Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.
A property is expected to earn net rent of €12 000 next year. The rent is expected to grow by 2% per year forever. The required return is 4% per year. Find the value PV of the property today.
A loan of €150 000 is repaid with equal monthly payments over 15 years at 5.4% per year, compounded monthly. How much interest I is paid in total over the life of the loan?
A project costs €40 000 today and returns €12 000 at the end of each year for 6 years. The cost of capital is 6%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.
A project costs €150 000 today. Its cash flow is €15 000 at the end of year 1 and grows by 3% per year until the last cash flow at the end of year 10. The cost of capital is 12%. Find the net present value NPV.
A bond with a face value of €1 000 pays an annual coupon of 3% and matures in 4 years. Its yield to maturity changes from 5.5% to 5%. By how much ΔP does the bond price change?
A bond with 11 years to maturity (face value €1 000, annual coupon 5%) sells for €1 185. The issuer can call it in 6 years at a call price of €1 050. Find the yield to call YTC in percent.
A share with a beta of 0.9 has a required return of 8.85%. The risk-free rate is 3%. What expected market return rm does CAPM imply?
A company has 2 000 000 shares at €53.00 each and 5 000 bonds trading at €1 020 each. The pre-tax cost of debt is 6% and the tax rate is 25%. The share’s beta is 1.2, the risk-free rate 3.5% and the market return 7.5%. Find the WACC.
A product sells for €32 per unit with a variable cost of €20 per unit. Fixed costs are €48 000 per month. How many units Q must be sold per month to make a profit of €6 000?
A company sells two products. Product A sells for €18 with a variable cost of €12; product B sells for €23 with a variable cost of €15. They sell in a fixed mix of 3 units of A for every 2 of B. Fixed costs are €17 000 per month. Find the total number of units Q (A and B together) needed to break even.
Answer: PV = 600000 €
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: I = 69182 €
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: DPB = 3.834 years
Payback period: the time until the cumulative cash flows repay the initial investment. Cash is assumed to come in evenly during each year, so the last part is a fraction of a year. The simple payback ignores the time value of money; the discounted payback period uses the discounted cash flows instead.
The discounted payback is always longer than the simple payback (3.333 years).
Answer: NPV = −55451 €
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.
Answer: ΔP = 16.71 €
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: YTC = 2.433 %
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 to maturity is 3.001%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.
Answer: rm = 9.5 %
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 = 8.126 %
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 = 4500 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).
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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