Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €5 000 in 9 years. An account pays 3.6% per year, compounded monthly. How much must you deposit today? Find the present value PV.
You save €200 at the end of every month for 29 years. The account pays 3.6% per year, compounded monthly. How much have you saved at the end? Find FV.
A machine costs €100 000 today. It generates €31 000 at the end of each year for 4 years and can then be sold for €10 000. The cost of capital is 12%. Find the net present value NPV.
A project costs €200 000 today. Its cash flow is €48 000 at the end of year 1 and grows by 3% per year until the last cash flow at the end of year 5. The cost of capital is 12%. Find the net present value NPV.
You buy a bond with 3 years to maturity (face value €1 000, annual coupon 5%) when its yield to maturity is 5.5%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 4.5%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 and an annual coupon of 6% matures in 8 years and sells for €910. Find its yield to maturity YTM in percent.
A company has a debt-to-equity ratio (at market values) of 0.25. Its cost of equity is 8%, its pre-tax cost of debt is 4% and the tax rate is 25.8%. Find the WACC.
A company's shares have a beta of 1.2 at its current debt-to-equity ratio of 0.2. It plans to raise its debt-to-equity ratio to 1. The tax rate is 20%, the risk-free rate 2.5% and the market risk premium 4.5%. Find the cost of equity re after the change.
A product can be made in two ways. Method A has fixed costs of €30 000 per year and a variable cost of €25 per unit. Method B, more automated, has fixed costs of €38 000 per year and a variable cost of €23 per unit. At what yearly volume Q do both methods cost the same?
A product sells for €11 per unit, with a variable cost of €6 per unit and fixed costs of €5 000 per year. Budgeted sales are 1 400 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 3618 €
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: FV = 122408 €
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.
You paid in €69 600 yourself; the rest is interest.
Answer: NPV = 513 €
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: NPV = −17495 €
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 is negative, so the project earns less than the required return.
Answer: HPR = 7.385 %
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 = 7.539 %
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 = 6.994 %
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 = 10.88 %
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 = 4000 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).
Below this volume method A is cheaper; above it the automated method B is cheaper.
Answer: MS = 28.57 %
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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