Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €20 000 in 6 years. An account pays 4% per year, compounded monthly. How much must you deposit today? Find the present value PV.
You want €20 000 in 4 years. You save the same amount at the end of every month in an account paying 4.8% per year, compounded monthly. Find the monthly deposit A.
A project costs €250 000 today. It generates €25 000 at the end of the first year, and the cash flow then grows by 1% per year forever. The cost of capital is 12%. Find the net present value NPV.
A project costs €50 000 and returns €18 500, €32 500, €37 000 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 2% annual coupon in two half-yearly payments and matures in 9 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 and an annual coupon of 6% matures in 9 years and sells for €1 070. Find its yield to maturity YTM in percent.
A company has equity worth €3 000 000 and debt worth €1 000 000. Its cost of equity is 9%, its pre-tax cost of debt is 4% and the tax rate is 28%. Find the WACC.
A company has equity with a market value of €12 000 000 and a cost of equity of 11%. Its debt consists of 8 000 bonds (face value €1 000, annual coupon 6%, 5 years to maturity) that trade at €1 021 each. The tax rate is 30%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
A product can be made in two ways. Method A has fixed costs of €20 000 per year and a variable cost of €18 per unit. Method B, more automated, has fixed costs of €32 000 per year and a variable cost of €12 per unit. At what yearly volume Q do both methods cost the same?
A company sells two products. Product A sells for €16 with a variable cost of €10; product B sells for €45 with a variable cost of €25. They sell in a fixed mix of 4 units of A for every 1 of B. Fixed costs are €22 000 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: PV = 15739 €
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 = 378.8 €
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: NPV = −22727 €
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: IRR = 30.38 %
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 = 959.9 €
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.
The yield is above the coupon rate, so the bond sells at a discount.
Answer: YTM = 5.015 %
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 = 7.47 %
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 = 8.107 %
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 = 2000 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).
Below this volume method A is cheaper; above it the automated method B is cheaper.
Answer: Q = 2500 units
Break-even with a sales mix: when products sell in a fixed ratio, use the weighted average contribution margin: weight each product's p − v by its share of units. Break-even units = F/(weighted contribution margin), split again in the mix ratio.
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