Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
An annuity pays €2 000 at the end of each year for 15 years. The interest rate is 8% per year. Find the present value PV of the annuity.
You save €200 at the end of every month for 16 years. The account pays 7.2% per year, compounded monthly. How much have you saved at the end? Find FV.
You buy a bond with 11 years to maturity (face value €1 000, annual coupon 8%) when its yield to maturity is 2.5%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 2%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 pays a 4.5% annual coupon in two half-yearly payments and matures in 9 years. It sells for €1 037. Find its yield to maturity YTM as an annual rate (twice the half-yearly yield), in percent.
A share with a beta of 0.8 has a required return of 7.7%. The risk-free rate is 2.5%. What expected market return rm does CAPM imply?
A company has 4 000 000 shares at €36.00 each and 10 000 bonds trading at €1 000 each. The pre-tax cost of debt is 7% and the tax rate is 21%. The share’s beta is 1.3, the risk-free rate 4% and the market return 9.5%. Find the WACC.
An investment of €75 000 gives these cash flows at the end of years 1 to 4: €36 000, €20 000, €24 000, €34 000. The cost of capital is 15%. Find the net present value NPV.
A project costs €50 000 today. Its cash flow is €6 500 at the end of year 1 and grows by 3% per year until the last cash flow at the end of year 7. The cost of capital is 12%. Find the net present value NPV.
Answer: PV = 17119 €
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 = 71788 €
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 €38 400 yourself; the rest is interest.
Answer: HPR = 6.281 %
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 = 4.006 %
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: rm = 9 %
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 = 10.79 %
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 = 6647 €
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: NPV = −17957 €
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.
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