Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €10 000 at 4% per year with continuous compounding. Find the value FV after 17 years.
A mortgage of €300 000 is repaid with equal monthly payments over 25 years at 7.2% 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 €20 000 today and returns €8 000, €5 000, €6 500, €6 000, €7 000, €4 000 at the end of years 1 to 6. The cost of capital is 12%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.
A project costs €25 000 and returns €18 000, €10 000 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 5% and matures in 9 years. Its yield to maturity changes from 7% to 7.5%. By how much ΔP does the bond price change?
A bond with a face value of €1 000 pays a 2% annual coupon in two half-yearly payments and matures in 5 years. It sells for €723. Find its yield to maturity YTM as an annual rate (twice the half-yearly yield), in percent.
A company has equity worth €3 000 000 and debt worth €2 000 000. Its cost of equity is 13%, its pre-tax cost of debt is 4% and the tax rate is 25.8%. Find the WACC.
A company has equity with a market value of €10 000 000 and a cost of equity of 13%. Its debt consists of 8 000 bonds (face value €1 000, annual coupon 6%, 9 years to maturity) that trade at €1 071 each. The tax rate is 20%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
A product can be made in two ways. Method A has fixed costs of €10 000 per year and a variable cost of €18 per unit. Method B, more automated, has fixed costs of €13 000 per year and a variable cost of €12 per unit. At what yearly volume Q do both methods cost the same?
A company sells 3 000 units at €45 each, with a variable cost of €25 per unit and fixed costs of €40 000. By what percentage %ΔΠ does operating profit change if sales volume rises by 12%?
Answer: FV = 19739 €
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 = 125911 €
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 = 4.109 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 (3.083 years).
Answer: IRR = 8.774 %
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 = −29.17 €
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 = 9.002 %
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 = 8.987 %
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 = 8.847 %
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 = 500 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).
Below this volume method A is cheaper; above it the automated method B is cheaper.
Answer: %ΔΠ = 36 %
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 = 3 360·20 − 40 000 = €27 200.00.
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