Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
A credit card charges a nominal interest rate of 19.5% per year, compounded monthly. Find the effective annual rate EAR in percent.
A mortgage of €150 000 is repaid with equal monthly payments over 30 years at 5.4% per year, compounded monthly. How much is still owed just after the payment at the end of year 13? Find the outstanding balance B.
An investment of €30 000 gives these cash flows at the end of years 1 to 3: €8 000, €12 000, €13 000. The cost of capital is 15%. Find the net present value NPV.
Project A costs €30 000 and returns €17 000, €12 500, €6 000 at the end of years 1 to 3. Project B costs €35 000 and returns €9 500, €14 500, €18 000. 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 a 2% annual coupon in two half-yearly payments and matures in 8 years. The yield to maturity is 5% per year. Find the price P of the bond.
A bond trades at €935 with a yield to maturity of 3% (annual compounding) and a Macaulay duration of 8.5 years. Use the modified duration to estimate the new price P if the yield rises by 50 basis points.
A company is financed with ordinary shares worth €12 000 000, preferred shares worth €500 000 and debt worth €4 000 000. The cost of equity is 11%, the cost of preferred stock 6% and the pre-tax cost of debt 4.5%. The tax rate is 20%. Find the WACC.
A company has 5 000 000 shares at €12.00 each and 40 000 bonds trading at €960 each. The pre-tax cost of debt is 3% and the tax rate is 28%. The share’s beta is 1.2, the risk-free rate 2.5% and the market return 7.5%. Find the WACC.
A product sells for €35 with a variable cost of €25 per unit. Fixed costs are €50 000 per month and profit is taxed at 25%. How many units Q must be sold per month to earn €15 000 after tax?
A company sells two products. Product A sells for €9 with a variable cost of €5; product B sells for €18 with a variable cost of €10. They sell in a fixed mix of 4 units of A for every 1 of B. Fixed costs are €9 600 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: EAR = 21.34 %
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 EAR is always above the nominal rate when interest compounds more than once a year.
Answer: B = 112280 €
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 = −5422 €
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: r* = 5.142 %
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 5.142% both NPVs are €2 637.81.
Answer: P = 804.2 €
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 = 896.4 €
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 = 9.055 %
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 = 6.026 %
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 = 7000 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 = 2000 units
Break-even with a sales mix: when products sell in a fixed ratio, use the weighted average contribution margin: weight each product's p − v by its share of units. Break-even units = F/(weighted contribution margin), split again in the mix ratio.
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