Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €12 000 at a nominal rate of 6% per year, compounded monthly. Find the value FV of the investment after 4 years.
You deposit €3 000 at the start of every year for 20 years, the first deposit today. The account pays 4% per year. How much is in the account at the end of year 20? Find FV.
A project costs €20 000 today and returns €8 500, €4 500, €8 000, €8 000, €6 000, €5 500 at the end of years 1 to 6. The cost of capital is 12%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.
A project costs €50 000 and returns €28 000, €30 500, €19 500 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 pays a 4.5% annual coupon in two half-yearly payments and matures in 4 years. The yield to maturity is 2.5% per year. Find the price P of the bond.
A bond with 10 years to maturity (face value €1 000, annual coupon 8%) sells for €1 108. The issuer can call it in 3 years at a call price of €1 080. Find the yield to call YTC in percent.
A company has equity worth €10 000 000 and debt worth €3 000 000. Its cost of equity is 14%, its pre-tax cost of debt is 3.5% and the tax rate is 25%. Find the WACC.
A company has equity with a market value of €15 000 000 and a cost of equity of 11%. Its debt consists of 4 000 bonds (face value €1 000, annual coupon 6%, 6 years to maturity) that trade at €1 025 each. The tax rate is 21%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
A company sells 10 000 units at €33 each. The variable cost is €30 per unit and fixed costs are €15 000. Find the degree of operating leverage DOL at this sales level.
A product sells for €14 per unit, with a variable cost of €12 per unit and fixed costs of €1 000 per year. Budgeted sales are 750 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 15246 €
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 = 92908 €
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.
Each deposit is invested one year longer.
Answer: DPB = 3.615 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 (2.875 years).
Answer: IRR = 27.69 %
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 falls as the rate rises, so the IRR lies between a rate with positive NPV and one with negative NPV.
Linear interpolation gives a first estimate; repeating it (or a calculator's IRR function) gives the exact rate.
Answer: P = 1076 €
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: YTC = 6.43 %
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 6.498%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: WACC = 11.38 %
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 = 9.571 %
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 = 33.33 %
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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