Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
A credit card charges a nominal interest rate of 21% per year, compounded quarterly. Find the effective annual rate EAR in percent.
You save €500 at the end of every month for 24 years. The account pays 6% per year, compounded monthly. How much have you saved at the end? Find FV.
A machine costs €100 000 today. It generates €20 500 at the end of each year for 7 years and can then be sold for €15 000. The cost of capital is 8%. Find the net present value NPV.
A project costs €40 000 and returns €22 000, €26 500 at the end of years 1 to 2. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 pays a 6% annual coupon in two half-yearly payments and matures in 7 years. The yield to maturity is 3.5% 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 8 years. It sells for €663. Find its yield to maturity YTM as an annual rate (twice the half-yearly yield), in percent.
A company has equity worth €3 000 000 and debt worth €6 000 000. Its cost of equity is 12%, its pre-tax cost of debt is 7% and the tax rate is 20%. Find the WACC.
A company has 2 000 000 shares trading at €23.00; the book value of its equity is €13 800 000. It just paid a dividend of €2.00 per share, and dividends grow by 4% per year. Its debt has a market value of €27 600 000 and a pre-tax cost of 5%. The tax rate is 20%. Find the WACC, using the dividend growth model for the cost of equity.
A company sells 6 250 units at €28 each. The variable cost is €8 per unit and fixed costs are €100 000. Find the degree of operating leverage DOL at this sales level.
A company sells 10 000 units per month at €8 each, with a variable cost of €6 per unit and fixed costs of €10 000 per month. The variable cost rises by €1 per unit and the price stays the same. How many units Q must now be sold per month to keep the same profit?
Answer: EAR = 22.71 %
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.
The EAR is always above the nominal rate when interest compounds more than once a year.
Answer: FV = 320558 €
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 €144 000 yourself; the rest is interest.
Answer: NPV = 15483 €
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.
Answer: IRR = 13.41 %
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 = 1154 €
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 = 9 %
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 = 7.733 %
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 = 9.652 %
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 = 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: Q = 20000 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).
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