Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €2 000 at a nominal rate of 3.6% per year, compounded monthly. Find the value FV of the investment after 3 years.
You borrow €8 000 at 3% per year, compounded monthly, and repay €150 at the end of every month. How many months does it take until the loan is repaid (the last payment may be smaller)? Give the number of payments N.
A machine costs €30 000 today, lasts 9 years and costs €4 000 per year to run (paid at the end of each year). The cost of capital is 12%. Find its equivalent annual cost EAC.
A project costs €100 000 today. Its cash flow is €13 000 at the end of year 1 and grows by 4% per year until the last cash flow at the end of year 7. The cost of capital is 10%. Find the net present value NPV.
A bond with a face value of €1 000 pays a 4.5% annual coupon in two half-yearly payments and matures in 3 years. The yield to maturity is 2.5% per year. Find the price P of the bond.
A bond with a face value of €1 000 pays a 3% annual coupon in two half-yearly payments and matures in 4 years. It sells for €963. Find its yield to maturity YTM as an annual rate (twice the half-yearly yield), in percent.
A company has equity worth €8 000 000 and debt worth €1 000 000. Its cost of equity is 8%, its pre-tax cost of debt is 4% and the tax rate is 21%. Find the WACC.
A company has 4 000 000 shares trading at €54.50; the book value of its equity is €65 400 000. It just paid a dividend of €2.90 per share, and dividends grow by 2% per year. Its debt has a market value of €109 000 000 and a pre-tax cost of 3%. The tax rate is 21%. Find the WACC, using the dividend growth model for the cost of equity.
A company sells 5 000 units at €28 each. The variable cost is €8 per unit and fixed costs are €40 000. Find the degree of operating leverage DOL at this sales level.
A product sells for €9 per unit, with a variable cost of €4 per unit and fixed costs of €5 000 per year. Budgeted sales are 1 600 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 2228 €
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: N = 58 payments
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.
57 full payments leave a small balance, which the last, smaller payment clears.
Answer: EAC = 9630 €
Equivalent annual cost: spread the present value of all costs over the life as an equal yearly amount (an annuity): EAC = PV·r/(1 − (1 + r)⁻ⁿ). It lets you compare machines with different lives.
Answer: NPV = −29644 €
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: P = 1057 €
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: YTM = 4.01 %
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 = 7.462 %
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 = 5.742 %
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 = 1.667
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 = 37.5 %
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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