Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €12 000 at a nominal rate of 6% per year, compounded monthly. Find the value FV of the investment after 2 years.
A loan of €15 000 is repaid with equal monthly payments over 3 years at 5.4% per year, compounded monthly. How much interest I is paid in total over the life of the loan?
A machine costs €20 000 today, lasts 9 years and costs €3 000 per year to run (paid at the end of each year). The cost of capital is 10%. Find its equivalent annual cost EAC.
A project costs €25 000 and returns €18 000, €16 500, €14 500 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 pays a 6% annual coupon in two half-yearly payments and matures in 6 years. The yield to maturity is 7% per year. Find the price P of the bond.
A bond with a face value of €1 000 and an annual coupon of 4% matures in 9 years and sells for €835. Find its yield to maturity YTM in percent.
A company has a debt-to-equity ratio (at market values) of 0.6. Its cost of equity is 13%, its pre-tax cost of debt is 6% and the tax rate is 25.8%. Find the WACC.
A company's business has an unlevered (asset) beta of 1. It plans a debt-to-equity ratio of 0.75. The tax rate is 28%, the risk-free rate is 1% and the market risk premium is 6%. Find the cost of equity re at the new capital structure.
A product sells for €20 with a variable cost of €18 per unit. Fixed costs are €4 000 per month and profit is taxed at 20%. How many units Q must be sold per month to earn €1 000 after tax?
A company sells 7 500 units per month at €45 each, with a variable cost of €25 per unit and fixed costs of €100 000 per month. The variable cost rises by €4 per unit and the price stays the same. How many units Q must now be sold per month to keep the same profit?
Answer: FV = 13526 €
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: I = 1281 €
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: EAC = 6473 €
Equivalent annual cost: spread the present value of all costs over the life as an equal yearly amount (an annuity): EAC = PV·r/(1 − (1 + r)⁻ⁿ). It lets you compare machines with different lives.
Answer: IRR = 45.06 %
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.
NPV falls as the rate rises, so the IRR lies between a rate with positive NPV and one with negative NPV.
Linear interpolation gives a first estimate; repeating it (or a calculator's IRR function) gives the exact rate.
Answer: P = 951.7 €
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 above the coupon rate, so the bond sells at a discount.
Answer: YTM = 6.476 %
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 = 9.794 %
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: re = 10.24 %
Levered beta (Hamada): debt makes equity riskier. βL = βU·(1 + (1 − T)·D/E), where βU is the beta of the business without debt. To move a beta to another capital structure, unlever it first and then relever it at the new D/E.
Answer: Q = 2625 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: Q = 9375 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).
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