Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €5 000 in 7 years. An account pays 5% per year, compounded monthly. How much must you deposit today? Find the present value PV.
A loan of €20 000 is repaid with equal monthly payments over 3 years at 3% per year, compounded monthly. How much interest I is paid in total over the life of the loan?
An investment of €20 000 gives these cash flows at the end of years 1 to 3: €4 000, €5 000, €10 000. The cost of capital is 15%. Find the net present value NPV.
A project costs €40 000 and returns €24 500, €24 000 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 pays an annual coupon of 5% and matures in 4 years. Its yield to maturity changes from 7% to 6%. By how much ΔP does the bond price change?
A bond with a face value of €1 000 and an annual coupon of 4% matures in 5 years and sells for €955. Find its yield to maturity YTM in percent.
A company has equity worth €3 000 000 and debt worth €5 000 000. Its cost of equity is 8%, its pre-tax cost of debt is 7% and the tax rate is 25.8%. Find the WACC.
A company has 2 000 000 shares trading at €17.00; the book value of its equity is €10 200 000. It just paid a dividend of €1.50 per share, and dividends grow by 3% per year. Its debt has a market value of €20 400 000 and a pre-tax cost of 5%. The tax rate is 30%. Find the WACC, using the dividend growth model for the cost of equity.
A product can be made in two ways. Method A has fixed costs of €15 000 per year and a variable cost of €10 per unit. Method B, more automated, has fixed costs of €35 000 per year and a variable cost of €5 per unit. At what yearly volume Q do both methods cost the same?
A product sells for €26 per unit, with a variable cost of €20 per unit and fixed costs of €7 200 per year. Budgeted sales are 1 440 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 3526 €
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 = 938.5 €
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 = −6166 €
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 = 13.92 %
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 = 33.09 €
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 = 5.041 %
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 = 6.246 %
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 = 8.868 %
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 = 4000 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: MS = 16.67 %
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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