Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
A property is expected to earn net rent of €6 000 next year. The rent is expected to grow by 2% per year forever. The required return is 7% per year. Find the value PV of the property today.
You want €25 000 in 11 years. You save the same amount at the end of every month in an account paying 7.2% per year, compounded monthly. Find the monthly deposit A.
A bond with a face value of €1 000 pays a 6% annual coupon in two half-yearly payments and matures in 3 years. The yield to maturity is 5% per year. Find the price P of the bond.
A bond with a face value of €1 000 and an annual coupon of 5% matures in 10 years and sells for €925. Find its yield to maturity YTM in percent.
A company has equity worth €12 000 000 and debt worth €4 000 000. Its cost of equity is 12%, its pre-tax cost of debt is 5% and the tax rate is 25%. Find the WACC.
A company has 5 000 000 shares at €48.00 each and 40 000 bonds trading at €960 each. The pre-tax cost of debt is 5% and the tax rate is 20%. The share’s beta is 1.8, the risk-free rate 2% and the market return 6.5%. Find the WACC.
A machine costs €80 000 today, lasts 7 years and costs €2 000 per year to run (paid at the end of each year). The cost of capital is 10%. Find its equivalent annual cost EAC.
A project costs €40 000 and returns €14 000, €18 500, €13 000, €15 000 at the end of years 1 to 4. Cash flows can be reinvested at the cost of capital of 12%. Find the modified internal rate of return MIRR in percent.
Answer: PV = 120000 €
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 = 124.7 €
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: P = 1028 €
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 below the coupon rate, so the bond sells at a premium.
Answer: YTM = 6.02 %
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 = 9.938 %
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 = 9.259 %
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 = 18432 €
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 = 16 %
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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