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 €10 000 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 want €30 000 in 20 years. You save the same amount at the end of every month in an account paying 4.2% per year, compounded monthly. Find the monthly deposit A.
An investment of €20 000 gives these cash flows at the end of years 1 to 4: €9 000, €8 000, €7 000, €8 000. The cost of capital is 5%. Find the net present value NPV.
An investment of €80 000 returns €30 000 at the end of each year for 3 years. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 pays an annual coupon of 4% and matures in 5 years. Its yield to maturity changes from 5.5% to 6%. By how much ΔP does the bond price change?
A bond trades at €1 060 with a yield to maturity of 3% (annual compounding) and a Macaulay duration of 10 years. Use the modified duration to estimate the new price P if the yield falls by 75 basis points.
A company has equity worth €6 000 000 and debt worth €3 000 000. Its cost of equity is 13%, its pre-tax cost of debt is 4% and the tax rate is 25%. Find the WACC.
A company has 2 500 000 shares at €32.00 each and 5 000 bonds trading at €1 020 each. The pre-tax cost of debt is 7% and the tax rate is 30%. The share’s beta is 1.5, the risk-free rate 3% and the market return 7%. Find the WACC.
A product sells for €33 per unit with a variable cost of €18 per unit. Fixed costs are €60 000 per month. How many units Q must be sold per month to make a profit of €6 000?
A product sells for €14 per unit, with a variable cost of €12 per unit and fixed costs of €10 000 per year. Budgeted sales are 6 250 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 200000 €
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 = 79.97 €
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 = 8456 €
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 = 6.129 %
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: ΔP = −20.19 €
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 = 1137 €
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 = 9.667 %
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 = 8.754 %
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 = 4400 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 = 20 %
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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