Solutions to Chapter 12 The Cost of Capital The yield to maturity on the bonds (since maturity is now 19 years) is the interest rate that solves the following equation: 90 annuity factor(r, 19 years) + 1000/(1 + r) 19 = 1050 The solution can be obtained most easily from a financial calculator: Set n = 19, FV = 1000, PV = (-)1050, PMT = 90 Compute the interest rate as 8.46% The after-tax cost of debt is therefore 8.46% (1 – 35) = 5.50% r = DIV/P0 = $4/$40 = 10 = 10% WACC = rdebt (1 – Tc) + rpreferred + requity = 8.46% (1 – 35) + 10% + 12.5% = 9.90% r = DIV1/P0 + g = + g = + 05 = 1375 = 13.75% The total value of the firm is $80 million The weights for each security class are: Debt: Preferred: Common: D/V = 20/80 = 250 P/V = 10/80 = 125 E/V = 50/80 = 625 WACC = rdebt (1 – Tc) + rpreferred + requity = 25 8% (1 – 35) + 125 10% + 625 15% = 11.925% Executive Fruit should use the WACC of Geothermal, not its own WACC, when evaluating an investment in geothermal power production The risk of the project determines the discount rate, and in this case, Geothermal’s WACC is more reflective of the risk of the project in question The proper discount rate, therefore, is not 12.3% It is more likely to be 11.4% The flotation costs reduce the NPV of the project by $1.2 million Then new NPV is $2.5 million - $1.2 million, or $1.3 million Even so, project NPV is still positive, so the project should be undertaken 121 Copyright © 2006 McGraw-Hill Ryerson Limited The rate on Buildwell’s debt is percent The cost of equity capital is the required rate of return on equity, which can be calculated from the CAPM as 4% + 80 8% = 10.4% The weighted average cost of capital, with a tax rate of zero, is WACC = rdebt + requity = 30 (1 – 0) 5% + 70 10.4% = 8.78% IRR, which is 12%, exceeds the cost of capital Therefore, BCCI should accept the project The present value of the project cash flows is $100,000 annuity factor(8.78%, years) = $507,032 This is the most BCCI should pay for the project 10 Security Market value Explanation Debt Equity $ 5.5 million $15.0 million 1.10 par value of $5 million $30 per share 500,000 shares* Total $20.5 million *Number of shares = = 500,000 WACC = rdebt + requity = (1 – 4) 9% + 15% = 12.42% 11 Because the firm is all-equity financed, asset beta = equity beta = The WACC is the same as the cost of equity which may be calculated using the CAPM: requity = rf + (rm – rf) = 5% + × 10% = 13% 12 The 12.5% value calculated by the analyst is the current yield of the firm’s outstanding debt: interest payments/bond value This calculation neglects the fact that bonds selling at discounts from or premiums over par value provide expected returns determined in part by expected price appreciation or depreciation The analyst should be using yield to maturity instead of current yield to calculate cost of debt [This answer assumes the value of the debt provided is the market value If it is the book value, then 12.5% would be the average coupon rate of outstanding debt, which also would be a poor estimate of the required rate of 122 Copyright © 2006 McGraw-Hill Ryerson Limited return on the firm’s debt.] Furthermore, if the analyst were interested in the firm’s after-tax cost of debt, he or she would use (1 - t) × 12.5% 13 a Using the recent growth rate of 30% and the dividend yield of 2%, one estimate would be: DIV1/P0 + g = 02 + 30 = 32 = 32% In this calculation, we’ve assumed that the current dividend yield is the next expected dividend divided by the current price, DIV 1/P0 However, if the dividend yield was the most recent past dividend, DIV0/P0, then with 30% dividend growth, DIV1/P0 would be 02 × 1.3 = 026 and the estimated required rate of return would be 026 + = 326, or 32.6% Another estimate, based on the CAPM, would be r = rf + (rm – rf) = 4% + 1.2 8% = 13.6% 14 b The estimate of 32% seems far less reasonable It is based on an historic growth rate that is impossible to sustain No company can grow at 30% forever The [DIV1/P0 + g] rule requires that the growth rate of dividends per share must be viewed as highly stable over the foreseeable future In other words, it requires us to use the sustainable growth rate a The 9% coupon bond has a yield to maturity of 10% and sells for 93.86% of face value: n = 10, i = 10%, PMT = 90, FV = 1000, compute PV = $938.55 The market value of the issue is therefore 9386 $20 million = $18.77 million The 10% coupon bond sells for 92.8% of par value, and has a yield to maturity of 11.0%: n = 15, PV = ()928, PMT = 100, FV = 1000, compute i = 11.00% The market value of the issue is 928 $25 million = $23.20 million The weighted average before-tax cost of debt is therefore 10% + 11% = 10.55% 123 Copyright © 2006 McGraw-Hill Ryerson Limited b 15 The after-tax cost of debt is (1 – 35) 10.55% = 6.86% The bonds must be selling below par value, because the YTM is greater than the coupon rate The price per $1000 par value is 80 annuity factor(9%, 10 years) + 1000/1.0910 = $935.82 The total market value of the bonds is $10 million par value = $9.36 million Book value of the preferred shares is $2 million and the par value per share is $20 Thus there are 100,000 shares of preferred stock (=$2 million/$20 per share) Preferred shares are selling at $15 per share, for total market value of $1.5 million The market value of million common shares selling at $20/share is $20 million The book value of the common shares is sum of the common stock plus retained earnings, which also happens to equal $20 million Therefore, the market value capital structure is: Bonds Preferred Stock Common Stock Total 16 Dollars 9.36 million 1.50 million 20.00 million 30.86 million Percent 30.3% 4.9% 64.8% 100.0% The yield to maturity on debt is r debt = 9% The rate on preferred stock is rpreferred = $2/$15 = 133 = 13.3% 124 Copyright © 2006 McGraw-Hill Ryerson Limited The rate on common stock is requity = rf + (rm – rf) = 4% + 1.5 7% = 14.5% Using the capital structure derived in the previous problem, we can calculate WACC as: WACC = rdebt + requity + rpreferred = 303 (1 – 4) 9% + 648 14.5% + 049 13.3% = 11.68% 17 The IRR on the computer project is less than the WACC of firms in the computer industry Therefore, the project should be rejected However, the WACC of the firm (based on its existing mix of projects) is only 11.68% If the firm uses this figure as the hurdle rate, it will incorrectly go ahead with the venture in home computers The discount rate for a project is determined by the risk of the project University Product's WACC is irrelevant to the analysis of the investment in the computer project 18 a r = rf + (rm – rf) = 4% + 1.5 7% = 14.5% b Total market value of Muskoka Real Estate is $6 million and the market value of the debt is $2 million Thus the market value of its equity is $6 - $2, or $4 million The current capital structure is 1/3 debt, 2/3 equity Weighted average beta = + 1.5 = 1.0 c WACC = rdebt + requity = (1 – 4) 4% + 14.5% = 10.47% d If the company wishes to expand its present business then the WACC is a reasonable estimate of the discount rate since the risk of the proposed project is similar to the risk of the existing projects Use a discount rate of 10.47% e The WACC of optical projects should be based on the risk of those projects Using a beta of 1.2, the discount rate for the new venture is r = + 1.2 = 12.4% 19 a Equity Market value = 10 million shares × $15/share = $150 million rE = rf + rf + (rm – rf) = 2.5% + 1.2×6.5% = 10.3% Debt Market value per bond = semi-annual coupon payment × PVIFA(6-month r B, no of payments) 125 Copyright © 2006 McGraw-Hill Ryerson Limited + face value × PVIF(6-month r B, no of periods to maturity) rB = required rate of return on 10-year Gov't debt + 95 basis points Required rate of return on 10-year Gov't debt = (1 + 04/2) - =.0404 rB = 0404 + 0095 = 0.0499 = 4.99% 6-month required rate of return = (1.0499)1/2 - = 0246 Face value = 1000 Coupon payment= 06/2 × 1000 = 30 No of payments = payments/year × 10 years = 20 Market value per bond = 30 PVIFA(.0246, 20) + 1000 × PVIF(.0246, 20) = $1,084.5 Market value of all bonds = 20,000 bonds × $1,084.5 = $21,690,000 WACC Calculation Market Value Market Weight Debt $ 21,690,000 126 Before-tax Required Rate of Return 4.99% Equity Total $150,000,000 $171,690,000 874 10.3% b After-tax Required Rate of Return Weight × After-tax Return (1-.35)×4.99% = 3.24% 10.3% 408% 9.002% 9.41% U = To find debt , use the CAPM: rdebt = rf + debt (rm – rf) debt = = = 383 U = = = 1.13 As expected, the unlevered beta is lower than the levered equity beta With no debt in the capital structure, the equity is less risky c Relever the equity beta to reflect the new capital structure of 50% debt: levered = U + [U - debt] × D/E × (1 - TC) = 1.13 + [1.13 - 383] × 5/.5 × (1 - 35) = 1.62 As expected, moving from a debt/equity ratio of 126/.874, or about 144 to 5/.5, or 1, increases the riskiness of the equity The levered equity beta increases from 1.2 to 1.62 The new required rate of return to equity is: rE = rf + (rm – rf) = 2.5% + 1.62ì6.5% = 13.03% 12ư6 Copyright â 2006 McGraw-Hill Ryerson Limited The new WACC is: WACC = rdebt × (1 - TC) + requity = 4.99% (1 - 35) + 