Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
An annuity pays €1 500 at the end of each year for 4 years. The interest rate is 7% per year. Find the present value PV of the annuity.
A mortgage of €300 000 is repaid with equal monthly payments over 20 years at 4.2% per year, compounded monthly. How much is still owed just after the payment at the end of year 12? Find the outstanding balance B.
A bond with a face value of €1 000 matures in 10 years and pays annual coupons. It sells for €923 when the yield to maturity is 5%. Find the coupon rate c in percent.
A bond trades at €1 010 with a yield to maturity of 6% (annual compounding) and a Macaulay duration of 10.4 years. Use the modified duration to estimate the new price P if the yield rises by 75 basis points.
A company has equity worth €10 000 000 and debt worth €6 000 000. Its cost of equity is 14%, its pre-tax cost of debt is 7% and the tax rate is 20%. Find the WACC.
A company has equity with a market value of €8 000 000 and a cost of equity of 13%. Its debt consists of 2 000 bonds (face value €1 000, annual coupon 5%, 10 years to maturity) that trade at €1 040 each. The tax rate is 28%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
An investment of €30 000 gives these cash flows at the end of years 1 to 4: €8 000, €12 000, €11 000, €8 000. The cost of capital is 6%. 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: PV = 5081 €
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: B = 150597 €
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.
Whatever is still owed must be exactly paid off by the remaining payments.
Answer: c = 4.003 %
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 = 935.7 €
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 = 10.85 %
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 = 10.99 %
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 = 3800 €
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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