maintenance fees in year 3.

Also that the system can be soldat the end of year 5 for $6,000.


NPV Example

Initialinvestment Maintenance

Operatingcosts

Residualvalue

Net cashflow

($100,000)

$31,000

$31,000

$31,000

$31,000

$ 37,000

Year

($100,000)

$ 31,000

$ 31,000

$ 19,000

$ 31,000

($12,000)

$6,000$31,000

0

1

2

3

4

5


NPV Example

YearPV of $1,

Factor at 14%

0.877

0.769

0.675

0.592

0.519

0

1

2

3

4

5

Net cashflow

$ 37,000

($100,000)

$ 31,000

$ 31,000

$ 19,000

$ 31,000

PresentValue

27,187

23,839

12,825

18,352

19,203

($100,000)

Net present value $ 1,406


Net Present Value

· NPV = the total PV of the annual net cash flows less the initial outlay.

NPV = - IO FCFt(1 + k) t

n

t=1S


End of chapter 2!


Comparing Simple and Compound Interest

– Simple interest refers to interest earned only on the original capital investment amount.

FV = PV(1 + r x n)(e.g. $100 for 3 years at 10% Simple Interest will accumulate to $130)

– Compound interest refers to interest earned on both the initial capital investment and on the interest reinvested from prior periods.

FV = PV(1 + r ) n

– In finance it is almost always compound interest that is used


– Making use of Example 1: $100 invested at 10% for 3 years

The accumulated value of this investment at the end of three years can be split into two components:

– original principal: $100– interest earned: $33.10

– Using simple interest, the total interest earned would only have been $30. The other $3.10 is from compounding that is, interest on interest.

Comparing Simple and Compound Interest


Example

– What will $1 000 amount to in 5 years’ time if interest is 12% per annum, compounded annually?

FV = PV(1 + r ) n


Future values at different interest ratesFuture value of $100 at various interest ratesNumber of

periods5% 10% 15% 20%

1 $105.00 $110.00 $115.00 $120.00

2 $110.25 $121.00 $132.25 $144.00

3 $115.76 $133.10 $152.09 $172.80

4 $121.55 $146.41 $174.90 $207.36

5 $127.63 $161.05 $201.14 $248.83

• For a given number of periods the higher the interest rate the higher the future value.

• For a given interest rate the longer the period the amount accumulates for the greater the future value



– Example: – The present value of a stream of two annual cash flows of $1000

beginning in one years time, where the interest rate is 8% p.a. for the first year and 10% p.a. for the second year is:

where:PV = present value of future multiple cash flowsr1 = rate for first cash flowr2 = rate for second cash flown = the number of periods

)^-nr)^-n)(1r)^-n(1r)^-n(1rPMT(1

)^-n)r)^-n(1r)^-n(1rPMT(1

^-n))^-n(1.0rr PMT(1

)^-nr PMT(1 PV

4321

321

21

1

topic 2 1

Documents

cash flow

cash flow

cash flow

multiple cash

assets cash

cash flows

continuously

future cash