Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €10 000 at 7% per year with continuous compounding. Find the value FV after 12 years.
You save €150 at the end of every month for 20 years. The account pays 3% per year, compounded monthly. How much have you saved at the end? Find FV.
An investment of €75 000 gives these cash flows at the end of years 1 to 3: €26 000, €25 000, €31 000. The cost of capital is 6%. Find the net present value NPV.
A project costs €25 000 and returns €9 500, €13 000, €9 000 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 an annual coupon of 4.5% and matures in 12 years. Its yield to maturity changes from 4% to 2.5%. By how much ΔP does the bond price change?
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. It sells for €1 076. Find its yield to maturity YTM as an annual rate (twice the half-yearly yield), in percent.
A company is financed with ordinary shares worth €4 000 000, preferred shares worth €2 000 000 and debt worth €3 000 000. The cost of equity is 13%, the cost of preferred stock 7% and the pre-tax cost of debt 4%. The tax rate is 20%. Find the WACC.
A company has 5 000 000 shares trading at €37.00; the book value of its equity is €92 500 000. It just paid a dividend of €3.00 per share, and dividends grow by 5% per year. Its debt has a market value of €185 000 000 and a pre-tax cost of 4.5%. The tax rate is 20%. Find the WACC, using the dividend growth model for the cost of equity.
A product sells for €26 per unit with a variable cost of €18 per unit. Fixed costs are €40 000 per month. How many units Q must be sold per month to make a profit of €6 000?
A company sells two products. Product A sells for €15 with a variable cost of €10; product B sells for €22 with a variable cost of €10. They sell in a fixed mix of 4 units of A for every 1 of B. Fixed costs are €12 800 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: FV = 23164 €
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 = 49245 €
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 €36 000 yourself; the rest is interest.
Answer: NPV = −2194 €
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 = 12.59 %
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: ΔP = 158.2 €
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 = 2.492 %
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 = 8.4 %
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 = 8.557 %
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 = 5750 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 = 2000 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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