## Solutions

1. We can calculate the present value of the cash inflows in several ways. We can discount each cash inflow separately at the required rate of return of 12 percent and then sum the present values. We can also find the present value of a four-year annuity of C\$3 million, add to it the present value of the t = 5 cash flow of C\$10 million, and subtract the t = 0 outflow of C\$13 million. Or we can compute the present value of a five-year annuity of C\$3 million, add to it the present value of a cash inflow of C\$7 million = C\$10 million — C\$3 million dated t = 5, and subtract the t = 0 outflow of C\$13 million. For this last approach, we illustrate the keystrokes for many financial calculators.

Notation Used on Most Calculators

PV compute

PMT FV

Numerical Value for This Problem

3,000,000 7,000,000

We find that the PV of the inflows is C\$14,786,317.

A. Therefore, NPV = C\$14,786,317 - C\$13,000,000 = C\$1,786,317.

B. Waldrup should undertake this project because it has a positive NPV.

A. The internal rate of return is the discount rate that makes the NPV equal zero:

To show that the investment project has an IRR of 13.51 percent we need to show that its NPV equals zero, ignoring rounding errors. Substituting the project's cash flows and IRR = 0.1351 into the above equation,

Given that the cash flows were of the magnitude of millions, the amount C\$734 differs negligibly from 0. B. The internal rate of return is unaffected by any change in any external rate, including the increase in Waldrup's opportunity cost of capital.

3. A. NPV is the sum of the present values of all the cash flows associated with the investment, where inflows are signed positive and outflows are signed negative. This problem has only one outflow, an initial expenditure of \$10 million at t = 0. The projected cash inflows from this advertising project form a perpetuity. We calculate the present value of a perpetuity as CF/r, where CF is the level annual cash flow and r is the discount rate. Using the opportunity cost of capital of 12.5 percent as the discount rate, we have

NPV = -\$10,000,000 + 1,600,000/0.125 = -\$10,000,000 + 12,800,000 = \$2,800,000

B. In this case, the cash inflows are a perpetuity. Therefore, we can solve for the internal rate of return as follows:

Initial investment = (Annual cash inflow)/IRR 10,000,000 = 1,600,000/IRR IRR = 16 percent

C. Yes, Bestfoods should spend \$10 million on advertising. The NPV of \$2.8 million is positive. The IRR of 16 percent is also in excess of the 12.5 percent opportunity cost of capital.

4. Using the IRR function in a spreadsheet or an IRR-enabled financial calculator, we enter the individual cash flows and apply the IRR function. We illustrate how we can solve for IRR in this particular problem using a financial calculator without a dedicated IRR function. The cash flows from t = 1 through t = 6 can be treated as a six-year, \$4 million annuity with \$7 million - \$4 million = \$3 million, entered as a future amount at t = 6.

Notation Used on Most Calculators

%i compute

Numerical Value for This Problem

-15,000,000 4,000,000 3,000,000

A. The IRR of the project is 18.25 percent.

B. Because the project's IRR is less than the hurdle rate of 19 percent, the company should not undertake the project.

5. A. Company A. Let CF = £300,000 be the amount of the perpetuity. Then with r = 0.12, the NPV in acquiring Company A would be

NPV = CF0 + CF/r = -£2,000,000 + £300,000/0.12 = £500,000

Company B.LetCF = £435,000 be the amount ofthe perpetuity. Then with r = 0.12, the NPV in acquiring Company B would be

NPV = CF0 + CF/r = -£3,000,000 + £435,000/0.12 = £625,000

Both Company A and Company B would be positive NPV acquisitions, but West-cott—Smith cannot purchase both because the total purchase price of £5 million exceeds its budgeted amount of £4 million. Because Company B's NPV of £625,000 is higher than Company A's NPV of £500,000, Westcott-Smith should purchase Company B according to the NPV rule.

B. Company A. Using the notation from Part A, IRR is defined by the expression NPV = -Investment + CF/IRR = 0. Thus -£2,000,000 + £300,000/IRR = 0 and solving for IRR,

Company B.

IRR = £435,000/£3,000,000 = 0.145 or 14.5 percent

Both Company A and Company B have IRRs that exceed Westcott-Smith's opportunity cost of 12 percent, but Westcott-Smith cannot purchase both because of its budget constraint. According to the IRR rule, Westcott-Smith should purchase Company A because its IRR of 15 percent is higher than Company B's IRR of 14.5 percent.

C. Westcott-Smith should purchase Company B. When the NPV and IRR rules conflict in ranking mutually exclusive investments, we should follow the NPV rule because it directly relates to shareholder wealth maximization.

6. A. The money-weighted rate of return is the discount rate that equates the present value of inflows to the present value of outflows. Outflows:

150 shares purchased x \$156.30 per share = \$23,445

Inflows:

150 shares x \$10 dividend per share = \$1,500 100 shares sold x \$165 per share = \$16,500

50 shares remaining x \$15 dividend per share = \$750 50 shares sold x \$170 per share = \$8,500

PV(Inflows)

The last line is the equation for calculating the money-weighted rate of return on Wilson's portfolio.

