Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €40 000 in 7 years. An account pays 2% per year, compounded monthly. How much must you deposit today? Find the present value PV.
A loan of €150 000 is repaid with equal monthly payments over 25 years at 3% per year, compounded monthly. How much interest I is paid in total over the life of the loan?
An investment of €75 000 gives these cash flows at the end of years 1 to 3: €33 000, €17 000, €28 000. The cost of capital is 5%. Find the net present value NPV.
An investment of €10 000 returns €1 500 at the end of each year for 10 years. Find its internal rate of return IRR in percent.
You buy a bond with 3 years to maturity (face value €1 000, annual coupon 6%) when its yield to maturity is 4%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 4.5%. Find your holding period return HPR in percent.
A bond trades at €1 095 with a yield to maturity of 4% (annual compounding) and a Macaulay duration of 4.4 years. Use the modified duration to estimate the new price P if the yield rises by 100 basis points.
A share with a beta of 0.8 has a required return of 7.4%. The risk-free rate is 3%. What expected market return rm does CAPM imply?
A company has 5 000 000 shares at €33.00 each and 20 000 bonds trading at €980 each. The pre-tax cost of debt is 7% and the tax rate is 28%. The share’s beta is 1.3, the risk-free rate 2% and the market return 6.5%. Find the WACC.
A product can be made in two ways. Method A has fixed costs of €25 000 per year and a variable cost of €18 per unit. Method B, more automated, has fixed costs of €33 000 per year and a variable cost of €14 per unit. At what yearly volume Q do both methods cost the same?
A product sells for €22 per unit, with a variable cost of €12 per unit and fixed costs of €20 000 per year. Budgeted sales are 3 000 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 34778 €
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: I = 63395 €
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: NPV = −3964 €
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: IRR = 8.144 %
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: HPR = 3.087 %
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 = 1049 €
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: rm = 8.5 %
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 = 7.552 %
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 = 2000 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).
Below this volume method A is cheaper; above it the automated method B is cheaper.
Answer: MS = 33.33 %
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).
ExamProblems · examproblems.com/courses/corporate-finance