Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €2 000 at a nominal rate of 3.6% per year, compounded monthly. Find the value FV of the investment after 6 years.
A loan of €25 000 is repaid with equal monthly payments over 4 years. The interest rate is 3.6% per year, compounded monthly. Find the monthly payment A.
An investment of €30 000 gives these cash flows at the end of years 1 to 3: €15 000, €12 000, €14 000. The cost of capital is 15%. Find the net present value NPV.
A project costs €80 000 today. Its cash flow is €18 000 at the end of year 1 and grows by 4% per year until the last cash flow at the end of year 4. The cost of capital is 8%. Find the net present value NPV.
A bond with a face value of €1 000 pays a 7% annual coupon in two half-yearly payments and matures in 11 years. The yield to maturity is 5.5% per year. Find the price P of the bond.
A bond with a face value of €1 000 and an annual coupon of 4% matures in 10 years and sells for €1 130. Find its yield to maturity YTM in percent.
A company has equity worth €8 000 000 and debt worth €3 000 000. Its cost of equity is 9%, its pre-tax cost of debt is 6% 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 12%. Its debt consists of 5 000 bonds (face value €1 000, annual coupon 6%, 4 years to maturity) that trade at €1 054 each. The tax rate is 28%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
A product sells for €28 per unit with a variable cost of €18 per unit. Fixed costs are €25 000 per month. How many units Q must be sold per month to make a profit of €5 000?
A company sells 1 250 units at €40 each, with a variable cost of €25 per unit and fixed costs of €7 500. By what percentage %ΔΠ does operating profit change if sales volume rises by 10%?
Answer: FV = 2481 €
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: A = 560 €
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.
The present value of N equal payments A is A·(1 − (1 + i)⁻ᴺ)/i. Set it equal to the loan and solve for A.
Answer: NPV = 1322 €
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: NPV = −16946 €
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 negative, so the project earns less than the required return.
Answer: P = 1123 €
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 = 2.514 %
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 = 7.76 %
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 = 8.975 %
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 = 3000 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 = 16.67 %
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 = 1 375·15 − 7 500 = €13 125.00.
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