# Expected Present Value

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Please may someone help me with this question

Calculate the expected present value of the following medical procedures

using 13% interest per annum and assuming the events are

uniformly distributed over the given periods. itemize

2. Caesarean section (C-section) for a woman currently 2 months

pregnant. The likelihood that a C-section will be needed is 45%,

and the expected timing of the C-section is sometime during the

9th month of pregnancy. The cost of the procedure is R8 500.

answered Apr 24, 2018 by (250 points)
edited Apr 24, 2018

Hi There.

Thank you so much for the Question.

The Expected Present Value (EPV) of a series of cashflows is quite simply the expectation of the present value of the cash flows. In other words, one needs to find the present value of a cashflow and then multiply it by the likelihood of that cash flow occuring.  So for a cashflow of value A at any given time in the future:

$$EPV(A) = P( Cash \ Flow \ A\ occuring)*Present \ Value(Cash \ Flow \ A)$$

In the question you have provided , there is a further concept that is explored : Events are uniformly distributed over the given periods.This is an assumption with regard to the timing of events,i.e. when events are expected to happen on average. What this assumption means is that, if some cash flow is expected to happen during a given period, we will assume that the likelihood of it occuring during that period is uniformly distributed and hence it is expected to occur  on average halfway through the period( the period only being the time that the cash  flow can/is expected to happen. ).

Aside: Note that the convention of halfway through the period for uniform distributed events comes from the fact that expected value of the uniform distribution is halfway between the length of the interval ( You will encounter more of this in your statistics courses)

To finally answer the question :

1)Finding the Present Value

The Question tells us that the woman is currently 2 months pregnant and that the expected timing of the C-Section will be sometime during the 9th month of pregnancy.  We therefore, for the purposes of finding our relevent "n" in order to calculate the present value, need to consider the following timeline:

[=================== {{}{}{}}]

|-----|-----|-----|-----|-----|-----|-----|-----|-----|
0    1      2    3     4     5     6     7     8     9

Since we are told to assume that the timing of events are uniformly distributed and that the cash flow is expected to occur sometime during the 9th month of pregnancy(between months 8 and 9),we know that the cashflow will therefore occur halfway between months 8 and  9. Our "n" is therefore the length of time between middle of  month 8 and 9 and month 2.

$$\therefore n = 8.5 - 2 = 6.5$$  , where n is in months!

The rest follows :
$$PV= 8500v^ \frac{6.5} {12} _ {13\%}$$
2)Expected Present Value :
$$EPV=0.45* 8500v^ \frac{6.5} {12} _ {13\%}$$
$$EPV = 0.45*8500(1.13)^\frac{ -6.5}{12} = R3579.98$$

Note that the cash flow is only expected to occur during the 9th month, and hence our assumption of the timing of the event being uniformly distributed only applies to the 9th month ( between month 8 and 9 ), when the cash flow is expected to happen.