Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
You invest €12 000 at a nominal rate of 4% per year, compounded quarterly. Find the value FV of the investment after 3 years.
You save €250 at the end of every month for 23 years. The account pays 6% per year, compounded monthly. How much have you saved at the end? Find FV.
A bond with a face value of €1 000 pays an annual coupon of 6% and matures in 6 years. Its yield to maturity changes from 8% to 6%. 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 4 years and sells for €1 075. Find its yield to maturity YTM in percent.
A company has a debt-to-equity ratio (at market values) of 1. Its cost of equity is 14%, its pre-tax cost of debt is 4.5% and the tax rate is 21%. Find the WACC.
A company has equity with a market value of €6 000 000 and a cost of equity of 10%. Its debt consists of 8 000 bonds (face value €1 000, annual coupon 7%, 11 years to maturity) that trade at €895 each. The tax rate is 25.8%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
A machine costs €40 000 today, lasts 5 years and costs €2 000 per year to run (paid at the end of each year). The cost of capital is 5%. Find its equivalent annual cost EAC.
A project costs €20 000 and returns €6 500, €9 500, €11 000 at the end of years 1 to 3. Cash flows can be reinvested at the cost of capital of 10%. Find the modified internal rate of return MIRR in percent.
Answer: FV = 13522 €
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 = 148063 €
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 €69 000 yourself; the rest is interest.
Answer: ΔP = 92.46 €
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 = 2.029 %
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 = 8.777 %
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 = 7.994 %
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: EAC = 11239 €
Equivalent annual cost: spread the present value of all costs over the life as an equal yearly amount (an annuity): EAC = PV·r/(1 − (1 + r)⁻ⁿ). It lets you compare machines with different lives.
Answer: MIRR = 13.59 %
Modified IRR: compound every inflow forward to the end of the project at the reinvestment rate (usually the cost of capital) to get the terminal value TV. MIRR is the rate that turns the initial investment into TV: MIRR = (TV/C₀)^(1/n) − 1.
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