Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €12 000 at a nominal rate of 3.6% per year, compounded monthly. Find the value FV of the investment after 2 years.
A loan of €200 000 is repaid with equal monthly payments over 30 years. The interest rate is 3% per year, compounded monthly. Find the monthly payment A.
A machine costs €50 000 today. It generates €7 000 at the end of each year for 7 years and can then be sold for €5 000. The cost of capital is 12%. Find the net present value NPV.
A project costs €10 000 and returns €4 500, €7 500 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 10 years. The yield to maturity is 3.5% per year. Find the price P of the bond.
A bond with 15 years to maturity (face value €1 000, annual coupon 9%) sells for €1 235. The issuer can call it in 6 years at a call price of €1 090. Find the yield to call YTC in percent.
A company has equity worth €5 000 000 and debt worth €5 000 000. Its WACC is 6.08%, its pre-tax cost of debt is 3% and the tax rate is 28%. Find the cost of equity re.
A company has equity with a market value of €8 000 000 and a cost of equity of 12%. Its debt consists of 2 000 bonds (face value €1 000, annual coupon 5%, 4 years to maturity) that trade at €1 055 each. The tax rate is 25%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
A company sells 7 500 units at €12 each. The variable cost is €6 per unit and fixed costs are €15 000. Find the degree of operating leverage DOL at this sales level.
A company sells 1 250 units per month at €18 each, with a variable cost of €6 per unit and fixed costs of €12 000 per month. The variable cost rises by €2 per unit and the price stays the same. How many units Q must now be sold per month to keep the same profit?
Answer: FV = 12894 €
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 = 843.2 €
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 = −15792 €
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 = 11.98 %
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 = 958.1 €
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: YTC = 5.587 %
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 to maturity is 6.501%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: re = 10 %
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 = 10.04 %
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: DOL = 1.5
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.
Answer: Q = 1500 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).
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