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DEPENDENCE AND

INDEPENDENCE OF

CASHFLOWS OVERTIME

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INTRODUCTION

Risk is the variability of possible outcomes.

Investment proposals entail different degrees of

risk, statistical techniques are used as analyticaltools for handling risky investments.

Risk is expressed interms of the dispersion of theprobability distribution of possible NPV or IRR

and is measured by S.D.

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PROBABILITY

Probability is defined as a measure ofsomeones opinion about the likelihood thatan event will occur.

The probability of all events lies between zeroand one

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PROBABILITY

Probability theory plays an important role inanalysing risk in capital budgeting .

The risk of an investment project can be

described by calculating the expected NPVand SD. The risk can be measured under the

assumptions of serial independence of cash

flows overtime and their dependence.

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INDEPENDENCE OF CASHFLOWS

OVERTIME

The independence of cash flows overtimemeans that the probability distributions for thefuture periods are not dependent on eachother.

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ASSUMPTION OF

INDEPENDENCE

With the independence of cash flowsovertime, the outcome in period t does notdepend on what happened in period

t-1 In other words, there is no causative

relationship between cash flows from period

to period.

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EXPECTED NET PRESENT VALUE

Once probabilities are assigned to the futurecash flows, the next step is to find outexpected NPV.

The ENPV can be found out by multiplyingthe monetary values of the possible events

(cash flows) by their probabilities.

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EXPECTED NET PRESENT VALUE

ENPV=Sum of P.V. of expected net cashflows

ENPV=nENCFt/(1+k)tt=0

Expected NCF can be calculated as

ENCFt=NCFjt * Pjt WHERE

NCFjt =Net cashflow for jth event in period t

Pjt= Probability of net cashflow for jth event in period t

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RISK-FREE RATE FOR

DISCOUNTING

It is important to note that the discount rateshould be risk-free when we use probabilityinformation for cashflow distributions.

The probability of the occurrence of a givenset of cashflows will be high or low dependingon whether the risk is high or low.

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RISK-FREE RATE FOR

DISCOUNTING

If cost of capital is used as the discount rate,it would result in double counting of risk. (

Cost of capital incorporates both a risk-freerate and a premium for risk)

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STANDARD DEVIATION

The S.D of net cashflows for each period canbe expressed as-

t=n(NCFjt-ENCFt)2 Pjt

t=S.D. of net cashflows in period t

NCFjt=net cashflow

ENCFt=Expected value of the net cashflow

Pjt =Probability associated with each cashflow

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STANDARD DEVIATION

Using S.Ds for various periods, we candevelop a measure of risk for the projectunder the assumption of independence of

cashflows overtime.

= 2/(1+Rf)2t or

= 2/(1+kf)2t

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STANDARDIZING THE

DISPERSION

The expected value and S.D. of theprobability distribution of possible NPVs givesus a considerable amount of information by

which to evaluate the risk of the investmentproposal.

If the probability distribution is approximatelynormal (bell-shaped), we are able to calculatethe probability of a proposal providing NPVsof less than or more than a specified amount.

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STANDARDIZING THE

DISPERSION

The normal distribution can be used to furtheranalyse the risk element in capital budgeting.

The probability is found by determining the

area under the curve to the left or right of aparticular point of interest.

S=X-NPV/ X= outcome in which we are interested

=S.D