Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €1 000 at 6% per year with continuous compounding. Find the value FV after 6 years.
You want €75 000 in 13 years. You save the same amount at the end of every month in an account paying 3.6% per year, compounded monthly. Find the monthly deposit A.
An investment of €75 000 gives these cash flows at the end of years 1 to 4: €34 000, €17 000, €16 000, €28 000. The cost of capital is 5%. Find the net present value NPV.
A project costs €50 000 today. Its cash flow is €7 500 at the end of year 1 and grows by 5% per year until the last cash flow at the end of year 6. The cost of capital is 10%. Find the net present value NPV.
You buy a bond with 6 years to maturity (face value €1 000, annual coupon 6%) when its yield to maturity is 5%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 6%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 and an annual coupon of 7% matures in 5 years and sells for €920. Find its yield to maturity YTM in percent.
A company is financed with ordinary shares worth €6 000 000, preferred shares worth €500 000 and debt worth €5 000 000. The cost of equity is 14%, the cost of preferred stock 7.5% and the pre-tax cost of debt 5%. The tax rate is 28%. Find the WACC.
A company's business has an unlevered (asset) beta of 1.1. It plans a debt-to-equity ratio of 0.6. The tax rate is 21%, the risk-free rate is 1% and the market risk premium is 4.5%. Find the cost of equity re at the new capital structure.
A product sells for €24 with a variable cost of €4 per unit. Fixed costs are €10 000 per month and profit is taxed at 20%. How many units Q must be sold per month to earn €4 000 after tax?
A product sells for €10 per unit, with a variable cost of €8 per unit and fixed costs of €5 000 per year. Budgeted sales are 4 000 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 1433 €
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 = 377.7 €
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: NPV = 9658 €
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 positive, so the project earns more than the required return.
Answer: NPV = −13467 €
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 = 0.8797 %
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 = 9.06 %
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 = 9.196 %
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 = 8.296 %
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 = 750 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: MS = 37.5 %
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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