Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €8 000 at a nominal rate of 2% per year, compounded monthly. Find the value FV of the investment after 8 years.
You save €250 at the end of every month for 22 years. The account pays 2.4% per year, compounded monthly. How much have you saved at the end? Find FV.
A project costs €20 000 today and returns €8 000, €6 000, €4 000, €4 500, €5 000, €8 000 at the end of years 1 to 6. The cost of capital is 10%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.
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 €6 000, €9 500, €12 000. Find the crossover rate r* at which both projects have the same NPV, in percent.
A bond with a face value of €1 000 matures in 13 years and pays annual coupons. It sells for €646 when the yield to maturity is 6%. Find the coupon rate c in percent.
A bond with 10 years to maturity (face value €1 000, annual coupon 7%) sells for €1 154. The issuer can call it in 5 years at a call price of €1 070. Find the yield to call YTC in percent.
A company has equity worth €2 000 000 and debt worth €4 000 000. Its cost of equity is 10%, its pre-tax cost of debt is 5% and the tax rate is 20%. Find the WACC.
A company has 1 000 000 shares at €47.00 each and 10 000 bonds trading at €960 each. The pre-tax cost of debt is 7% and the tax rate is 20%. The share’s beta is 1.2, the risk-free rate 2% and the market return 8.5%. Find the WACC.
A company sells 12 000 units at €26 each. The variable cost is €18 per unit and fixed costs are €32 000. Find the degree of operating leverage DOL at this sales level.
A product sells for €16 per unit, with a variable cost of €4 per unit and fixed costs of €9 600 per year. Budgeted sales are 960 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 9387 €
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 = 86831 €
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 €66 000 yourself; the rest is interest.
Answer: DPB = 4.544 years
Payback period: the time until the cumulative cash flows repay the initial investment. Cash is assumed to come in evenly during each year, so the last part is a fraction of a year. The simple payback ignores the time value of money; the discounted payback period uses the discounted cash flows instead.
The discounted payback is always longer than the simple payback (3.444 years).
Answer: r* = 9.059 %
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 9.059% both NPVs are €740.11.
Answer: c = 2.001 %
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 = 4.742 %
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 5.005%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: WACC = 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 = 9.088 %
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 = 16.67 %
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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