Question

Superfast Bikes is thinking of developing a new composite road bike. Development will take six years and the cost is $ 210 800 per year. Once in production, the bike is expected to make $ 286 072 per year for 10 years. The cash inflows begin at the end of year 7. Assuming the cost of capital is 9.7 %​: a. Calculate the NPV of this investment opportunity. Should the company make the​ investment? b. Calculate the IRR and use it to determine the maximum deviation allowable in the cost of capital estimate to leave the decision unchanged. c. How long must development last to change the​ decision? Assume the cost of capital is 13.8 %. d. Calculate the NPV of this investment opportunity. Should the company make the​ investment? e. How much must this cost of capital estimate deviate to change the​ decision? f. How long must development last to change the​ decision?

Answer 1

Since the horizon of the project is long, we should resort to excel. Presented below is the snapshot my model. The cells containing the answers are shown in yellow color. Adjacent to each cell continuing answer, i have also provided the formula used in that cell, so that your understand the mathematics and can do this yourself.

Part (a) NPV = $ 95,532.28 as shown in the yellow colored cells.

Since NPV is positive, the firm should take this investment

Part (b) IRR = 11.15%

The maximum deviation allowable in the cost of capital estimate to leave the decision unchanged = IRR - Cost of capital = 11.15% - 9.70% = 1.45%. So the allowed deviation in cost of capital is +1.45%

Part (c)

Let's say the development lasts for N years. Let R be the cost of capital.

The entire cash flows can be split into two series:

1. The first one being annuity of the cost of development per annum (C_o) over N years. PV of all outflows at t = 0, will be = C_o / R x [1 - (1 + R)^-N] = 210,800 / 9.70% x [1 - (1 + 9.70%)^-N] =  2,173,195.88 x (1 - 1.097^-N)
2. The second one being the annuity of the cash inflows per annum, C_i starting at the end of N+1 and lasting for 10 years. PV of these annuities at the end of year N = C_i / R x [1 - (1 + R)^-10] = 286,072 / 9.70% x [1 - (1 + 9.70%)^-10] = $1,780,672.67. This is the PV at the end of year N. PV of this at t = 0 will be =  1,780,672.67 / (1 + R)^N =  1,780,672.67 x (1 + 9.70%)^-N =  1,780,672.67 x 1.097^-N

Now, for decision to change, NPV ≤ 0

Or, PV of inflows - PV of outflows ≤ 0

Or, 1,780,672.67 x 1.097^-N - 2,173,195.88 x (1 - 1.097^-N) ≤ 0

Or,  3,953,868.54 x 1.097^-N ≤  2,173,195.88

Or, 1.097^-N ≤  2,173,195.88 / 3,953,868.54 = 0.549637868

Take log to the base "e" on both sides:

-N x ln (1.097) ≤ ln (0.549637868)

Hence, N ≥ - ln (0.549637868) / ln (1.097) =  6.46

Hence, N should be 7 years, that is, if development lasts for 7 years, it will lead to change the​ decision.

Part (d)

Cost of capital is now 13.8%. So, we have to rework the entire exercise at this rate of 13.8%. Please see the screenshot below.

NPV = - $ 131,824.94

Since, NPV is negative, the firm should not take this investment.

Part (e)

The maximum deviation allowable in the cost of capital estimate to leave the decision unchanged = IRR - Cost of capital = 11.15% - 13.80% = -2.65%. So the allowed deviation in cost of capital is -2.65%

Part (f)

Let's say the development lasts for N years. Let R be the cost of capital.

The entire cash flows can be split into two series:

1. The first one being annuity of the cost of development per annum (C_o) over N years. PV of all outflows at t = 0, will be = C_o / R x [1 - (1 + R)^-N] = 210,800 / 13.80% x [1 - (1 + 13.80%)^-N] =   1,527,536.23 x (1 - 1.138^-N)
2. The second one being the annuity of the cash inflows per annum, C_i starting at the end of N+1 and lasting for 10 years. PV of these annuities at the end of year N = C_i / R x [1 - (1 + R)^-10] = 286,072 / 13.80% x [1 - (1 + 13.80%)^-10] = $ 1,503,905.08 . This is the PV at the end of year N. PV of this at t = 0 will be =   1,503,905.08 / (1 + R)^N =   1,503,905.08 x (1 + 13.80%)^-N =  1,503,905.08 x 1.138^-N

Now, for decision to change, NPV ≥ 0

Or, PV of inflows - PV of outflows ≥ 0

Or, 1,503,905.08 x 1.138^-N - 1,527,536.23 x (1 - 1.138^-N) ≥ 0

Or,   3,031,441.32 x 1.138^-N ≥   1,527,536.23  

Or, 1.138^-N  ≥  1,527,536.23 / 3,031,441.32 = 0.503897675

Take log to the base "e" on both sides:

-N x ln (1.138) ≥ ln (0.503897675)

Hence, N ≤ -ln (0.503897675) / ln (1.138) = 5.300

Hence, N should be 5 years or lesser, that is, if development lasts for 5 years, it will lead to change the​ decision.