Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
You need €50 000 in 10 years. An account pays 3% per year, compounded quarterly. How much must you deposit today? Find the present value PV.
A loan of €250 000 is repaid with equal monthly payments over 30 years. The interest rate is 3.6% per year, compounded monthly. Find the monthly payment A.
An investment of €30 000 gives these cash flows at the end of years 1 to 4: €15 000, €9 000, €9 000, €13 000. The cost of capital is 5%. Find the net present value NPV.
A project costs €50 000 and returns €36 000, €23 000, €27 500 at the end of years 1 to 3. Find its internal rate of return IRR in percent.
You buy a bond with 4 years to maturity (face value €1 000, annual coupon 6%) when its yield to maturity is 9%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 8%. Find your holding period return HPR in percent.
A bond with a face value of €1 000 and an annual coupon of 3% matures in 6 years and sells for €925. Find its yield to maturity YTM in percent.
A share with a beta of 1.5 has a required return of 12%. The risk-free rate is 3%. What expected market return rm does CAPM imply?
A company has 2 500 000 shares at €27.50 each and 40 000 bonds trading at €980 each. The pre-tax cost of debt is 3% and the tax rate is 21%. The share’s beta is 0.6, the risk-free rate 2% and the market return 9%. Find the WACC.
A product can be made in two ways. Method A has fixed costs of €15 000 per year and a variable cost of €12 per unit. Method B, more automated, has fixed costs of €23 000 per year and a variable cost of €8 per unit. At what yearly volume Q do both methods cost the same?
A company sells two products. Product A sells for €8 with a variable cost of €5; product B sells for €33 with a variable cost of €25. They sell in a fixed mix of 2 units of A for every 3 of B. Fixed costs are €15 000 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: PV = 37082 €
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 = 1137 €
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: NPV = 10919 €
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 = 35.74 %
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.
Answer: HPR = 11.7 %
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 = 4.452 %
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 = 9 %
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 = 4.809 %
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: Q = 2000 units
Cost-volume-profit: each unit sold earns a contribution margin, price minus variable cost per unit. Profit = Q·(p − v) − F, where F is the fixed cost. At the break-even point profit is zero, so Q = F/(p − v).
Below this volume method A is cheaper; above it the automated method B is cheaper.
Answer: Q = 2500 units
Break-even with a sales mix: when products sell in a fixed ratio, use the weighted average contribution margin: weight each product's p − v by its share of units. Break-even units = F/(weighted contribution margin), split again in the mix ratio.
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