Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
An annuity pays €1 000 at the end of each year for 20 years. The interest rate is 10% per year. Find the present value PV of the annuity.
A loan of €200 000 is repaid with equal monthly payments over 30 years. The interest rate is 3% per year, compounded monthly. Find the monthly payment A.
A bond with a face value of €1 000 pays a 4.5% annual coupon in two half-yearly payments and matures in 8 years. The yield to maturity is 3% per year. Find the price P of the bond.
A bond trades at €1 060 with a yield to maturity of 4% (annual compounding) and a Macaulay duration of 8.2 years. Use the modified duration to estimate the new price P if the yield rises by 150 basis points.
A company has equity worth €4 000 000 and debt worth €3 000 000. Its cost of equity is 10%, its pre-tax cost of debt is 3.5% and the tax rate is 25%. Find the WACC.
A company has 2 500 000 shares at €37.00 each and 10 000 bonds trading at €1 050 each. The pre-tax cost of debt is 4% and the tax rate is 21%. The share’s beta is 0.8, the risk-free rate 2% and the market return 8%. Find the WACC.
An investment of €45 000 gives these cash flows at the end of years 1 to 3: €11 000, €14 000, €19 000. The cost of capital is 6%. Find the net present value NPV.
A project costs €10 000 and returns €7 000, €7 000, €6 000 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
Answer: PV = 8514 €
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 = 843.2 €
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: P = 1106 €
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 below the coupon rate, so the bond sells at a premium.
Answer: P = 934.6 €
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 = 6.839 %
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 = 6.429 %
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 = −6210 €
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 = 46.05 %
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.
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