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 €12 000 next year. The rent is expected to grow by 3% per year forever. The required return is 6% per year. Find the value PV of the property today.
A loan of €8 000 is repaid with equal monthly payments over 6 years. The interest rate is 7.2% per year, compounded monthly. Find the monthly payment A.
An investment of €75 000 gives these cash flows at the end of years 1 to 3: €31 000, €22 000, €23 000. The cost of capital is 5%. Find the net present value NPV.
A project costs €10 000 and returns €4 500, €7 000 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 5% and matures in 4 years. Its yield to maturity changes from 2.5% to 3.5%. By how much ΔP does the bond price change?
A bond with a face value of €1 000 pays an annual coupon of 3% and matures in 6 years. The yield to maturity is 4%. Find the Macaulay duration D in years.
A company has equity worth €8 000 000 and debt worth €5 000 000. Its WACC is 7.82%, its pre-tax cost of debt is 6% and the tax rate is 28%. Find the cost of equity re.
A company has 5 000 000 shares at €16.00 each and 5 000 bonds trading at €1 050 each. The pre-tax cost of debt is 6% and the tax rate is 30%. The share’s beta is 1.3, the risk-free rate 1.5% and the market return 7%. Find the WACC.
A product sells for €20 per unit with a variable cost of €8 per unit. Fixed costs are €12 000 per month. How many units Q must be sold per month to make a profit of €12 000?
A company sells 12 500 units per month at €53 each, with a variable cost of €45 per unit and fixed costs of €40 000 per month. The variable cost rises by €4 per unit and the price stays the same. How many units Q must now be sold per month to keep the same profit?
Answer: PV = 400000 €
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 = 137.2 €
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.
The present value of N equal payments A is A·(1 − (1 + i)⁻ᴺ)/i. Set it equal to the loan and solve for A.
Answer: NPV = −5653 €
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: IRR = 9.139 %
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 = −38.95 €
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 = 5.566 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: re = 10.01 %
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 = 8.376 %
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 = 2000 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: Q = 25000 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).
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