Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
An annuity pays €1 000 at the end of each year for 11 years. The interest rate is 7% per year. Find the present value PV of the annuity.
A mortgage of €250 000 is repaid with equal monthly payments over 25 years at 2.4% per year, compounded monthly. How much is still owed just after the payment at the end of year 2? Find the outstanding balance B.
A machine costs €50 000 today. It generates €18 500 at the end of each year for 3 years and can then be sold for €5 000. The cost of capital is 6%. Find the net present value NPV.
Project A costs €10 000 and returns €6 000, €4 000, €2 000 at the end of years 1 to 3. Project B costs €15 000 and returns €4 000, €6 000, €8 000. Find the crossover rate r* at which both projects have the same NPV, in percent.
A bond with a face value of €1 000 matures in 7 years and pays annual coupons. It sells for €1 240 when the yield to maturity is 4%. Find the coupon rate c in percent.
A bond with a face value of €1 000 and an annual coupon of 7% matures in 5 years and sells for €1 040. Find its yield to maturity YTM in percent.
A company has equity worth €12 000 000 and debt worth €1 000 000. Its cost of equity is 12%, its pre-tax cost of debt is 7% and the tax rate is 25%. Find the WACC.
A company has 2 500 000 shares at €22.50 each and 20 000 bonds trading at €1 000 each. The pre-tax cost of debt is 7% and the tax rate is 20%. The share’s beta is 1.5, the risk-free rate 2% and the market return 7.5%. Find the WACC.
A product sells for €35 with a variable cost of €30 per unit. Fixed costs are €7 500 per month and profit is taxed at 20%. How many units Q must be sold per month to earn €4 000 after tax?
A company sells 8 000 units at €19 each, with a variable cost of €15 per unit and fixed costs of €20 000. By what percentage %ΔΠ does operating profit change if sales volume rises by 15%?
Answer: PV = 7499 €
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: B = 235043 €
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.
Whatever is still owed must be exactly paid off by the remaining payments.
Answer: NPV = 3649 €
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* = 5.567 %
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 5.567% both NPVs are €972.77.
Answer: c = 7.999 %
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: YTM = 6.049 %
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 = 11.48 %
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 = 9.03 %
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: Q = 2500 units
Cost-volume-profit: each unit sold earns a contribution margin, price minus variable cost per unit. Profit = Q·(p − v) − F, where F is the fixed cost. At the break-even point profit is zero, so Q = F/(p − v).
Answer: %ΔΠ = 40 %
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 = 9 200·4 − 20 000 = €16 800.00.
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