Question

Question 5: Suppose we have an investment project with initial expenditure equal to 10,000 SAR and the distribution of cash flows are as follows: Period 1 Period 2 CF 3000 4000 5000 5000 6000 7000 8000 4000 7000 .30 6000 Period 1 Period 1 3000 5000 6000 8000 7000 Total Period 2 Period 1 4000 5000 7000 4000 6000 Total

1-Calculate the expected value of NPV if the risk-free rate equal to 8%.

2-If cash flows are considered independent. Calculate the variance and standard deviation of the NPV.

3-What is the probability that the NPV of the project is less than or equal to zero?

4-If cash flows are considered totally dependent. Calculate the variance and standard deviation of the NPV.

5-What is the probability that the profitability index is greater than 2?

finance : investment decision under uncertainty

*Project selection for several periods - a state of complete independence*

Answer 1

Solution:

1) Expected Value of NPV

NPV=(Cash flow​/(1+i)t )−initial investment

where:i=Required return or discount rate

t=Number of time periods​

Expected Cash Flow = Cash Flow * Probability

Period 1
3000	.10	300
5000	.20	1000
6000	.25	1500
8000	.15	1200
7000	.30	2100
Total	6100 SAR

Period 2
4000	.15	600
5000	.20	1000
7000	.25	1750
4000	.20	800
6000	.20	1200
Total	5350 SAR

NPV = (Expected Cash Flow Period 1 * Dicount Rate at 8%) + (Expected Cash Flow Period 2 * Dicount Rate at 8%) - Initial Investment

= (6100 * .92593) + (5350 *.85734) - 10,000

= 5648.173 + 4586.769 - 10000

= 234.942 SAR

Thus, the Expected Value of NPV at 8% risk free rate is 234.942 SAR

2) Variance and Standard Deviation of NPV

Period 1:

Variance = (Cash Flow - Total Expected Cash)2 * Probability

(3000 - 6100)2 *.10	961000
( 5000 - 6100)2 *.20	242000
( 6000- 6100)2 *.25	2500
( 8000- 6100)2 *.15	541500
( 7000- 6100)2 *.30	243000
Total	1990000

Standard Deviation of Period 1 = √1990000

= 1410.674

Period 2:

(4000 - 5350) * .15	273375
(5000 - 5350) * .20	24500
(7000 - 5350) * .25	680625
(4000 - 5350) * .20	364500
(6000 - 5350) * .20	84500
Total	1427500

Standard Deviation of Period 2 = √1427500

= 1194.780

Standard Deviation of NPV = √({[1410.674]2 / [1+0.08]2} + {[1194.780]2 / [1+0.08]2})

= √({1706105.22485} + {1223850.5216})

= 1711.7114 SAR

Variance of NPV = Square of Standard deviation

= 1711.7114 * 1711.7114

= 2929955.74645 SAR

Thus, the standard deviation of NPV is 1711.7114 SAR and the variance of NPV is 2929955.74645 SAR

3) Probability that the NPV of the project is less than or equal to zero

Z = (0- {234.942/1711.7114}).

= -0.1373

According to the Z table, the probability of NPV being zero is .44828 that is 44.828%. THus, the probability of NPV being zero is .8 - .44828 = 0.35172 or 35.172%.

4) Standard Deviation and Variance when totally dependent

Standard Deviation of NPV = ({√1990000} / 1.08) + {√1427500} / {1.08}2)

= 1306.17925737 + 1024.33154537

= 2330.51080274 SAR

Variance of NPV = Square of standard deviation

= 2330.51080274 * 2330.51080274

= 5431280.60618 SAR

Thus, the standard deviation of NPV when cash flows are dependent is 2330.51080274 and variance of NPV when dependent is 5431280.60618 SAR

5) Probability that the profitability index is greater than 2

Probability Index grater than 2 = p(Z grater than or equal to (10000-234.492)/2330.51080274

= P(Z ≥ 4.19)

= P(Z ≤ -4.19)

= 0.00001

= .001%

Thus, Probability that the profitability index is greater than 2 is 0.001%