Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €25 000 in 6 years. An account pays 5% per year, compounded monthly. How much must you deposit today? Find the present value PV.
A mortgage of €150 000 is repaid with equal monthly payments over 20 years at 6% per year, compounded monthly. How much is still owed just after the payment at the end of year 6? Find the outstanding balance B.
A project costs €200 000 today. It generates €12 000 at the end of the first year, and the cash flow then grows by 4% per year forever. The cost of capital is 9%. Find the net present value NPV.
A project costs €10 000 and returns €6 000, €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 a 3% annual coupon in two half-yearly payments and matures in 5 years. The yield to maturity is 9% per year. Find the price P of the bond.
A bond with a face value of €1 000 and an annual coupon of 7% matures in 7 years and sells for €1 085. Find its yield to maturity YTM in percent.
A company has equity worth €6 000 000 and debt worth €4 000 000. Its cost of equity is 10%, its pre-tax cost of debt is 4% and the tax rate is 20%. Find the WACC.
A company has 2 000 000 shares trading at €42.50; the book value of its equity is €42 500 000. It just paid a dividend of €2.00 per share, and dividends grow by 3% per year. Its debt has a market value of €68 000 000 and a pre-tax cost of 3%. The tax rate is 25%. Find the WACC, using the dividend growth model for the cost of equity.
A product sells for €14 per unit with a variable cost of €6 per unit. Fixed costs are €32 000 per month. How many units Q must be sold per month to make a profit of €2 000?
A company sells 1 280 units at €40 each, with a variable cost of €20 per unit and fixed costs of €16 000. By what percentage %ΔΠ does operating profit change if sales volume falls by 20%?
Answer: PV = 18532 €
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 = 121948 €
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 = 40000 €
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 = 18.88 %
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 = 762.6 €
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: YTM = 5.504 %
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 = 7.28 %
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 = 5.359 %
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 = 4250 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 = −53.33 %
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 = 1 024·20 − 16 000 = €4 480.00.
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