Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
You need €40 000 in 4 years. An account pays 3% per year, compounded monthly. How much must you deposit today? Find the present value PV.
A loan of €200 000 is repaid with equal monthly payments over 25 years. The interest rate is 4.2% per year, compounded monthly. Find the monthly payment A.
A bond with a face value of €1 000 pays an annual coupon of 8% and matures in 5 years. Its yield to maturity changes from 4% to 3.5%. By how much ΔP does the bond price change?
A bond with a face value of €1 000 and an annual coupon of 7% matures in 5 years and sells for €920. Find its yield to maturity YTM in percent.
A company has equity worth €2 000 000 and debt worth €5 000 000. Its cost of equity is 10%, its pre-tax cost of debt is 3.5% and the tax rate is 20%. Find the WACC.
A company has equity with a market value of €15 000 000 and a cost of equity of 10%. Its debt consists of 5 000 bonds (face value €1 000, annual coupon 4%, 4 years to maturity) that trade at €1 018 each. The tax rate is 21%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
A machine costs €30 000 today. It generates €6 000 at the end of each year for 6 years and can then be sold for €3 000. The cost of capital is 10%. Find the net present value NPV.
A project costs €80 000 today. Its cash flow is €14 500 at the end of year 1 and grows by 3% per year until the last cash flow at the end of year 6. The cost of capital is 12%. Find the net present value NPV.
Answer: PV = 35482 €
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: A = 1078 €
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.
The present value of N equal payments A is A·(1 − (1 + i)⁻ᴺ)/i. Set it equal to the loan and solve for A.
Answer: ΔP = 25.1 €
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 = 9.06 %
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 = 4.857 %
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 = 8.169 %
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 = −2175 €
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.
Answer: NPV = −16352 €
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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