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 3% per year, compounded quarterly. Find the value FV of the investment after 9 years.
A loan of €200 000 is repaid with equal monthly payments over 20 years. The interest rate is 2.4% per year, compounded monthly. Find the monthly payment A.
You buy a bond with 7 years to maturity (face value €1 000, annual coupon 4%) when its yield to maturity is 7%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 6.5%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 and an annual coupon of 8% matures in 6 years and sells for €1 050. Find its yield to maturity YTM in percent.
A company has equity worth €4 000 000 and debt worth €5 000 000. Its cost of equity is 8%, its pre-tax cost of debt is 3.5% and the tax rate is 28%. Find the WACC.
A company has 5 000 000 shares trading at €22.00; the book value of its equity is €66 000 000. It just paid a dividend of €0.80 per share, and dividends grow by 3% per year. Its debt has a market value of €55 000 000 and a pre-tax cost of 4%. The tax rate is 25%. Find the WACC, using the dividend growth model for the cost of equity.
A project costs €400 000 today. It generates €24 000 at the end of the first year, and the cash flow then grows by 3% per year forever. The cost of capital is 8%. Find the net present value NPV.
A project costs €25 000 and returns €15 500, €12 000 at the end of years 1 to 2. Find its internal rate of return IRR in percent.
Answer: FV = 15704 €
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 = 1050 €
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: HPR = 9.621 %
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 = 6.953 %
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 = 4.956 %
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 = 5.497 %
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 = 80000 €
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 = 6.901 %
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.
ExamProblems · examproblems.com/courses/investments-and-valuation