Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €50 000 in 6 years. An account pays 4.8% per year, compounded quarterly. How much must you deposit today? Find the present value PV.
A loan of €20 000 is repaid with equal monthly payments over 5 years. The interest rate is 4.8% per year, compounded monthly. Find the monthly payment A.
An investment of €45 000 gives these cash flows at the end of years 1 to 3: €18 000, €14 000, €9 000. The cost of capital is 5%. Find the net present value NPV.
A project costs €40 000 and returns €29 000, €16 500, €16 000 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 matures in 8 years and pays annual coupons. It sells for €1 194 when the yield to maturity is 5%. Find the coupon rate c in percent.
A bond with a face value of €1 000 and an annual coupon of 4% matures in 5 years and sells for €1 070. Find its yield to maturity YTM in percent.
A company has equity worth €5 000 000 and debt worth €2 000 000. Its WACC is 10.43%, its pre-tax cost of debt is 5% and the tax rate is 20%. Find the cost of equity re.
A company's business has an unlevered (asset) beta of 0.6. It plans a debt-to-equity ratio of 1.5. The tax rate is 25%, the risk-free rate is 3.5% and the market risk premium is 6%. Find the cost of equity re at the new capital structure.
A product sells for €26 with a variable cost of €18 per unit. Fixed costs are €40 000 per month and profit is taxed at 20%. How many units Q must be sold per month to earn €20 000 after tax?
A company sells two products. Product A sells for €14 with a variable cost of €8; product B sells for €37 with a variable cost of €25. They sell in a fixed mix of 3 units of A for every 2 of B. Fixed costs are €8 400 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: PV = 37552 €
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 = 375.6 €
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 = −7384 €
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: IRR = 28.7 %
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: c = 8.002 %
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 = 2.494 %
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: re = 13 %
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 = 11.15 %
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 = 8125 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 = 1000 units
Break-even with a sales mix: when products sell in a fixed ratio, use the weighted average contribution margin: weight each product's p − v by its share of units. Break-even units = F/(weighted contribution margin), split again in the mix ratio.
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