Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
You need €20 000 in 4 years. An account pays 3.6% per year, compounded monthly. How much must you deposit today? Find the present value PV.
A loan of €250 000 is repaid with equal monthly payments over 30 years. The interest rate is 7.2% per year, compounded monthly. Find the monthly payment A.
A bond with a face value of €1 000 matures in 10 years and pays annual coupons. It sells for €1 291 when the yield to maturity is 3.5%. Find the coupon rate c in percent.
A bond trades at €915 with a yield to maturity of 5% (annual compounding) and a Macaulay duration of 11.6 years. Use the modified duration to estimate the new price P if the yield falls by 25 basis points.
A company has a debt-to-equity ratio (at market values) of 0.4. Its cost of equity is 13%, its pre-tax cost of debt is 4.5% and the tax rate is 28%. Find the WACC.
A company has 2 000 000 shares trading at €35.00; the book value of its equity is €28 000 000. It just paid a dividend of €1.40 per share, and dividends grow by 2% per year. Its debt has a market value of €70 000 000 and a pre-tax cost of 4.5%. The tax rate is 30%. Find the WACC, using the dividend growth model for the cost of equity.
A machine costs €50 000 today. It generates €6 500 at the end of each year for 8 years and can then be sold for €7 500. The cost of capital is 12%. Find the net present value NPV.
An investment of €20 000 returns €6 000 at the end of each year for 4 years. Find its internal rate of return IRR in percent.
Answer: PV = 17321 €
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 = 1697 €
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.
The present value of N equal payments A is A·(1 − (1 + i)⁻ᴺ)/i. Set it equal to the loan and solve for A.
Answer: c = 6.999 %
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 = 940.3 €
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 = 10.21 %
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 = 4.615 %
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: NPV = −14681 €
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.
Answer: IRR = 7.714 %
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.
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