Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
You invest €5 000 at a nominal rate of 2% per year, compounded quarterly. Find the value FV of the investment after 8 years.
A loan of €15 000 is repaid with equal monthly payments over 3 years. The interest rate is 3.6% per year, compounded monthly. Find the monthly payment A.
A bond with a face value of €1 000 matures in 5 years and pays annual coupons. It sells for €860 when the yield to maturity is 8%. Find the coupon rate c in percent.
A bond with a face value of €1 000 pays an annual coupon of 4% and matures in 3 years. The yield to maturity is 3%. Find the Macaulay duration D in years.
A company has a debt-to-equity ratio (at market values) of 0.5. Its cost of equity is 11%, its pre-tax cost of debt is 4% and the tax rate is 21%. Find the WACC.
A company has equity with a market value of €12 000 000 and a cost of equity of 12%. Its debt consists of 4 000 bonds (face value €1 000, annual coupon 6%, 10 years to maturity) that trade at €1 038 each. The tax rate is 28%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
A project costs €400 000 today. It generates €43 000 at the end of the first year, and the cash flow then grows by 2% per year forever. The cost of capital is 11%. Find the net present value NPV.
An investment of €20 000 returns €4 000 at the end of each year for 9 years. Find its internal rate of return IRR in percent.
Answer: FV = 5865 €
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 = 440.2 €
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 = 4.494 %
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: D = 2.888 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 = 8.387 %
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 = 9.933 %
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 = 77778 €
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 positive, so the project earns more than the required return.
Answer: IRR = 13.7 %
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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