Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €20 000 at a nominal rate of 6% per year, compounded monthly. Find the value FV of the investment after 4 years.
A loan of €10 000 is repaid with equal monthly payments over 3 years at 3.6% per year, compounded monthly. How much interest I is paid in total over the life of the loan?
An investment of €60 000 gives these cash flows at the end of years 1 to 4: €22 000, €17 000, €14 000, €28 000. The cost of capital is 12%. Find the net present value NPV.
A project costs €10 000 and returns €6 500, €5 500, €5 500 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 matures in 9 years and pays annual coupons. It sells for €1 078 when the yield to maturity is 3%. Find the coupon rate c in percent.
A bond with a face value of €1 000 pays an annual coupon of 5% and matures in 4 years. The yield to maturity is 3%. Find the Macaulay duration D in years.
A company has equity worth €5 000 000 and debt worth €2 000 000. Its cost of equity is 11%, its pre-tax cost of debt is 3.5% and the tax rate is 21%. Find the WACC.
A company has 2 000 000 shares trading at €28.50; the book value of its equity is €22 800 000. It just paid a dividend of €2.40 per share, and dividends grow by 5% per year. Its debt has a market value of €45 600 000 and a pre-tax cost of 4.5%. The tax rate is 21%. Find the WACC, using the dividend growth model for the cost of equity.
A product sells for €40 per unit with a variable cost of €20 per unit. Fixed costs are €30 000 per month. How many units Q must be sold per month to make a profit of €6 000?
A product sells for €49 per unit, with a variable cost of €45 per unit and fixed costs of €10 000 per year. Budgeted sales are 4 500 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 25410 €
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 = 564.7 €
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 = 954.6 €
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 positive, so the project earns more than the required return.
Answer: IRR = 35.53 %
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 = 4.002 %
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: D = 3.734 years
Duration: the Macaulay duration is the weighted average time at which a bond pays its cash flows, with the present values of the cash flows as weights: D = Σ t·PVₜ/P. The modified duration D/(1 + y) estimates the percentage price change for a small change in yield: %ΔP ≈ −Dmod·Δy.
Answer: WACC = 8.647 %
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 = 9.27 %
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 = 1800 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 = 44.44 %
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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