Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €20 000 at a nominal rate of 4.8% per year, compounded monthly. Find the value FV of the investment after 10 years.
A mortgage of €250 000 is repaid with equal monthly payments over 25 years at 4.8% per year, compounded monthly. How much is still owed just after the payment at the end of year 7? Find the outstanding balance B.
A project costs €200 000 today. It generates €14 000 at the end of the first year, and the cash flow then grows by 3% per year forever. The cost of capital is 9%. Find the net present value NPV.
An investment of €25 000 returns €5 500 at the end of each year for 8 years. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 pays a 2% annual coupon in two half-yearly payments and matures in 5 years. The yield to maturity is 3.5% per year. Find the price P of the bond.
A bond with a face value of €1 000 and an annual coupon of 6% matures in 10 years and sells for €1 160. Find its yield to maturity YTM in percent.
A company has equity worth €8 000 000 and debt worth €4 000 000. Its cost of equity is 10%, its pre-tax cost of debt is 7% and the tax rate is 21%. Find the WACC.
A company has 5 000 000 shares trading at €40.00; the book value of its equity is €80 000 000. It just paid a dividend of €2.40 per share, and dividends grow by 5% per year. Its debt has a market value of €120 000 000 and a pre-tax cost of 5%. The tax rate is 30%. Find the WACC, using the dividend growth model for the cost of equity.
A company sells 6 250 units at €32 each. The variable cost is €12 per unit and fixed costs are €100 000. Find the degree of operating leverage DOL at this sales level.
A company sells 12 500 units per month at €47 each, with a variable cost of €45 per unit and fixed costs of €10 000 per month. The variable cost rises by €1 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 = 32291 €
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 = 206924 €
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 = 33333 €
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 = 14.61 %
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.
The annuity factor falls as the rate rises; bracket the rate and narrow down.
Answer: P = 931.7 €
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 = 4.025 %
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 = 8.51 %
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 = 8.375 %
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 = 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 = 25000 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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