Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
You invest €1 000 at a nominal rate of 2% per year, compounded monthly. Find the value FV of the investment after 10 years.
You deposit €1 000 at the start of every year for 8 years, the first deposit today. The account pays 5% per year. How much is in the account at the end of year 8? Find FV.
A bond with a face value of €1 000 pays an annual coupon of 4% and matures in 7 years. Its yield to maturity changes from 3% to 1.5%. By how much ΔP does the bond price change?
A bond with a face value of €1 000 and an annual coupon of 4% matures in 10 years and sells for €1 130. Find its yield to maturity YTM in percent.
A company has a debt-to-equity ratio (at market values) of 0.6. Its cost of equity is 8%, its pre-tax cost of debt is 6% and the tax rate is 25.8%. Find the WACC.
A company has 2 000 000 shares at €27.00 each and 20 000 bonds trading at €980 each. The pre-tax cost of debt is 6% and the tax rate is 21%. The share’s beta is 1.1, the risk-free rate 2% and the market return 7.5%. Find the WACC.
An investment of €60 000 gives these cash flows at the end of years 1 to 4: €25 000, €24 000, €13 000, €27 000. The cost of capital is 9%. Find the net present value NPV.
A project costs €20 000 and returns €13 500, €9 000 at the end of years 1 to 2. Find its internal rate of return IRR in percent.
Answer: FV = 1221 €
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 = 10027 €
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.
Each deposit is invested one year longer.
Answer: ΔP = 102.7 €
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: YTM = 2.514 %
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: WACC = 6.669 %
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 = 7.169 %
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 = 12302 €
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 = 8.844 %
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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