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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
