Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
An annuity pays €5 000 at the end of each year for 11 years. The interest rate is 6% per year. Find the present value PV of the annuity.
A mortgage of €250 000 is repaid with equal monthly payments over 25 years at 7.2% per year, compounded monthly. How much is still owed just after the payment at the end of year 15? Find the outstanding balance B.
An investment of €30 000 gives these cash flows at the end of years 1 to 3: €7 000, €8 000, €13 000. The cost of capital is 9%. Find the net present value NPV.
A project costs €10 000 and returns €4 000, €5 500, €4 500 at the end of years 1 to 3. Cash flows can be reinvested at the cost of capital of 8%. Find the modified internal rate of return MIRR 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 2.5% 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 €1 055. Find its yield to maturity YTM in percent.
A company has equity worth €2 000 000 and debt worth €6 000 000. Its cost of equity is 10%, its pre-tax cost of debt is 3% and the tax rate is 28%. Find the WACC.
A company has 5 000 000 shares at €41.00 each and 20 000 bonds trading at €1 050 each. The pre-tax cost of debt is 3.5% and the tax rate is 28%. The share’s beta is 1.3, the risk-free rate 3.5% and the market return 10.5%. Find the WACC.
A company sells 1 500 units at €20 each. The variable cost is €8 per unit and fixed costs are €12 000. Find the degree of operating leverage DOL at this sales level.
A product sells for €20 per unit, with a variable cost of €12 per unit and fixed costs of €4 000 per year. Budgeted sales are 800 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 39434 €
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 = 153572 €
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 = −6806 €
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: MIRR = 14.74 %
Modified IRR: compound every inflow forward to the end of the project at the reinvestment rate (usually the cost of capital) to get the terminal value TV. MIRR is the rate that turns the initial investment into TV: MIRR = (TV/C₀)^(1/n) − 1.
Answer: P = 1361 €
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 below the coupon rate, so the bond sells at a premium.
Answer: YTM = 5.945 %
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 = 4.12 %
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 = 11.66 %
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 = 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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