B. We can solve for the money-weighted return by entering -23,445, 18,000, and 9,250 in a spreadsheet or calculator with an IRR function. In this case, we can also solve for money-weighted rate of return as the real root of the quadratic equation 18,000x + 9,250x2 — 23,445 = 0, wherex = 1/(1 + r). Byanymethod, thesolution is r = 0.120017 or approximately 12 percent.

C. The time-weighted rate of return is the solution to (1 + Time-weighted rate of return)2 = (1 + r1)(1 + r2), where r1 and r2 are the holding period returns in the first and second years, respectively. The value of the portfolio at t = 0 is \$23,445. At t = 1, there are inflows of sale proceeds of \$16,500 and \$1,500 in dividends, or \$18,000 in total. The balance of 50 shares is worth \$8,250 = 50 shares x \$165 per share. So at t = 1 the valuation is \$26,250 = \$18,000 + \$8,250. Thus r1 = (\$26,250 — \$23,445)/\$23,445 = 0.119642 for the first year

The amount invested at t = 1 is \$8,250 = (50 shares)(\$165 per share). At t = 2, \$750 in dividends are received, as well as sale proceeds of \$8,500 (50 shares sold x \$170 per share). So at t = 2, the valuation is \$9,250 = \$750 + \$8,500. Thus r2 = (\$9,250 — \$8,250)/\$8,250 = 0.121212 for the second year

Time-weighted rate of return = s/( 1.119642)(1.121212) — 1 = 0.1204 or approximately 12 percent.

D. If Wilson is a private investor with full discretionary control over the timing and amount ofwithdrawals and additions to his portfolios, then the money-weighted rate of return is an appropriate measure of portfolio returns.

E. If Wilson is an investment manager whose clients exercise discretionary control over the timing and amount of withdrawals and additions to the portfolio, then the time-weighted rate of return is the appropriate measure of portfolio returns. Time-weighted rate of return is standard in the investment management industry.

7. Similarities. The time-weighted returns for Luongo's and Weaver's investments will be equal, because the time-weighted return is not sensitive to additions or withdrawals of funds. Even though Weaver purchased another share at €110, the return earned by Luongo and Weaver each year for the time-weighted return calculation is the same.

PV(Outflows) =

Differences. The money-weighted returns for Luongo and Weaver will differ because they take into account the timing of additions and withdrawals. During the two-year period, Weaver owned more shares of the stock during the year that it did poorly (the stock return for Year 1 is (110 + 5 — 100)/100 = 15 percent and for Year 2 it is (100 + 5 — 110)/110 = —4.55 percent). As a consequence, the money-weighted return for Weaver (1.63 percent) is less than that of Luongo (5.00 percent). The money-weighted return reflects the timing of additions and withdrawals. Note, the cash flows for the money-weighted returns for Luongo and Weaver are (for t = 0, 1, and 2) Luongo: — 100, +5, +105; Weaver: —100, —105, +210.

8. In this solution, F stands for face value, P stands for price, and D stands for the discount from face value (D = F — P).

A. Use the discount yield formula (Equation 2-3), r^D = D/F x 360/t:

rgD = (\$1,500/\$100,000) x (360/120) = 0.0150 x 3 = 0.045

The T-bill's bank discount yield is 4.5 percent a year.

B. Use your answer from Part A and the money market yield formula (Equation 2-6), rMM = (360 x rBD)/(360 — t x rBD):

rMM = (360 x 0.045)/(360 — 120 x 0.045) = 0.04568

The T-bill's bank discount yield is 4.57 percent a year.

C. Calculate the holding period yield (using Equation 2-4), then compound it forward to one year. First, the holding period yield (HPY) is

HPY = ———-- = (100,000 - 98,500)/98,500 = 0.015228

Next, compound the 120-day holding period yield, a periodic rate, forward to one year using Equation 2-5:

Effective annual yield = (1.015228)365/120 — 1 = 0.047044

The T-bill's effective annual yield is 4.7 percent a year.

9. A. In the United States, T-bill yields are quoted on a bank discount basis. The bank discount yield is not a meaningful measure of the return for three reasons: First, the yield is based on the face value of the bond, not on its purchase price. Returns from investments should be evaluated relative to the amount that is invested. Second, the yield is annualized based on a 360-day year rather than a 365-day year. Third, the bank discount yield annualizes with simple interest, which ignores the opportunity to earn interest on interest (compound interest). B. The money market yield is superior to the bank discount yield because the money market yield is computed relative to the purchase price (not the face value).

C. The T-bill yield can be restated on a money market basis by multiplying the bank discount yield by the ratio of the face value to the purchase price. Cavell could divide the annualized yield by 4 to compute the 90-day holding period yield. This is a more meaningful measure of the return that she will actually earn over 90 days (assuming that she holds the T-bill until it matures).