Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €50 000 in 3 years. An account pays 3% per year, compounded monthly. How much must you deposit today? Find the present value PV.
You want €20 000 in 6 years. You save the same amount at the end of every month in an account paying 4.8% per year, compounded monthly. Find the monthly deposit A.
A machine costs €40 000 today. It generates €11 500 at the end of each year for 4 years and can then be sold for €4 000. The cost of capital is 12%. Find the net present value NPV.
Project A costs €20 000 and returns €11 500, €8 000, €4 500 at the end of years 1 to 3. Project B costs €22 000 and returns €7 000, €8 500, €11 500. Find the crossover rate r* at which both projects have the same NPV, in percent.
You buy a bond with 11 years to maturity (face value €1 000, annual coupon 4%) when its yield to maturity is 9%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 10%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 and an annual coupon of 5% 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 €6 000 000. Its cost of equity is 8%, its pre-tax cost of debt is 4.5% and the tax rate is 20%. Find the WACC.
A company has 5 000 000 shares at €27.00 each and 20 000 bonds trading at €960 each. The pre-tax cost of debt is 7% and the tax rate is 25%. The share’s beta is 1.1, the risk-free rate 2% and the market return 7%. Find the WACC.
A company sells 3 000 units at €26 each. The variable cost is €18 per unit and fixed costs are €12 000. Find the degree of operating leverage DOL at this sales level.
A product sells for €33 per unit, with a variable cost of €30 per unit and fixed costs of €6 000 per year. Budgeted sales are 2 300 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 45702 €
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 = 240.2 €
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: NPV = −2528 €
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.
Answer: r* = 6.579 %
Crossover rate: the discount rate at which two projects have the same NPV. It is the IRR of the difference in their cash flows (B − A). Below it the project with the later, larger cash flows has the higher NPV; above it the other one wins.
Check: at 6.579% both NPVs are €1 549.92.
Answer: HPR = 1.756 %
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 = 3.452 %
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 = 5.6 %
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 = 7.22 %
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 = 2
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 = 13.04 %
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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