Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €1 000 at 5% per year with continuous compounding. Find the value FV after 18 years.
A loan of €30 000 is repaid with equal monthly payments over 3 years at 4.8% per year, compounded monthly. How much interest I is paid in total over the life of the loan?
A project costs €250 000 today. It generates €16 000 at the end of the first year, and the cash flow then grows by 3% per year forever. The cost of capital is 10%. Find the net present value NPV.
A project costs €20 000 and returns €13 500, €12 000 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 matures in 13 years and pays annual coupons. It sells for €1 100 when the yield to maturity is 4%. Find the coupon rate c in percent.
A bond with a face value of €1 000 and an annual coupon of 4% matures in 7 years and sells for €865. Find its yield to maturity YTM in percent.
A company has equity worth €8 000 000 and debt worth €4 000 000. Its cost of equity is 11%, its pre-tax cost of debt is 4.5% and the tax rate is 30%. Find the WACC.
A company has 2 000 000 shares at €55.00 each and 20 000 bonds trading at €1 000 each. The pre-tax cost of debt is 4% and the tax rate is 30%. The share’s beta is 1.3, the risk-free rate 1.5% and the market return 8.5%. Find the WACC.
A product sells for €14 with a variable cost of €8 per unit. Fixed costs are €12 000 per month and profit is taxed at 20%. How many units Q must be sold per month to earn €6 000 after tax?
A product sells for €19 per unit, with a variable cost of €15 per unit and fixed costs of €16 000 per year. Budgeted sales are 6 000 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 2460 €
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: I = 2272 €
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 = −21429 €
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 = 18.24 %
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: c = 5.001 %
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 = 6.458 %
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 = 8.383 %
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 = 9.4 %
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: Q = 3250 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: MS = 33.33 %
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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