13.03% = 8.14% 20 a The annual cash flows expected from Premier Pizza are: Revenues - operating costs - capital expenditures Percent of Revenues Revenues Operating costs Capital expenditures Annual after-tax cash flows 75 05 Annual Cash Flow $10 million 75 × 10 million = 7.5 million 05 × 10 million = 0.5 million $2 million The annual cash flows from Premier Pizza are a constant growing perpetuity Using the perpetuity formula, the present value of the cash flows is: PV(annual cash flows) = annual cash flow required rate of return - growth rate Required rate of return on cash flows: WACC = rdebt × (1 - TC) + requity = × 06 × (1-.35) + × 15 = 1167 PV(annual cash flows) = $2 million = $26.0756 million 1167 - 04 If Boris and Isabelle offer $26.0756 million for Premier Pizza, the NPV of their investment will be zero The offer is the investment in the firm: NPV = - investment + PV(future cash flows from firm) = -26.0756 million - 26.0756 million = The project will earn the required rate of return, just enough return to compensate all investors This is the maximum amount they should offer If they can purchase Premier Pizza for less than $26.0756 million, the investment will earn more than the required rate of return, with the bulk of the extra return going to the equity investors in the project b Fresh Foods current WACC rEQUITY = rf + (rm – rf) = 03 + × 07 = 086 rDEBT = 05 D/V = 4, E/V = WACC = rdebt × (1 - TC) + requity = × 05 × (1 - 35) + × 086 = 0646 127 Copyright © 2006 McGraw-Hill Ryerson Limited Using Fresh Food's current WACC as the required rate of return on Premier Pizza gives a valuation of: PV(annual cash flows) = $2 million = $81.3 million 0646 - 04 This says that Premier Pizza is worth over times more owned by Fresh Foods than by Boris and Isabelle, even though both groups expect the same cash flows! The difference comes solely from the different discount rates Question: What is the appropriate discount rate for Fresh Foods to use in valuing Premier Pizza? If you use Fresh Foods' current WACC, you are assuming that riskiness of the pizza manufacturing business is the same as the riskiness of the grocery retail business Why should it necessarily be the case? Without further investigation, the better assumption would be that the appropriate discount rate for Fresh Foods to use for Premier Pizza's cash flows is 11.67%, the rate used by Boris and Isabella We know that Boris and Isabelle investigated the risks of the business before determining the appropriate discount rate Under this assumption, the maximum Fresh Foods should be willing to pay is also $26.0756 million 21 If Big Oil does not pay taxes, the after-tax and before-tax costs of debt are identical WACC would then become: WACC = rdebt + requity = 243 9% + 757 13.5% = 12.41% If Big Oil issues new equity and uses the proceeds to pay off all of its debt, the cost of equity will fall There is no longer any leverage, so the equity becomes safer and commands a lower risk premium In fact, with all-equity financing, the cost of equity would be the same as the firm’s WACC, which is 12.41%, lower than the previous value of 13.5% (We use the WACC derived in the absence of interest tax shields since, for the all-equity firm, there is no interest tax shield.) 22 The net effect of Big Oil’s transaction is to leave the firm with $200 million more debt (because of the borrowing) and $200 million less equity (because of the dividend payout) Total assets and business risk are unaffected The WACC will remain unaffected, since business risk is unchanged However, the cost of equity will rise With the now higher leverage, the business risk is spread over a smaller equity base, so each share is now riskier The new financing mix for the firm would be E = 1,000 and D = 585.7 Therefore, = = 369 and = = 631 128 Copyright © 2006 McGraw-Hill Ryerson Limited If the cost of debt is still 9%, then we can solve for the new cost of equity as follows We use the fact that, even at the new financing mix, WACC must still be 12.41% WACC = rdebt + requity = 369 9% + 631 requity = 12.41% We solve to find that requity = 14.40% 23 Even if the WACC were lower when the firm’s tax rate is higher, this does not imply that the firm would be worth more The after-tax cash flows that the firm would generate for its owners also would be lower and this would reduce the value of the firm, even if those cash flows were discounted at a lower rate If the tax authority is collecting more income from the firm, the value of the firm will fall 24 This reasoning is faulty in that it implicitly treats the