Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €25 000 in 5 years. An account pays 3.6% per year, compounded quarterly. How much must you deposit today? Find the present value PV.
You want €25 000 in 16 years. You save the same amount at the end of every month in an account paying 4.2% per year, compounded monthly. Find the monthly deposit A.
An investment of €30 000 gives these cash flows at the end of years 1 to 4: €10 000, €10 000, €8 000, €6 000. The cost of capital is 7.000000000000001%. Find the net present value NPV.
A project costs €25 000 and returns €11 500, €15 500, €9 000 at the end of years 1 to 3. 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 11 years. The yield to maturity is 7% per year. Find the price P of the bond.
A bond trades at €975 with a yield to maturity of 3% (annual compounding) and a Macaulay duration of 3.9 years. Use the modified duration to estimate the new price P if the yield falls by 25 basis points.
A company is financed with ordinary shares worth €12 000 000, preferred shares worth €1 000 000 and debt worth €4 000 000. The cost of equity is 9%, the cost of preferred stock 6% and the pre-tax cost of debt 3%. The tax rate is 30%. Find the WACC.
A company has 2 000 000 shares at €60.00 each and 5 000 bonds trading at €980 each. The pre-tax cost of debt is 5% and the tax rate is 25%. The share’s beta is 0.9, the risk-free rate 1.5% and the market return 6.5%. Find the WACC.
A company sells 1 500 units at €30 each. The variable cost is €25 per unit and fixed costs are €2 500. Find the degree of operating leverage DOL at this sales level.
A company sells 5 000 units per month at €24 each, with a variable cost of €12 per unit and fixed costs of €30 000 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: PV = 20899 €
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 = 91.54 €
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 = −812.1 €
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 = 21.45 %
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 = 620.8 €
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: P = 984.2 €
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 = 7.2 %
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 = 5.912 %
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 = 6000 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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