Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €3 000 at a nominal rate of 3.6% per year, compounded monthly. Find the value FV of the investment after 5 years.
You save €500 at the end of every month for 30 years. The account pays 6% per year, compounded monthly. How much have you saved at the end? Find FV.
An investment of €20 000 gives these cash flows at the end of years 1 to 4: €5 000, €8 000, €9 000, €4 000. The cost of capital is 8%. Find the net present value NPV.
A project costs €50 000 and returns €28 500, €33 500 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 3% and matures in 6 years. Its yield to maturity changes from 3.5% to 5.5%. By how much ΔP does the bond price change?
A bond trades at €1 080 with a yield to maturity of 6% (annual compounding) and a Macaulay duration of 6.2 years. Use the modified duration to estimate the new price P if the yield rises by 50 basis points.
A company has a debt-to-equity ratio (at market values) of 0.4. Its cost of equity is 10%, its pre-tax cost of debt is 3% and the tax rate is 28%. Find the WACC.
A company has 4 000 000 shares trading at €40.50; the book value of its equity is €97 200 000. It just paid a dividend of €2.20 per share, and dividends grow by 4% per year. Its debt has a market value of €81 000 000 and a pre-tax cost of 3%. The tax rate is 28%. Find the WACC, using the dividend growth model for the cost of equity.
A company sells 1 880 units at €14 each. The variable cost is €6 per unit and fixed costs are €12 000. Find the degree of operating leverage DOL at this sales level.
A product sells for €50 per unit, with a variable cost of €45 per unit and fixed costs of €6 000 per year. Budgeted sales are 1 800 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 3591 €
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 = 502258 €
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 €180 000 yourself; the rest is interest.
Answer: NPV = 1573 €
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 = 15.17 %
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 = −98.25 €
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: P = 1048 €
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.
Duration is a linear estimate; for large yield changes convexity makes the true price a little higher.
Answer: WACC = 7.76 %
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 = 7.153 %
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: DOL = 4.947
Operating leverage: fixed costs make profit react more strongly than sales. The degree of operating leverage DOL = contribution margin/operating profit tells you that a 1% change in sales changes profit by DOL %. It is very high just above break-even and falls as sales grow.
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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