Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
An annuity pays €3 000 at the end of each year for 13 years. The interest rate is 5% per year. Find the present value PV of the annuity.
A mortgage of €150 000 is repaid with equal monthly payments over 25 years at 2.4% per year, compounded monthly. How much is still owed just after the payment at the end of year 5? Find the outstanding balance B.
A project costs €400 000 today. It generates €34 000 at the end of the first year, and the cash flow then grows by 3% per year forever. The cost of capital is 10%. Find the net present value NPV.
A project costs €50 000 and returns €21 000, €32 500 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 matures in 6 years and pays annual coupons. It sells for €908 when the yield to maturity is 8%. Find the coupon rate c in percent.
A bond with 12 years to maturity (face value €1 000, annual coupon 6%) sells for €1 089. The issuer can call it in 3 years at a call price of €1 060. Find the yield to call YTC in percent.
A share with a beta of 1.4 has a required return of 10.2%. The risk-free rate is 2.5%. What expected market return rm does CAPM imply?
A company has 5 000 000 shares at €57.50 each and 20 000 bonds trading at €1 000 each. The pre-tax cost of debt is 3% and the tax rate is 30%. The share’s beta is 1.5, the risk-free rate 1.5% and the market return 5.5%. Find the WACC.
A company sells 1 200 units at €18 each. The variable cost is €12 per unit and fixed costs are €4 800. Find the degree of operating leverage DOL at this sales level.
A product sells for €23 per unit, with a variable cost of €20 per unit and fixed costs of €1 500 per year. Budgeted sales are 750 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 28181 €
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: B = 126731 €
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.
Whatever is still owed must be exactly paid off by the remaining payments.
Answer: NPV = 85714 €
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 = 4.313 %
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 = 6.01 %
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: YTC = 4.662 %
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 to maturity is 4.996%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: rm = 8 %
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 = 7.149 %
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 = 33.33 %
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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