Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
An annuity pays €1 000 at the end of each year for 11 years. The interest rate is 10% per year. Find the present value PV of the annuity.
You borrow €20 000 at 3% per year, compounded monthly, and repay €250 at the end of every month. How many months does it take until the loan is repaid (the last payment may be smaller)? Give the number of payments N.
A bond with a face value of €1 000 pays a 4% annual coupon in two half-yearly payments and matures in 10 years. The yield to maturity is 7% per year. Find the price P of the bond.
A bond with a face value of €1 000 and an annual coupon of 8% matures in 4 years and sells for €950. Find its yield to maturity YTM in percent.
A share with a beta of 1.4 has a required return of 9%. The risk-free rate is 2%. What expected market return rm does CAPM imply?
A company has 4 000 000 shares at €45.00 each and 20 000 bonds trading at €1 000 each. The pre-tax cost of debt is 6% and the tax rate is 25.8%. The share’s beta is 0.6, the risk-free rate 1.5% and the market return 7.5%. Find the WACC.
A machine costs €100 000 today. It generates €35 500 at the end of each year for 3 years and can then be sold for €15 000. The cost of capital is 12%. Find the net present value NPV.
An investment of €20 000 returns €4 000 at the end of each year for 9 years. Find its internal rate of return IRR in percent.
Answer: PV = 6495 €
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: N = 90 payments
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.
89 full payments leave a small balance, which the last, smaller payment clears.
Answer: P = 786.8 €
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.
The yield is above the coupon rate, so the bond sells at a discount.
Answer: YTM = 9.562 %
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 = 7 %
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 = 5.035 %
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 = −4058 €
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: IRR = 13.7 %
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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