Financial Calculation

8/8/2019 Financial Calculation

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Applied Quantitative Methods

Financial Calculations


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CE61001-1-Applied Quantitative Methods Financial Calculations Slide 2 (of 69)

Topic & Structure of the lesson

Introduction

Discounting, Present Values

NPV

IRR

Annuities

Perpetual annuities

Loan Repayment


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At the end of this topic , You should beable to

Use the discounting process to find the presentvalue of future streams of income

To use Net Present Value (NPV) and Internal Rate ofReturn (IRR) to compare alternative investmentprojects.

Know what annuities are

Define a perpetuity

Learning Outcomes


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CE61001-1-Applied Quantitative Methods Financial Calculations Slide 4(of 69)

Key Terms you must be able to use

Ifyou have mastered this topic, you should be able to use the following termscorrectly in your assignments and exams:


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Recap:

Simple interest Simple interest is where interest is added based only on the

original sum deposited.

Example:

Suppose we deposit $500 at the beginning ofyear 1 in an account thatpays simple interest at 6% p.a., how much is in the account at the end of5 years ?

Financial Arithmetic


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Compound interest Compound interest is where interest is added based not only on

the original sum deposited, but also on any interest added to date.

Example:

Suppose we deposit $500 at the beginning ofyear 1 in an accountthat pays compound interest at 6% p.a. how much is in the accountat the end of5 years ?


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Depreciation A car is purchased for $14750. If it depreciates in value at 3% per

quarter, what is its value after 3 years ?


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Effective Rates of Appreciation or Depreciation

Example:

A machine costing $20000 has a value of $6000

after 8 years. What has been the effective monthlyrate of depreciation ?


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Annual Percentage rate The APR is the equivalent compound rate that needs to be applied

just once at the end ofa year that is the same as applying thegiven compound rate per period for a number ofperiods in one

year. Example:

Determine the APRfor a loan on which an interest rate of10% isapplied every halfyear.


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APR Equivalent Compound Rates

1APR1rn

!Example:

A loan for which monthly repayments are required hasan APR of 10%. What is the equivalent monthlycompound rate ?


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Quick Review Question


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Introduction

A means ofassessing whether an investment

project is worthwhile or notInvestment decisions are long run decisions

where consumptions and investmentopportunities are balanced over time in the

hope that investment will generate extraincome in the future.

Investment Appraisal


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The decisions to invest is based on factorslike:

Investors beliefs in the futureBased on forecasts of internal and external factors

including costs, revenues, inflation and interestrates, taxation and others.


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The alternatives available

Analysts should analyse thealternatives available and thenanalyse the data using the mostappropriate techniques


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Investors attitudes to risk

Uncertainty and risk are important factors tobe considered in investment appraisal

The decision makers attitude to risk is ofcritical importance but is extremely difficultto quantify. In general, decisions makers arerisk averters.


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Investment appraisal method

Can be categorised into

Traditional methods

Payback period (PBP)

Accounting Rate ofReturn

Discounted Cash flow techniques

Net Present Value

Internal Rate ofReturn


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Present Value

Present values It is a technique that enables a future

cash flow to be represented in the

equivalent of todays money terms. It is a simple re-arrangement of the

formula given for calculating anaccrued amount using compound

interest


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The present value of amount A, payablein n years time, subject to a discount rateof

100i % is given by:

where: P = present value

A = amount, payable in nyears timeI = discount rate

n = number of time periods

ni1

AP

!


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Using a discount rate of 12% p.a., find the presentvalue of $10000 to be received in 20 years?



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Net Present Value (NPV)

Net Present Value is a means ofexamining the return from a proposedproject in order to decide whether the

project is viable. NPV compares the value of a dollar

today to the value of that same dollar inthe future, taking inflation and returns into

account.


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What it means: It's the net result of amultiyear investment expressed in today'sdollars.

The NPV is calculated as the presentvalue of the project's cash inflows minusthe present value of the project's cashoutflows


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To do the calculation you apply adiscount % rate to the future earnings.

Discount the future net cash flows from

the investment at the required rate ofreturn

Subtract the initial amount invested from

sum of the discounted cash flows.


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The NPV can also be used to comparethe returns from a number of differentinvestments in order to establish which is

the better option to choose. The investment showing the greater NPV is

the better one to opt for.


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If the Net esent

Value is . . . hen the oject is . . .

ositive . . .Accepta le since it p o ises a

etu n eate than the equi ed

ate of etu n.

Ze o . . .

Accepta le since it p o ises a

etu n equal to the equi ed ate

of etu n.

Ne ative . . .Not accepta le since it

p o ises a etu n less than the

equi ed ate of etu n.

Decision Criterion using NPV


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NPV With Unequal Cash Flows

Present

alue of

et Cash Flows 1 Factor P of et Cash Flows

Year A B C at 10% A B C

1 5,000 8,000 1,000 0 9091 4,546 7,273 909

2 5,000 5,000 5,000 0 8264 4,132 4,132 4,132

3 5,000 2,000 9,000 0 7513 3,757 1,503 6,762

Total 15,000 15,000 15,000 12,435 12,908 11,803

Amount invested (12,000) (12,000) (12,000)

et Present alue 435 908 (197)

Although all pro ects re uire the same investment andhave the same total net cash flows, pro ect B has a highernet present value because of a larger net cash flow inyear 1


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Using Net Present Value

Advantages:

Disadvantages:


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Internal Rate ofReturn (IRR)

The Internal Rate of Return (IRR) is defined asthe discount rate that makes the project have azero Net Present Value (NPV).

The interest rate the makes

Presentvalue of

cash inflows

Presentvalue of

cash outflows=

The net present value equal zero.


