Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
You invest €10 000 at 3% per year with continuous compounding. Find the value FV after 3 years.
A loan of €150 000 is repaid with equal monthly payments over 20 years. The interest rate is 6% per year, compounded monthly. Find the monthly payment A.
A bond with a face value of €1 000 pays a 4% annual coupon in two half-yearly payments and matures in 11 years. The yield to maturity is 9% per year. Find the price P of the bond.
A bond with a face value of €1 000 and an annual coupon of 6% matures in 5 years and sells for €900. Find its yield to maturity YTM in percent.
A company is financed with ordinary shares worth €8 000 000, preferred shares worth €500 000 and debt worth €4 000 000. The cost of equity is 14%, the cost of preferred stock 8% and the pre-tax cost of debt 4.5%. The tax rate is 25.8%. Find the WACC.
A company has 4 000 000 shares at €10.50 each and 10 000 bonds trading at €960 each. The pre-tax cost of debt is 4% and the tax rate is 25.8%. The share’s beta is 1.5, the risk-free rate 4% and the market return 10.5%. Find the WACC.
An investment of €20 000 gives these cash flows at the end of years 1 to 3: €10 000, €7 000, €5 000. The cost of capital is 5%. Find the net present value NPV.
A project costs €100 000 today. Its cash flow is €27 500 at the end of year 1 and grows by 4% per year until the last cash flow at the end of year 4. The cost of capital is 8%. Find the net present value NPV.
Answer: FV = 10942 €
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 = 1075 €
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: P = 655.4 €
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 = 8.54 %
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 = 10.35 %
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 = 11.74 %
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 = 192.2 €
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: NPV = −3668 €
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.
ExamProblems · examproblems.com/courses/investments-and-valuation