Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
An annuity pays €500 at the end of each year for 5 years. The interest rate is 10% per year. Find the present value PV of the annuity.
You save €50 at the end of every month for 28 years. The account pays 5.4% per year, compounded monthly. How much have you saved at the end? Find FV.
A bond with a face value of €1 000 pays a 7% annual coupon in two half-yearly payments and matures in 9 years. The yield to maturity is 9% per year. Find the price P of the bond.
A bond trades at €900 with a yield to maturity of 5% (annual compounding) and a Macaulay duration of 7.8 years. Use the modified duration to estimate the new price P if the yield rises by 25 basis points.
A company has equity worth €4 000 000 and debt worth €4 000 000. Its cost of equity is 12%, its pre-tax cost of debt is 5% and the tax rate is 21%. Find the WACC.
A company has 1 000 000 shares at €46.50 each and 10 000 bonds trading at €1 000 each. The pre-tax cost of debt is 4% and the tax rate is 30%. The share’s beta is 0.8, the risk-free rate 3% and the market return 7%. Find the WACC.
A machine costs €60 000 today. It generates €22 500 at the end of each year for 3 years and can then be sold for €9 000. The cost of capital is 6%. Find the net present value NPV.
An investment of €25 000 returns €4 500 at the end of each year for 7 years. Find its internal rate of return IRR in percent.
Answer: PV = 1895 €
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: FV = 39116 €
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.
You paid in €16 800 yourself; the rest is interest.
Answer: P = 878.4 €
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 = 883.3 €
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 = 7.975 %
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 = 5.598 %
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 = 7699 €
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 = 6.136 %
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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