Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
A property is expected to earn net rent of €12 000 next year. The rent is expected to grow by 1% per year forever. The required return is 6% per year. Find the value PV of the property today.
A loan of €15 000 is repaid with equal monthly payments over 6 years. The interest rate is 3.6% per year, compounded monthly. Find the monthly payment A.
A machine costs €50 000 today. It generates €8 500 at the end of each year for 6 years and can then be sold for €10 000. The cost of capital is 10%. Find the net present value NPV.
Project A costs €20 000 and returns €11 500, €8 000, €4 500 at the end of years 1 to 3. Project B costs €25 000 and returns €6 500, €10 500, €14 000. Find the crossover rate r* at which both projects have the same NPV, in percent.
A bond with a face value of €1 000 pays a 5% annual coupon in two half-yearly payments and matures in 6 years. The yield to maturity is 3% per year. Find the price P of the bond.
A bond with a face value of €1 000 pays an 8% annual coupon in two half-yearly payments and matures in 7 years. It sells for €1 113. Find its yield to maturity YTM as an annual rate (twice the half-yearly yield), in percent.
A company is financed with ordinary shares worth €8 000 000, preferred shares worth €2 000 000 and debt worth €4 000 000. The cost of equity is 14%, the cost of preferred stock 6.5% and the pre-tax cost of debt 4.5%. The tax rate is 21%. Find the WACC.
A company has 5 000 000 shares at €28.00 each and 20 000 bonds trading at €1 000 each. The pre-tax cost of debt is 3.5% and the tax rate is 30%. The share’s beta is 1.3, the risk-free rate 2.5% and the market return 7.5%. Find the WACC.
A company sells 3 000 units at €12 each. The variable cost is €8 per unit and fixed costs are €10 000. Find the degree of operating leverage DOL at this sales level.
A company sells 750 units at €30 each, with a variable cost of €15 per unit and fixed costs of €7 500. By what percentage %ΔΠ does operating profit change if sales volume rises by 20%?
Answer: PV = 240000 €
Time value of money: money today is worth more than the same amount later, because it can earn interest. With interest rate i per period, an amount grows as FV = PV·(1 + i)n. A series of equal payments (an annuity) is valued by adding up each payment, which gives the annuity formulas.
Answer: A = 232 €
Time value of money: money today is worth more than the same amount later, because it can earn interest. With interest rate i per period, an amount grows as FV = PV·(1 + i)n. A series of equal payments (an annuity) is valued by adding up each payment, which gives the annuity formulas.
The present value of N equal payments A is A·(1 − (1 + i)⁻ᴺ)/i. Set it equal to the loan and solve for A.
Answer: NPV = −7336 €
Net present value: discount every cash flow back to today with the required rate of return r, then add them up: NPV = Σ CFₜ/(1 + r)ᵗ. A project with NPV > 0 earns more than r and adds value. The internal rate of return (IRR) is the rate at which NPV = 0.
Answer: r* = 8.269 %
Crossover rate: the discount rate at which two projects have the same NPV. It is the IRR of the difference in their cash flows (B − A). Below it the project with the later, larger cash flows has the higher NPV; above it the other one wins.
Check: at 8.269% both NPVs are €992.18.
Answer: P = 1109 €
Bond pricing: a bond’s price is the present value of its cash flows, discounted at the yield to maturity (YTM): the coupons form an annuity and the face value is paid back at maturity. P = C·(1 − (1 + y)⁻ⁿ)/y + F/(1 + y)ⁿ. When the yield is above the coupon rate the bond trades below face value, and the other way round.
The yield is below the coupon rate, so the bond sells at a premium.
Answer: YTM = 5.999 %
Bond pricing: a bond’s price is the present value of its cash flows, discounted at the yield to maturity (YTM): the coupons form an annuity and the face value is paid back at maturity. P = C·(1 − (1 + y)⁻ⁿ)/y + F/(1 + y)ⁿ. When the yield is above the coupon rate the bond trades below face value, and the other way round.
Answer: WACC = 9.944 %
Cost of capital: CAPM gives the return shareholders require: re = rf + β·(rm − rf). The weighted average cost of capital mixes equity and debt by their market values: WACC = E/V·re + D/V·rd·(1 − T), with V = E + D. Interest is tax-deductible, hence the (1 − T).
Answer: WACC = 8.181 %
Cost of capital: CAPM gives the return shareholders require: re = rf + β·(rm − rf). The weighted average cost of capital mixes equity and debt by their market values: WACC = E/V·re + D/V·rd·(1 − T), with V = E + D. Interest is tax-deductible, hence the (1 − T).
Answer: DOL = 6
Operating leverage: fixed costs make profit react more strongly than sales. The degree of operating leverage DOL = contribution margin/operating profit tells you that a 1% change in sales changes profit by DOL %. It is very high just above break-even and falls as sales grow.
Answer: %ΔΠ = 60 %
Operating leverage: fixed costs make profit react more strongly than sales. The degree of operating leverage DOL = contribution margin/operating profit tells you that a 1% change in sales changes profit by DOL %. It is very high just above break-even and falls as sales grow.
Check: new profit = 900·15 − 7 500 = €6 000.00.
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