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Example:



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The graph below was plotted for a wide range ofrates until the IRR was found that yields an NPVequal to zero (at the intercept with the x-axis).


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Calculation of IRR can be obtained using

Graphical approach

Linear interpolation

Using IRR criteria:Choose that project which has thehighest IRR


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Using Graphical Approach

Calculate two values of NPV with therespective discount rate: NPV + and NPV

Plot these two points on the graph of NPCagainst discount rate

Draw a line between these two points.

The points where the line crosses the x-axis(which NPV = 0) in the IRR


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Example:

A company can buy either of two machines used tomanufacture a particular component that it suppliesto the motor car industry. The cost of the twomachines and the estimated year end net cash flowsare given below: Using present values and assumingan interest rate of 7% decide which machine shouldbe purchased.


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Using LinearInterpolation

2N1N

1r

2N

2r1

N

IRR

!


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Example:

A company can buy either of two machines used tomanufacture a particular component that it suppliesto the motor car industry. The cost of the twomachines and the estimated year end net cash flowsare given below: Using present values and assumingan interest rate of 7% decide which machine shouldbe purchased.


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IRR for Machine A

1= 12904 r 1= 7%

2= -2088 r2= 12%

11.95%IRR

0.1195208812904

0.0720880.1212904IRR

!

!

vv!


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IRR for Machine B

1= 8094 r 1= 7%

2= -1601 r2= 12%

11.17%IRR

0.111760110948

0.0760110.120948IRR

!

!

vv

!


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Calculate the IRR of a project that costs$100 million and yields $106 million inone year when the opportunity cost of

capital is 7% ?

Irr = 6%

Since the irr of the project is less than theopportunity cost of capital, it should berejected.

100

irr1

1060

!



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1 Compute present value factor12,000 5,000 per year = 2 4000

2 Using present value of annuity table

Pro ect life = 3 yearsInitial cost = 12,000

Annual net cash inflows = 5,000

Determine the IRR for this pro ect

Projects with equal annual cash flows


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Using present value of annuity table

Periods 10% 12% 14%1 0.90909 0.89286 0.877192 1.73554 1.69005 1.646663 2.48685 2.40183 2.321634 3.16987 3.03735 2.91371

5 3.79079 3.60478 3.43308

In that row,

locate theinterest factor

closest inamount to thepresent value

factor.


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Periods 10% 12% 14%1 0.90909 0.89286 0.877192 1.73554 1.69005 1.646663 2.48685 2.40183 2.321634 3.16987 3.03735 2.913715 3.79079 3.60478 3.43308

IRR is the

interest rateof the columnin which the

present value

factor is found

IRR isapproximately

12%.


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Ifcash inflows are unequal, trial and errorsolution will result ifpresent value tablesare used.

Sophisticated business calculators andelectronic spreadsheets can be used toeasily solve these problems.

IRR with unequal cash flows


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Both NPV and IRR allow judgments to bemade at the current value of money. (+)

NPV very important to set the right

interest rate (market interest, inflation &risk)

IRR difficult to compute solving nthorder polynomial

Pros and Cons


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NPV is the preferred method for mostsituations but often several methods areused to assess an investment opportunity.


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Annuities

Annuities it is a se uence of fixed e ual payments

(or receipts) made over uniform timeintervals

it can be due (paid at beginning of period) ordinary (paid at end of period)

commonly used to repay debts as investments to meet fixed known

commitments(known as sinking funds)


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Cost of Annuities

the total sum of the present values of the futurepayments

when the first payment is going to start one time periodaway

where C = cost of annuityP = annuity

i = interest rate

? Ai

ni11

C

!


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? Ain

i11i1P

!

when the first payment is going to startimmediately


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Accumulated value ofannuity the accumulated value at year n, with first payment

starting one period from now.

The accumulated value at end ofyear n, with firstpayment starting immediately.

? Ai1

ni)(1PS !

? A i1i

1ni)(1PS

!


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Loan repayment

Loan can be repaid by

amortisation method

sinking fund method

Loan Repayment Methods


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Amortisation method

Loan is repaid by using a series ofinstalments payments

Payment is made at one time period away

Payment is made immediately


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Amortisation schedule

it is a specification, period by period (normallyyear by year) of the state of the debt

It is usual to show for each year amount of debt outstanding at beginning of year

interest paid

annual payment, and, optionally,

amount of principal repaid


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Sinking Funds Method

defined as annuity invested in order to meeta known commitment at some future date

Commonly used for repayment ofdebts

to provide funds to purchase a new asset whenthe existing one is fully depreciated.


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Repayment can be

made at end of year

made at beginning of year

where interest is serviced periodically overthe loan

where interest is not serviced and forms partof the loan at the end of the period


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Sinking fund schedule

it is used to show for each year

for the debt

the regular repayment (deposit) interest earned

for the fund

amount in fund

the outstanding amount

interest paid


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Perpetual Annuities

A perpetual annuity has no statedperiod over which the annual paymentsare made but rather provides an

indefinite number of payments (usuallyuntil the death of the investor)

Its theoretical value will be the sum to

infinity of these discounted annualpayments.


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Total value =

...r1r1r1r1

432

Provided r is numerically less than 1 thesesuccessive terms will get progressivelysmaller and the total value will have a

finite sum.


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The value of a perpetual annuity paying $Aat the end of every year when the financialinterest rate is equal to r is

So a perpetual annuity paying $2000annually, when the interest rate is 8%, will

have a theoretical (maximum) value of,

rAS !

g

2500008.0

2000!


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Follow Up Assignment


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Summary of Main Teaching Points


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Financial Calculation