Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
An annuity pays €500 at the end of each year for 15 years. The interest rate is 6% per year. Find the present value PV of the annuity.
A loan of €250 000 is repaid with equal monthly payments over 20 years. The interest rate is 7.2% per year, compounded monthly. Find the monthly payment A.
A machine costs €20 000 today, lasts 4 years and costs €1 500 per year to run (paid at the end of each year). The cost of capital is 8%. Find its equivalent annual cost EAC.
A project costs €40 000 and returns €30 000, €22 500 at the end of years 1 to 2. Find its internal rate of return IRR in percent.
You buy a bond with 7 years to maturity (face value €1 000, annual coupon 7%) when its yield to maturity is 5.5%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 6.5%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 and an annual coupon of 8% matures in 5 years and sells for €960. Find its yield to maturity YTM in percent.
A company has equity worth €5 000 000 and debt worth €4 000 000. Its cost of equity is 8%, its pre-tax cost of debt is 4.5% and the tax rate is 28%. Find the WACC.
A company has 5 000 000 shares trading at €36.00; the book value of its equity is €54 000 000. It just paid a dividend of €2.90 per share, and dividends grow by 2% per year. Its debt has a market value of €108 000 000 and a pre-tax cost of 3%. The tax rate is 28%. Find the WACC, using the dividend growth model for the cost of equity.
A product sells for €23 per unit with a variable cost of €15 per unit. Fixed costs are €20 000 per month. How many units Q must be sold per month to make a profit of €10 000?
A company sells 5 000 units per month at €19 each, with a variable cost of €15 per unit and fixed costs of €10 000 per month. The variable cost rises by €2 per unit and the price stays the same. How many units Q must now be sold per month to keep the same profit?
Answer: PV = 4856 €
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 = 1968 €
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 present value of N equal payments A is A·(1 − (1 + i)⁻ᴺ)/i. Set it equal to the loan and solve for A.
Answer: EAC = 7538 €
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: IRR = 21.35 %
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: HPR = 0.8257 %
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 = 9.029 %
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 = 5.884 %
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 = 7.195 %
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 = 3750 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: Q = 10000 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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