Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
A property is expected to earn net rent of €6 000 next year. The rent is expected to grow by 4% per year forever. The required return is 9% per year. Find the value PV of the property today.
You save €200 at the end of every month for 18 years. The account pays 4.8% per year, compounded monthly. How much have you saved at the end? Find FV.
A project costs €40 000 today and returns €12 000, €12 500, €16 000, €15 500, €14 000, €14 500 at the end of years 1 to 6. The cost of capital is 8%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.
A project costs €10 000 and returns €6 500, €6 500 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 matures in 15 years and pays annual coupons. It sells for €948 when the yield to maturity is 5%. Find the coupon rate c in percent.
A bond with a face value of €1 000 pays an annual coupon of 4.5% and matures in 4 years. The yield to maturity is 7%. Find the Macaulay duration D in years.
A company has equity worth €6 000 000 and debt worth €4 000 000. Its cost of equity is 10%, its pre-tax cost of debt is 5% and the tax rate is 28%. Find the WACC.
A company has 2 000 000 shares trading at €54.00; the book value of its equity is €64 800 000. It just paid a dividend of €2.80 per share, and dividends grow by 2% per year. Its debt has a market value of €32 400 000 and a pre-tax cost of 3%. The tax rate is 21%. 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 €18 per unit. Method B, more automated, has fixed costs of €40 000 per year and a variable cost of €13 per unit. At what yearly volume Q do both methods cost the same?
A product sells for €38 per unit, with a variable cost of €18 per unit and fixed costs of €80 000 per year. Budgeted sales are 7 200 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 120000 €
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 = 68427 €
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 €43 200 yourself; the rest is interest.
Answer: DPB = 3.48 years
Payback period: the time until the cumulative cash flows repay the initial investment. Cash is assumed to come in evenly during each year, so the last part is a fraction of a year. The simple payback ignores the time value of money; the discounted payback period uses the discounted cash flows instead.
The discounted payback is always longer than the simple payback (2.969 years).
Answer: IRR = 19.43 %
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 = 4.499 %
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 = 3.736 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 = 7.44 %
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 = 6.154 %
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 = 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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