Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €5 000 at 4% per year with continuous compounding. Find the value FV after 8 years.
You want €10 000 in 10 years. You save the same amount at the end of every month in an account paying 7.2% per year, compounded monthly. Find the monthly deposit A.
A machine costs €60 000 today. It generates €8 500 at the end of each year for 7 years and can then be sold for €15 000. The cost of capital is 12%. Find the net present value NPV.
A project costs €200 000 today. Its cash flow is €33 500 at the end of year 1 and grows by 2% per year until the last cash flow at the end of year 6. The cost of capital is 12%. Find the net present value NPV.
You buy a bond with 9 years to maturity (face value €1 000, annual coupon 7%) when its yield to maturity is 9%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 9.5%. Find your holding period return HPR in percent.
A bond trades at €925 with a yield to maturity of 3% (annual compounding) and a Macaulay duration of 8.6 years. Use the modified duration to estimate the new price P if the yield rises by 50 basis points.
A share with a beta of 0.6 has a required return of 5.9%. The risk-free rate is 2%. What expected market return rm does CAPM imply?
A company has 5 000 000 shares at €21.00 each and 5 000 bonds trading at €980 each. The pre-tax cost of debt is 3.5% and the tax rate is 25.8%. The share’s beta is 0.9, the risk-free rate 3.5% and the market return 8.5%. Find the WACC.
A company sells 3 000 units at €14 each. The variable cost is €4 per unit and fixed costs are €10 000. Find the degree of operating leverage DOL at this sales level.
A product sells for €45 per unit, with a variable cost of €30 per unit and fixed costs of €12 000 per year. Budgeted sales are 960 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 6886 €
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: A = 57.14 €
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 = −14423 €
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.
Answer: NPV = −56134 €
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: HPR = 6.144 %
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 = 886.4 €
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: rm = 8.5 %
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 = 7.759 %
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 = 1.5
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 = 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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