Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
You need €10 000 in 2 years. An account pays 5% per year, compounded monthly. How much must you deposit today? Find the present value PV.
You deposit €1 000 at the start of every year for 5 years, the first deposit today. The account pays 4% per year. How much is in the account at the end of year 5? Find FV.
A bond with a face value of €1 000 matures in 4 years and pays annual coupons. It sells for €945 when the yield to maturity is 3.5%. Find the coupon rate c in percent.
A bond with a face value of €1 000 and an annual coupon of 4% matures in 9 years and sells for €865. Find its yield to maturity YTM in percent.
A company has equity worth €3 000 000 and debt worth €6 000 000. Its cost of equity is 10%, its pre-tax cost of debt is 5% and the tax rate is 21%. Find the WACC.
A company has 5 000 000 shares at €25.50 each and 40 000 bonds trading at €960 each. The pre-tax cost of debt is 5% and the tax rate is 28%. The share’s beta is 1.2, the risk-free rate 3.5% and the market return 9.5%. Find the WACC.
An investment of €30 000 gives these cash flows at the end of years 1 to 3: €15 000, €13 000, €9 000. The cost of capital is 8%. Find the net present value NPV.
An investment of €10 000 returns €2 000 at the end of each year for 8 years. Find its internal rate of return IRR in percent.
Answer: PV = 9050 €
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 = 5633 €
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: c = 2.003 %
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: YTM = 5.983 %
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 = 5.967 %
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 = 9.057 %
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 = 2179 €
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 = 11.81 %
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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