Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €20 000 in 7 years. An account pays 5% per year, compounded monthly. How much must you deposit today? Find the present value PV.
A mortgage of €200 000 is repaid with equal monthly payments over 20 years at 4.2% per year, compounded monthly. How much is still owed just after the payment at the end of year 5? Find the outstanding balance B.
An investment of €20 000 gives these cash flows at the end of years 1 to 4: €6 000, €10 000, €8 000, €6 000. The cost of capital is 9%. Find the net present value NPV.
A project costs €50 000 and returns €21 000, €35 500 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 pays an 8% annual coupon in two half-yearly payments and matures in 8 years. The yield to maturity is 7% per year. Find the price P of the bond.
A bond trades at €1 035 with a yield to maturity of 6% (annual compounding) and a Macaulay duration of 7.1 years. Use the modified duration to estimate the new price P if the yield falls by 150 basis points.
A company has equity worth €5 000 000 and debt worth €3 000 000. Its WACC is 10.02%, its pre-tax cost of debt is 4.5% and the tax rate is 25%. Find the cost of equity re.
A company has 2 500 000 shares at €42.50 each and 20 000 bonds trading at €1 050 each. The pre-tax cost of debt is 7% and the tax rate is 21%. The share’s beta is 1.1, the risk-free rate 3.5% and the market return 9%. Find the WACC.
A product sells for €37 per unit with a variable cost of €25 per unit. Fixed costs are €60 000 per month. How many units Q must be sold per month to make a profit of €12 000?
A company sells 2 250 units at €51 each, with a variable cost of €45 per unit and fixed costs of €9 000. By what percentage %ΔΠ does operating profit change if sales volume rises by 20%?
Answer: PV = 14104 €
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 = 164474 €
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 = 4349 €
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 = 7.839 %
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: P = 1060 €
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 below the coupon rate, so the bond sells at a premium.
Answer: P = 1139 €
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: re = 14.01 %
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 = 8.887 %
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 = 6000 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: %ΔΠ = 60 %
Operating leverage: fixed costs make profit react more strongly than sales. The degree of operating leverage DOL = contribution margin/operating profit tells you that a 1% change in sales changes profit by DOL %. It is very high just above break-even and falls as sales grow.
Check: new profit = 2 700·6 − 9 000 = €7 200.00.
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