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 20% per year, compounded monthly. Find the effective annual rate EAR in percent.
You save €200 at the end of every month for 27 years. The account pays 4.8% per year, compounded monthly. How much have you saved at the end? Find FV.
A project costs €250 000 today. It generates €24 000 at the end of the first year, and the cash flow then grows by 4% per year forever. The cost of capital is 12%. Find the net present value NPV.
A project costs €25 000 and returns €12 000, €17 000 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 an annual coupon of 6% and matures in 12 years. Its yield to maturity changes from 5% to 4.5%. By how much ΔP does the bond price change?
A bond with a face value of €1 000 and an annual coupon of 7% matures in 6 years and sells for €1 100. Find its yield to maturity YTM in percent.
A company has a debt-to-equity ratio (at market values) of 0.5. Its cost of equity is 13%, its pre-tax cost of debt is 3% and the tax rate is 25%. Find the WACC.
A company has 1 000 000 shares at €38.00 each and 5 000 bonds trading at €980 each. The pre-tax cost of debt is 6% and the tax rate is 21%. The share’s beta is 0.8, the risk-free rate 4% and the market return 11%. Find the WACC.
A company sells 2 250 units at €14 each. The variable cost is €4 per unit and fixed costs are €15 000. Find the degree of operating leverage DOL at this sales level.
A product sells for €34 per unit, with a variable cost of €30 per unit and fixed costs of €6 000 per year. Budgeted sales are 1 950 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: EAR = 21.94 %
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: FV = 132261 €
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 €64 800 yourself; the rest is interest.
Answer: NPV = 50000 €
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: IRR = 9.884 %
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: ΔP = 48.15 €
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 = 5.028 %
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 = 9.417 %
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: WACC = 9.045 %
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: DOL = 3
Operating leverage: fixed costs make profit react more strongly than sales. The degree of operating leverage DOL = contribution margin/operating profit tells you that a 1% change in sales changes profit by DOL %. It is very high just above break-even and falls as sales grow.
Answer: MS = 23.08 %
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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