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 5% per year, compounded monthly. Find the value FV of the investment after 10 years.
A mortgage of €300 000 is repaid with equal monthly payments over 25 years at 4.2% per year, compounded monthly. How much is still owed just after the payment at the end of year 14? Find the outstanding balance B.
An investment of €45 000 gives these cash flows at the end of years 1 to 4: €9 000, €12 000, €22 000, €22 000. The cost of capital is 6%. Find the net present value NPV.
An investment of €20 000 returns €4 000 at the end of each year for 6 years. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 pays a 4% annual coupon in two half-yearly payments and matures in 7 years. The yield to maturity is 5% per year. Find the price P of the bond.
A bond trades at €940 with a yield to maturity of 3% (annual compounding) and a Macaulay duration of 10.2 years. Use the modified duration to estimate the new price P if the yield falls by 75 basis points.
A share with a beta of 0.6 has a required return of 4.9%. The risk-free rate is 1%. What expected market return rm does CAPM imply?
A company has 4 000 000 shares trading at €40.50; the book value of its equity is €64 800 000. It just paid a dividend of €1.80 per share, and dividends grow by 2% per year. Its debt has a market value of €162 000 000 and a pre-tax cost of 3%. The tax rate is 25.8%. 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 €20 000 per year and a variable cost of €10 per unit. Method B, more automated, has fixed costs of €29 000 per year and a variable cost of €4 per unit. At what yearly volume Q do both methods cost the same?
A product sells for €18 per unit, with a variable cost of €8 per unit and fixed costs of €10 000 per year. Budgeted sales are 1 600 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 32940 €
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: B = 170677 €
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.
Whatever is still owed must be exactly paid off by the remaining payments.
Answer: NPV = 10068 €
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 = 5.472 %
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.
The annuity factor falls as the rate rises; bracket the rate and narrow down.
Answer: P = 941.5 €
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.
The yield is above the coupon rate, so the bond sells at a discount.
Answer: P = 1010 €
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 = 7.5 %
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 = 4.38 %
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 = 1500 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 = 37.5 %
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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