Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €20 000 at 2% per year with continuous compounding. Find the value FV after 11 years.
A mortgage of €250 000 is repaid with equal monthly payments over 30 years at 7.2% per year, compounded monthly. How much is still owed just after the payment at the end of year 4? Find the outstanding balance B.
A machine costs €40 000 today. It generates €9 500 at the end of each year for 5 years and can then be sold for €4 000. The cost of capital is 12%. Find the net present value NPV.
A project costs €10 000 and returns €6 000, €6 000 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 6% and matures in 9 years. Its yield to maturity changes from 9% to 11%. By how much ΔP does the bond price change?
A bond with a face value of €1 000 and an annual coupon of 5% matures in 10 years and sells for €890. Find its yield to maturity YTM in percent.
A company has equity worth €3 000 000 and debt worth €1 000 000. Its cost of equity is 9%, its pre-tax cost of debt is 6% and the tax rate is 20%. Find the WACC.
A company has 4 000 000 shares at €20.00 each and 40 000 bonds trading at €1 050 each. The pre-tax cost of debt is 3% and the tax rate is 25.8%. The share’s beta is 0.6, the risk-free rate 1% and the market return 7%. Find the WACC.
A product sells for €28 with a variable cost of €8 per unit. Fixed costs are €16 000 per month and profit is taxed at 25%. How many units Q must be sold per month to earn €3 000 after tax?
A product sells for €9 per unit, with a variable cost of €6 per unit and fixed costs of €4 500 per year. Budgeted sales are 2 400 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 24922 €
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 = 239081 €
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 = −3485 €
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.
Answer: IRR = 13.07 %
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 = −96.99 €
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: YTM = 6.532 %
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: WACC = 7.95 %
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 = 3.783 %
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 = 1000 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 = 37.5 %
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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