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 €10 000 next year. The rent is expected to grow by 1% per year forever. The required return is 5% per year. Find the value PV of the property today.
You borrow €10 000 at 6% per year, compounded monthly, and repay €325 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.
You buy a bond with 9 years to maturity (face value €1 000, annual coupon 2%) when its yield to maturity is 6%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 5.5%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 and an annual coupon of 7% matures in 3 years and sells for €950. Find its yield to maturity YTM in percent.
A company has a debt-to-equity ratio (at market values) of 0.4. Its cost of equity is 14%, its pre-tax cost of debt is 7% and the tax rate is 28%. Find the WACC.
A company has equity with a market value of €6 000 000 and a cost of equity of 11%. Its debt consists of 8 000 bonds (face value €1 000, annual coupon 7%, 6 years to maturity) that trade at €932 each. The tax rate is 20%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.
An investment of €45 000 gives these cash flows at the end of years 1 to 3: €12 000, €22 000, €20 000. The cost of capital is 6%. Find the net present value NPV.
A project costs €20 000 and returns €9 000, €10 000, €11 500 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
Answer: PV = 250000 €
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 = 34 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.
33 full payments leave a small balance, which the last, smaller payment clears.
Answer: HPR = 9.665 %
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 = 8.974 %
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 = 11.44 %
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 = 8.67 %
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 = 2693 €
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 = 23.34 %
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 falls as the rate rises, so the IRR lies between a rate with positive NPV and one with negative NPV.
Linear interpolation gives a first estimate; repeating it (or a calculator's IRR function) gives the exact rate.
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