Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €8 000 at a nominal rate of 4.8% per year, compounded quarterly. Find the value FV of the investment after 9 years.
You save €100 at the end of every month for 14 years. The account pays 6% per year, compounded monthly. How much have you saved at the end? Find FV.
An investment of €45 000 gives these cash flows at the end of years 1 to 3: €15 000, €16 000, €11 000. The cost of capital is 10%. Find the net present value NPV.
A project costs €150 000 today. Its cash flow is €16 500 at the end of year 1 and grows by 2% per year until the last cash flow at the end of year 10. The cost of capital is 8%. Find the net present value NPV.
You buy a bond with 12 years to maturity (face value €1 000, annual coupon 6%) when its yield to maturity is 2.5%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 2%. Find your holding period return HPR in percent.
A bond trades at €910 with a yield to maturity of 6% (annual compounding) and a Macaulay duration of 4 years. Use the modified duration to estimate the new price P if the yield rises by 50 basis points.
A company has equity worth €8 000 000 and debt worth €1 000 000. Its cost of equity is 13%, its pre-tax cost of debt is 5% and the tax rate is 30%. Find the WACC.
A company has 1 000 000 shares trading at €11.00; the book value of its equity is €3 300 000. It just paid a dividend of €1.00 per share, and dividends grow by 3% per year. Its debt has a market value of €3 300 000 and a pre-tax cost of 3%. The tax rate is 25%. Find the WACC, using the dividend growth model for the cost of equity.
A company sells 4 800 units at €47 each. The variable cost is €45 per unit and fixed costs are €8 000. Find the degree of operating leverage DOL at this sales level.
A product sells for €15 per unit, with a variable cost of €12 per unit and fixed costs of €4 500 per year. Budgeted sales are 1 880 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 12291 €
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: FV = 26230 €
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.
You paid in €16 800 yourself; the rest is interest.
Answer: NPV = −9876 €
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: NPV = −30273 €
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: HPR = 6.803 %
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: P = 892.8 €
Duration: the Macaulay duration is the weighted average time at which a bond pays its cash flows, with the present values of the cash flows as weights: D = Σ t·PVₜ/P. The modified duration D/(1 + y) estimates the percentage price change for a small change in yield: %ΔP ≈ −Dmod·Δy.
Duration is a linear estimate; for large yield changes convexity makes the true price a little higher.
Answer: WACC = 11.94 %
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 = 10.03 %
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 = 6
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: MS = 20.21 %
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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