Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €40 000 in 2 years. An account pays 3% per year, compounded monthly. How much must you deposit today? Find the present value PV.
You save €150 at the end of every month for 7 years. The account pays 2.4% per year, compounded monthly. How much have you saved at the end? Find FV.
A project costs €250 000 today. It generates €11 000 at the end of the first year, and the cash flow then grows by 4% per year forever. The cost of capital is 9%. Find the net present value NPV.
Project A costs €20 000 and returns €10 500, €8 000, €4 500 at the end of years 1 to 3. Project B costs €22 000 and returns €5 500, €8 500, €11 500. Find the crossover rate r* at which both projects have the same NPV, in percent.
A bond with a face value of €1 000 pays a 6% annual coupon in two half-yearly payments and matures in 6 years. The yield to maturity is 3.5% per year. Find the price P of the bond.
A bond trades at €980 with a yield to maturity of 7% (annual compounding) and a Macaulay duration of 7.3 years. Use the modified duration to estimate the new price P if the yield rises by 100 basis points.
A company has equity worth €5 000 000 and debt worth €1 000 000. Its cost of equity is 10%, its pre-tax cost of debt is 6% and the tax rate is 25.8%. Find the WACC.
A company has 5 000 000 shares at €21.50 each and 40 000 bonds trading at €960 each. The pre-tax cost of debt is 5% and the tax rate is 21%. The share’s beta is 1.2, the risk-free rate 2% and the market return 8.5%. Find the WACC.
A company sells 3 000 units at €11 each. The variable cost is €6 per unit and fixed costs are €5 000. 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 €6 400 per year. Budgeted sales are 1 440 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 37673 €
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: FV = 13705 €
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 €12 600 yourself; the rest is interest.
Answer: NPV = −30000 €
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: r* = 3.155 %
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 3.155% both NPVs are €1 796.67.
Answer: P = 1134 €
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: P = 913.1 €
Duration: the Macaulay duration is the weighted average time at which a bond pays its cash flows, with the present values of the cash flows as weights: D = Σ t·PVₜ/P. The modified duration D/(1 + y) estimates the percentage price change for a small change in yield: %ΔP ≈ −Dmod·Δy.
Duration is a linear estimate; for large yield changes convexity makes the true price a little higher.
Answer: WACC = 9.075 %
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 = 8.26 %
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 = 1.5
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 = 44.44 %
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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