# Question 3.23: Assume that a project has a mean of Rs 40 and standard devia......

Assume that a project has a mean of Rs 40 and standard deviation of Rs 20. The management wants to determine the probability of the NPV under the following ranges:
(i) Zero or less,
(ii) Greater than zero,
(iii) Between the range of Rs 25 and Rs 45,
(iv) Between the range of Rs 15 and Rs 30.

Step-by-Step
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(i) Zero or less: The first step is to determine the difference between the expected outcome $X$ and the expected net present value $\overline{X}$ . The second step is to standardise the difference (as obtained in the first step) by the standard deviation of the possible net present values. Then, the resultant quotient is to be seen in statistical tables of the area under the normal curve. Such a table (Table Z) is given at the end of the book. The table contains values for various standard normal distribution functions. Z is the value which we obtain through the first two steps, that is:

$Z=\frac{0 \,-\, \mathrm{Rs\>40}}{\mathrm{Rs\>20}}= -2.0$

This is also illustrated in Fig. 3.3.

$\,$  The figure of –2 indicates that a NPV of 0 lies 2 standard deviation to the left of the expected value of the probability distribution of possible NPV. Table Z indicates that the probability of the value within the range of 0 to 40 is 0.4772. Since the area of the left-hand side of the normal curve is equal to 0.5, the probability of NPV being zero or less would be 0.0228, that is, 0.5 – 0.4772. It means that there is 2.28 per cent probability that the NPV of the project will be zero or less.
$\,$  (ii) Greater than zero: The probability for the NPV being greater than zero would be equal to 97.72 per cent, that is, 100 – 2.28 per cent probability of NPV being zero or less.
$\,$  (iii) Between the range of Rs 25 and Rs 45: The first step is to calculate the value of Z for two ranges:
(a) between Rs 25 and Rs 40, and (b) between Rs 40 and Rs 45. The second and the last step is to sum up the probabilities obtained for these values of Z:

$Z_1=\frac{\mathrm{Rs\>25}\,-\, \mathrm{Rs\>40}}{\mathrm{Rs\>20}}= -0.75$   $Z_2=\frac{\mathrm{Rs\>45}\, -\, \mathrm{Rs\>40}}{\mathrm{Rs\>20}}= +0.25$

$\,$  The area as per Table Z for the respective values of –0.75 and 0.25 is 0.2734 and 0.0987 respectively.
Summing up, we have 0.3721. In other words, there is 37.21 per cent probability of NPV being within the range of Rs 25 and Rs 45. (It maybe noted that the negative signs for the value of Z in any way does not affect the way Table Z is to be consulted. It simply reflects that the value lies to the left of the mean value).
$\,$  (iv) Between the range of Rs 15 and Rs 30:

$Z_1=\frac{\mathrm{Rs\>15}\,-\, \mathrm{Rs\>40}}{\mathrm{Rs\>20}}= -1.25$   $Z_2=\frac{\mathrm{Rs\>30} \,- \, \mathrm{Rs\>40}}{\mathrm{Rs\>20}}= -0.50$

$\,$  According to Table Z, the area for respective values –1.25 and –0.5 is 0.3944 and 0.1915. The probability of having value between Rs 15 and 40 is 39.44 per cent, while the probability of having value between Rs 30 and 40 = 19.15 per cent. Therefore, the probability of having value between Rs 15 and Rs 30 would be 20.29 per cent = (39.44 per cent – 19.15 per cent).
$\,$ The application of the probability distribution approach in evaluating risky projects is comprehensively illustrated in Example 3.24.

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