Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You invest €2 000 at 3% per year with continuous compounding. Find the value FV after 11 years.
You want €25 000 in 18 years. You save the same amount at the end of every month in an account paying 3.6% per year, compounded monthly. Find the monthly deposit A.
A project costs €30 000 today and returns €13 000, €13 000, €11 000, €11 000 at the end of years 1 to 4. The cost of capital is 5%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.
A project costs €50 000 and returns €15 500, €26 000, €19 000, €19 000 at the end of years 1 to 4. Cash flows can be reinvested at the cost of capital of 8%. Find the modified internal rate of return MIRR in percent.
A bond with a face value of €1 000 pays a 2% annual coupon in two half-yearly payments and matures in 7 years. The yield to maturity is 9% per year. Find the price P of the bond.
A bond with a face value of €1 000 pays an annual coupon of 3% and matures in 5 years. The yield to maturity is 5%. Find the Macaulay duration D in years.
A company has equity worth €12 000 000 and debt worth €5 000 000. Its cost of equity is 8%, its pre-tax cost of debt is 3% and the tax rate is 25.8%. Find the WACC.
A company has 2 500 000 shares at €20.50 each and 40 000 bonds trading at €1 020 each. The pre-tax cost of debt is 4.5% and the tax rate is 25.8%. The share’s beta is 0.6, the risk-free rate 3.5% and the market return 9.5%. Find the WACC.
A company sells 2 400 units at €25 each. The variable cost is €15 per unit and fixed costs are €12 000. Find the degree of operating leverage DOL at this sales level.
A product sells for €18 per unit, with a variable cost of €12 per unit and fixed costs of €6 000 per year. Budgeted sales are 1 250 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: FV = 2782 €
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 = 82.43 €
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: DPB = 2.613 years
Payback period: the time until the cumulative cash flows repay the initial investment. Cash is assumed to come in evenly during each year, so the last part is a fraction of a year. The simple payback ignores the time value of money; the discounted payback period uses the discounted cash flows instead.
The discounted payback is always longer than the simple payback (2.364 years).
Answer: MIRR = 15.63 %
Modified IRR: compound every inflow forward to the end of the project at the reinvestment rate (usually the cost of capital) to get the terminal value TV. MIRR is the rate that turns the initial investment into TV: MIRR = (TV/C₀)^(1/n) − 1.
Answer: P = 642.2 €
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: D = 4.702 years
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.
Answer: WACC = 6.302 %
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 = 5.433 %
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 = 2
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 %
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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