Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
An annuity pays €2 000 at the end of each year for 9 years. The interest rate is 4% per year. Find the present value PV of the annuity.
You borrow €12 000 at 4.2% per year, compounded monthly, and repay €225 at the end of every month. How many months does it take until the loan is repaid (the last payment may be smaller)? Give the number of payments N.
A machine costs €50 000 today, lasts 6 years and costs €1 500 per year to run (paid at the end of each year). The cost of capital is 5%. Find its equivalent annual cost EAC.
A project costs €40 000 and returns €17 500, €21 000, €24 500 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 pays a 7% annual coupon in two half-yearly payments and matures in 5 years. The yield to maturity is 5% per year. Find the price P of the bond.
A bond with a face value of €1 000 pays a 5% annual coupon in two half-yearly payments and matures in 10 years. It sells for €1 082. Find its yield to maturity YTM as an annual rate (twice the half-yearly yield), in percent.
A company has equity worth €12 000 000 and debt worth €1 000 000. Its cost of equity is 14%, its pre-tax cost of debt is 5% and the tax rate is 25.8%. Find the WACC.
A company has 4 000 000 shares at €28.00 each and 40 000 bonds trading at €1 050 each. The pre-tax cost of debt is 6% and the tax rate is 25.8%. The share’s beta is 0.8, the risk-free rate 4% and the market return 10%. Find the WACC.
A product sells for €17 per unit with a variable cost of €12 per unit. Fixed costs are €20 000 per month. How many units Q must be sold per month to make a profit of €20 000?
A product sells for €14 per unit, with a variable cost of €4 per unit and fixed costs of €5 000 per year. Budgeted sales are 630 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 14871 €
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: N = 60 payments
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.
59 full payments leave a small balance, which the last, smaller payment clears.
Answer: EAC = 11351 €
Equivalent annual cost: spread the present value of all costs over the life as an equal yearly amount (an annuity): EAC = PV·r/(1 − (1 + r)⁻ⁿ). It lets you compare machines with different lives.
Answer: IRR = 24.97 %
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 = 1088 €
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: YTM = 3.997 %
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 = 13.21 %
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 = 7.614 %
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 = 8000 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 = 20.63 %
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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