Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
A credit card charges a nominal interest rate of 15% per year, compounded monthly. Find the effective annual rate EAR in percent.
You borrow €15 000 at 4.2% per year, compounded monthly, and repay €675 at the end of every month. How many months does it take until the loan is repaid (the last payment may be smaller)? Give the number of payments N.
An investment of €45 000 gives these cash flows at the end of years 1 to 3: €11 000, €16 000, €19 000. The cost of capital is 15%. Find the net present value NPV.
A project costs €50 000 and returns €30 500, €24 000 at the end of years 1 to 2. 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 12 years. The yield to maturity is 8% per year. Find the price P of the bond.
A bond with a face value of €1 000 and an annual coupon of 8% matures in 3 years and sells for €1 025. Find its yield to maturity YTM in percent.
A company has equity worth €3 000 000 and debt worth €1 000 000. Its cost of equity is 11%, its pre-tax cost of debt is 7% and the tax rate is 21%. Find the WACC.
A company's business has an unlevered (asset) beta of 0.9. It plans a debt-to-equity ratio of 0.75. The tax rate is 25.8%, the risk-free rate is 1.5% and the market risk premium is 6.5%. Find the cost of equity re at the new capital structure.
A product sells for €31 per unit with a variable cost of €25 per unit. Fixed costs are €24 000 per month. How many units Q must be sold per month to make a profit of €6 000?
A company sells two products. Product A sells for €12 with a variable cost of €8; product B sells for €45 with a variable cost of €25. They sell in a fixed mix of 2 units of A for every 1 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: EAR = 16.08 %
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 EAR is always above the nominal rate when interest compounds more than once a year.
Answer: N = 24 payments
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.
23 full payments leave a small balance, which the last, smaller payment clears.
Answer: NPV = −10844 €
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 = 6.198 %
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 = 847.5 €
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 = 7.047 %
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 = 9.633 %
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 = 10.61 %
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 = 5000 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 = 900 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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