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 €20 000 next year. The rent is expected to grow by 1.5% per year forever. The required return is 3.5% per year. Find the value PV of the property today.
A mortgage of €250 000 is repaid with equal monthly payments over 25 years at 3.6% per year, compounded monthly. How much is still owed just after the payment at the end of year 19? Find the outstanding balance B.
A project costs €30 000 today and returns €12 000, €13 000, €10 500, €13 000, €11 000 at the end of years 1 to 5. The cost of capital is 6%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.
Project A costs €10 000 and returns €5 500, €4 000, €2 000 at the end of years 1 to 3. Project B costs €15 000 and returns €4 000, €6 000, €7 500. Find the crossover rate r* at which both projects have the same NPV, in percent.
A bond with a face value of €1 000 pays an annual coupon of 2% and matures in 8 years. Its yield to maturity changes from 7% to 5%. By how much ΔP does the bond price change?
A bond with a face value of €1 000 pays an annual coupon of 7% and matures in 6 years. The yield to maturity is 5.5%. Find the Macaulay duration D in years.
A company has equity worth €8 000 000 and debt worth €2 000 000. Its WACC is 10.85%, its pre-tax cost of debt is 3% and the tax rate is 25%. Find the cost of equity re.
A company has 2 000 000 shares at €54.00 each and 20 000 bonds trading at €1 050 each. The pre-tax cost of debt is 5% and the tax rate is 25.8%. The share’s beta is 0.6, the risk-free rate 4% and the market return 9%. Find the WACC.
A product sells for €33 per unit with a variable cost of €25 per unit. Fixed costs are €16 000 per month. How many units Q must be sold per month to make a profit of €10 000?
A company sells 6 000 units per month at €32 each, with a variable cost of €12 per unit and fixed costs of €80 000 per month. The variable cost rises by €5 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 = 1 × 10⁶ €
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 = 81806 €
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: DPB = 2.806 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 (2.476 years).
Answer: r* = 5.886 %
Crossover rate: the discount rate at which two projects have the same NPV. It is the IRR of the difference in their cash flows (B − A). Below it the project with the later, larger cash flows has the higher NPV; above it the other one wins.
Check: at 5.886% both NPVs are €446.54.
Answer: ΔP = 104.7 €
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 = 5.136 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 = 13 %
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 = 6.464 %
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 = 3250 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 = 8000 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).
ExamProblems · examproblems.com/courses/corporate-finance