Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
An annuity pays €1 000 at the end of each year for 9 years. The interest rate is 5% per year. Find the present value PV of the annuity.
You want €20 000 in 12 years. You save the same amount at the end of every month in an account paying 3% per year, compounded monthly. Find the monthly deposit A.
An investment of €20 000 gives these cash flows at the end of years 1 to 3: €7 000, €6 000, €5 000. The cost of capital is 7.000000000000001%. Find the net present value NPV.
Project A costs €10 000 and returns €6 000, €3 500, €2 500 at the end of years 1 to 3. Project B costs €15 000 and returns €5 000, €6 000, €7 500. Find the crossover rate r* at which both projects have the same NPV, in percent.
You buy a bond with 7 years to maturity (face value €1 000, annual coupon 5%) when its yield to maturity is 2.5%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 3.5%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 and an annual coupon of 3% matures in 8 years and sells for €935. Find its yield to maturity YTM in percent.
A company has a debt-to-equity ratio (at market values) of 0.25. Its cost of equity is 9%, its pre-tax cost of debt is 5% and the tax rate is 28%. Find the WACC.
A company has 5 000 000 shares at €44.00 each and 10 000 bonds trading at €1 000 each. The pre-tax cost of debt is 3.5% and the tax rate is 21%. The share’s beta is 0.8, the risk-free rate 2.5% and the market return 7.5%. Find the WACC.
A product sells for €40 per unit with a variable cost of €20 per unit. Fixed costs are €40 000 per month. How many units Q must be sold per month to make a profit of €10 000?
A product sells for €55 per unit, with a variable cost of €45 per unit and fixed costs of €10 000 per year. Budgeted sales are 1 300 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 7108 €
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 = 115.6 €
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 = −4136 €
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* = 9.352 %
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 9.352% both NPVs are €325.75.
Answer: HPR = −2.486 %
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 = 3.964 %
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 = 7.92 %
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 = 6.338 %
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: Q = 2500 units
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).
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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