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 3% per year forever. The required return is 7% per year. Find the value PV of the property today.
A loan of €20 000 is repaid with equal monthly payments over 3 years at 3.6% per year, compounded monthly. How much interest I is paid in total over the life of the loan?
A project costs €30 000 today and returns €11 000, €12 500, €10 000, €6 000, €10 500 at the end of years 1 to 5. The cost of capital is 5%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.
An investment of €50 000 returns €9 500 at the end of each year for 8 years. Find its internal rate of return IRR in percent.
A bond with a face value of €1 000 pays a 6% annual coupon in two half-yearly payments and matures in 3 years. The yield to maturity is 8% per year. Find the price P of the bond.
A bond with a face value of €1 000 and an annual coupon of 5% matures in 9 years and sells for €1 075. Find its yield to maturity YTM in percent.
A share with a beta of 1.6 has a required return of 11.9%. The risk-free rate is 1.5%. What expected market return rm does CAPM imply?
A company's shares have a beta of 1.3 at its current debt-to-equity ratio of 0.5. It plans to raise its debt-to-equity ratio to 1.25. The tax rate is 30%, the risk-free rate 1% and the market risk premium 5.5%. Find the cost of equity re after the change.
A product sells for €26 per unit with a variable cost of €18 per unit. Fixed costs are €4 000 per month. How many units Q must be sold per month to make a profit of €2 000?
A product sells for €32 per unit, with a variable cost of €30 per unit and fixed costs of €5 000 per year. Budgeted sales are 3 500 units. Find the margin of safety MS as a percentage of budgeted sales.
Answer: PV = 300000 €
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 = 1129 €
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 = 2.948 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 (2.65 years).
Answer: IRR = 10.37 %
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.
The annuity factor falls as the rate rises; bracket the rate and narrow down.
Answer: P = 947.6 €
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 = 3.992 %
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 = 8 %
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: re = 10.93 %
Levered beta (Hamada): debt makes equity riskier. βL = βU·(1 + (1 − T)·D/E), where βU is the beta of the business without debt. To move a beta to another capital structure, unlever it first and then relever it at the new D/E.
Answer: Q = 750 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: MS = 28.57 %
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).
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