discount rate for the project as the cost of debt if the project is debt financed, and as the cost of equity if the project is equity financed In fact, if the project poses risk comparable to the risk of the firm’s other projects, the proper discount rate is the firm’s cost of capital, which is a weighted average of the costs of both debt and equity 25 Internet: Calculating WACC of Canadian Companies Expected results: This question gives students enough information to get a rough estimate of companies' current WACC 26 Standard & Poor's: Calculate WACC of U.S Companies Tips: The free corporate bond credit spread data has been removed from Bonds Online Instead, they provide an example of credit spreads, dated June 30, 2004 Tell your students to assume that these credit spreads are still valid The new web address for spreads is: http://www.bondsonline.com/Search_Quote Center/Corporate_Agency_Bonds/Spreads/ Expected results: Similar to Problem 25, this question gives students the opportunity to estimate WACC for companies 129 Copyright © 2006 McGraw-Hill Ryerson Limited Solution to Minicase for Chapter 12 Bernice needs to explain to her boss, Mr Brinestone, that appropriate rates of return for cost of capital calculations are the rates of return that investors can earn on comparable risk investments in the capital market Mr Brinestone’s estimate of the cost of equity is his target value for the book return on equity; it is not the expected rate of return that investors demand on shares of stock with the same risk as Sea Shore Salt Bernice’s CAPM calculation indicates that the correct value for the equity rate is 11% This value is broadly consistent with the rate one would infer from the constant growth dividend discount model (which seems appropriate for a mature firm like this one with stable growth prospects) The dividend discount model implies a cost of equity of a bit more than 11 percent: requity = + g = + 067 = 117 = 11.7% This value is close to Bernice's value It is not surprising that the different methods for estimating the cost of equity yield different values However, it is reassuring that the values are similar For the rest of the solution we will use 11% as the cost of equity Mr Brinestone’s returns for other securities should be modified to reflect the expected returns these securities currently offer to investors The bank loan and bond issue offer pre-tax rates of 7.75% and 8%, respectively, as in Mr Brinestone’s memo It is acceptable to use the interest rates on the loan and the debt as the required rates of return given the information provided We are told the "bank charged interest at the current rate", which implies that the bank loan rate is a floating rate and hence changes when market rates change We are told that the bonds were just issued The convention is to set the bond coupon rate at the bond's required rate of return so that the bond will be sold at par value The preferred stock, however, is not selling at par, so Mr Brinestone’s assertion that the rate of return on preferred is 6% is incorrect In fact, with the preferred selling at $70 per share, the rate of return is rpreferred = = = 086 = 8.6% This makes sense: the pre-tax return on preferred should exceed that on the firm’s debt Finally, the weights used to calculate the WACC should reflect market, not book, values These are the prices that investors would pay to acquire the securities The market value weights are computed as follows: Amount Percent Rate of Comment (millions) of total return (%) Bank loan valued at face amount $120 17.91 7.75 Bond issue valued at par 80 11.94 8.0 Pfd stock $70 million shares 70 10.45 8.6 Common stock $40 10 million shares 400 59.70 11.0 $670 100.00 1210 Copyright © 2006 McGraw-Hill Ryerson Limited Therefore, the WACC, which serves as the corporate hurdle rate, should be 9%: WACC = 1791 7.75% (1 – 35) + 1194 8% (1 – 35) + 1045 8.6% + 5970 11% = 8.99% 1211 Copyright © 2006 McGraw-Hill Ryerson Limited ... Ryerson Limited Solution to Minicase for Chapter 12 Bernice needs to explain to her boss, Mr Brinestone, that appropriate rates of return for cost of capital calculations are the rates of return that... all-equity financed, asset beta = equity beta = The WACC is the same as the cost of equity which may be calculated using the CAPM: requity = rf + (rm – rf) = 5% + × 10% = 13% 12 The 12. 5% value... instead of current yield to calculate cost of debt [This answer assumes the value of the debt provided is the market value If it is the book value, then 12. 5% would be the average coupon rate of outstanding