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# NPV and Standard Deviation

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To determine the present value, we need to calculate the cash value of the two 7 million deposits. The discount rate is an 8% opportunity rate, and the present value of the 7 million deposit is:

Present Value= (Cash Flow at end of year 1/ (1+ interest)1 )+(Cash Flow at end of year 2/(1+interest)2 )

Value= (7/1.081) + (7/1.082)

Value of second alternative= 6.48 + 6.00= 12.48

The present value of the second option is \$12.48 million, which is higher than the \$12 million of the first option. The second alternative needs to be chosen over the first alternative since it provides a higher dollar value than the first alternative.

Part b

The use of a 12% discount rate will provide new present value for the second alterative as follows:

Present Value= (Cash Flow at end of year 1/ (1+ interest)1 )+(Cash Flow at end of year 2/(1+interest)2 )

Value= (7/1.121) + (7/1.122)

Value of second alternative= 6.25+ 5.58= 11.83

When the discount rate is 12%, the new present value of cash flows received under the second alternative becomes \$ 11.83 million. When the interest rate changes to 12 %, the first alternative is preferable since it has a higher cash flow of \$12 million.

Part C

This type of payment scenario is easily evidenced in hire purchase store. The cash value of goods are usually lower that the higher purchase price of the goods. The higher value is usually set to compensate for the opportunity interest rate over the extended time as well as other financial risks that could result from the late payment.

Problem 2

Part a

The expected net present value (ENPV) is provided by determining the expected present value and then subtracting the initial investment from the outcome (ENPV= EPV-Initial Investment). The expected values are obtained by summing up the probable outcomes of the cash flows. For instance in the year 1 expected cash flows will be computed as follow;

EV year 1= (0.2*50)+ (0.3*40) + (0.4*30) + (0.1*20)= 10 + 12 + 12 + 2= 36

After the expected values have been obtained, the values are discounted for each year. For instance the discounted value for year 1 will be \$ 33.33 million (36/ (1,081). The present values are then summed up to provide the ENPV of the project as illustrated in the table below;

Year 0

Year 1

Year 2

Year 3

Total

Probability

Cash Flow(\$ mil.)

Probability

Cash Flow(\$mil.)

Probability

Cash Flow(\$ mil.)

(\$mil)

0.2

50

0.1

60

0.3

70

0.3

40

0.2

50

0.4

60

0.4

30

0.3

40

0.1

50

0.1

20

0.4

30

0.2

40

Expected Value

36

40

58

Discount Factor @ 8%

0.9259

0.8573

0.7938

Present value (Expected Value × Discount Factor)

33.33

34.29

46.04

113.66

Initial Investment\$80 mil.

(80)

ENPV

33.66

The ENPV of Project A is \$ 33.66 million given a discount factor of 8%. The NPV of \$ 33.66 million is positive indicating that Project A is worth the investment.

The standard deviation will be obtained by computing the square root of the variances in each year and then summing them after they have been discounted.

Year 1

Probability

Cash Flow(\$ mil.)

Variance

( (cash flow-expected value(36) )2 × Probability)

0.2

50

(50-36)2 × (0.2)= 39.20

0.3

40

4.8

0.4

30

14.40

0.1

20

25.6

Total

84

Standard Deviation (√Variance)

9.17

Discount @ 8%

0.9259

Present Value of Standard Deviation

8.49

Year 2

Probability

Cash Flow(\$ mil.)

Variance

( (cash flow-expected value(40) )2 × Probability)

0.1

60

(60-40)2 × (0.1)= 40

0.2

50

20

0.3

40

0

0.4

30

40

Total

100

Standard Deviation (√variance)

10

Discount @ 8%

0.8573

Present Value of Standard Deviation

8.57

Year 3

Probability

Cash Flow(\$ mil.)

Variance

( (cash flow-expected value(58) )2 × Probability)

0.3

70

(70-58)2 × (0.3)= 43.2

0.4

60

1.6

0.1

50

6.4

0.2

40

64.8

Total

116

Standard Deviation (√variance)

10.77

Discount @ 8%

0.7938

Present Value of Standard Deviation

8.55

The standard deviation for project A is obtained by adding the standard deviation from year 1 to year 3 to obtain \$ 25.61 (8.49 + 8.57 + 8.55)

Project A has an ENPV of \$ 33.66 million and a standard deviation of \$. 25.61. The decision rule in terms of ENPV is that the project needs to have a positive ENPV to be implemented. An ENPV of zero is indifferent while a project with a negative ENPV should be rejected (Kayshap, 2014). San Diego LLC needs to implement project A since it will provide net positive cash flows.

Part B

Project A is preferable since it has a higher of \$.33.66 million compared to project B which has a lower ENPV of \$. 32 million. The preference for project A is only based on the ENPV value ignoring risk and assuming similar discount rates were used in the ENPV computation.

Part C

The coefficient of variation and the standard deviation are measures that indicate the likelihood of the actual values deviating from the estimated values. Higher coefficients of variation and standard deviation indicates increased chances of the actual values differing from the estimates. In terms of risk, the higher the coefficient and the higher the deviation, the higher the risk of the investment.

When risk is incorporated, Project B should be selected over project A since it has a lower standard deviation hence lower risk. Project B only has a standard deviation of \$ 10.5 million while project B has a larger standard deviation of \$. 25.61 million. In an extreme loss scenario, Project A will only provide an ENPV of \$8.05 million (33.66-25.61) compared to Project B’s \$21.5 million (32-10.5).

Reference

Kashyap, A. (2014). Capital Allocating Decisions: Time Value of Money. Asian Journal of Management, 5(1), 106-110.

March 15, 2023
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