Corporate Finance: cram pack

Formula sheets, the key method steps for every topic and 10 practice problems, with full solutions at the back.

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1Formulas and methods

Time value of money

Future value: FV = PV·(1 + i)ⁿ
Present value: PV = FV/(1 + i)ⁿ
Several periods per year: i = r/m, N = m·n
Annuity, present value: PV = PMT·[1 − (1 + i)⁻ⁿ]/i
Annuity, future value: FV = PMT·[(1 + i)ⁿ − 1]/i

Method

Draw a timeline with every cash flow and its timing.
Match the rate to the period: a monthly rate for monthly payments.
Use FV = PV·(1 + i)ⁿ for single amounts and the annuity formulas for equal payments.
Check the sign and the order of magnitude of the answer.

NPV & IRR

Net present value: NPV = −C₀ + Σ CF_t/(1 + r)^t
Equal cash flows: NPV = −C₀ + CF·[1 − (1 + r)⁻ⁿ]/r
Internal rate of return: the r at which NPV = 0
Decision: accept when NPV > 0 (or IRR > required return)

Method

List all cash flows with their year; the investment at year 0 is negative.
Discount each cash flow: CF_t/(1 + r)^t.
Add them up to get the NPV. Positive means the project earns more than r.
For the IRR, find the rate that makes NPV zero by trial or interpolation.

Bond pricing & yield

Bond price: P = C·(1 − (1 + y)⁻ⁿ)/y + F/(1 + y)ⁿ
Semi-annual: C/2, y/2 and 2n periods
Current yield: C/P
Price and yield: yield up → price down

Method

Find the coupon per period and the yield per period.
Price = present value of the coupons (an annuity) + present value of the face value.
For semi-annual coupons use half the coupon, half the yield and twice the periods.
Check: if the yield is above the coupon rate, the price is below face value.

CAPM & WACC

CAPM: r_e = r_f + β·(r_m − r_f)
WACC: E/V·r_e + D/V·r_d·(1 − T)
Firm value: V = E + D

Break-even analysis

Contribution margin: CM = p − v
Break-even quantity: Q_BE = FC/(p − v)
Target profit: Q = (FC + profit)/(p − v)
Profit: π = (p − v)·Q − FC
Margin of safety: (Q − Q_BE)/Q

2Practice problems

1.Time value of moneyMedium

An annuity pays €1 500 at the end of each year for 16 years. The interest rate is 3% per year. Find the present value PV of the annuity.

A = €1 500i = 3%n = 16

PV = €

2.Time value of moneyHard

You deposit €5 000 at the start of every year for 21 years, the first deposit today. The account pays 4% per year. How much is in the account at the end of year 21? Find FV.

A = €5 000i = 4%n = 21

FV = €

3.NPV & IRRMedium

A project costs €50 000 today and returns €16 000 at the end of each year for 4 years. The cost of capital is 8%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.

r = 8%CF₀ = −50 000CF₁ = 16 000CF₂ = 16 000CF₃ = 16 000CF₄ = 16 000

DPB = years

4.NPV & IRRHard

Project A costs €30 000 and returns €16 000, €12 000, €7 000 at the end of years 1 to 3. Project B costs €32 000 and returns €8 000, €12 500, €17 500. Find the crossover rate r* at which both projects have the same NPV, in percent.

A: −30 000, 16 000, 12 000, 7 000B: −32 000, 8 000, 12 500, 17 500

r* = %

5.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays a 4.5% annual coupon in two half-yearly payments and matures in 10 years. The yield to maturity is 5% per year. Find the price P of the bond.

F = €1 000coupon = 4.5%YTM = 5%n = 10

P = €

6.Bond pricing & yieldHard

A bond with a face value of €1 000 and an annual coupon of 4% matures in 8 years and sells for €1 145. Find its yield to maturity YTM in percent.

F = €1 000coupon = 4%n = 8P = €1 145

YTM = %

7.CAPM & WACCMedium

A company has equity worth €12 000 000 and debt worth €2 000 000. Its cost of equity is 10%, its pre-tax cost of debt is 6% and the tax rate is 21%. Find the WACC.

E = €12 000 000D = €2 000 000r_e = 10%r_d = 6%T = 21%

WACC = %

8.CAPM & WACCHard

A company's business has an unlevered (asset) beta of 1. It plans a debt-to-equity ratio of 0.25. The tax rate is 21%, the risk-free rate is 3.5% and the market risk premium is 7%. Find the cost of equity r_e at the new capital structure.

β_U = 1D/E = 0.25T = 21%r_f = 3.5%r_m − r_f = 7%

r_e = %

9.Break-even analysisMedium

A company sells 12 000 units at €45 each. The variable cost is €30 per unit and fixed costs are €60 000. Find the degree of operating leverage DOL at this sales level.

Q = 12 000p = €45v = €30F = €60 000

DOL =

10.Break-even analysisHard

A company sells two products. Product A sells for €17 with a variable cost of €12; product B sells for €35 with a variable cost of €25. They sell in a fixed mix of 3 units of A for every 2 of B. Fixed costs are €7 000 per month. Find the total number of units Q (A and B together) needed to break even.

p_A = €17, v_A = €12p_B = €35, v_B = €25A : B = 3 : 2F = €7 000

Q = units

3Solutions

1.Time value of money

Answer: PV = 18842 €

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.

Annuity factor

(1 − (1 + i)⁻ⁿ)/i = (1 − 0.6232)/0.03 = 12.56

Present value

PV = 1 500·12.56 = €18 841.65

Answer

PV = €18 841.65

2.Time value of money

Answer: FV = 166240 €

Ordinary annuity factor

((1 + 0.04)²¹ − 1)/0.04 = 31.97

Annuity due

Each deposit is invested one year longer.

FV = 5 000·31.97·1.04 = €166 239.85

Answer

FV = €166 239.85

3.NPV & IRR

Answer: DPB = 3.745 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.

Discounted and cumulative cash flows

year 1: 16 000/1.08 = 14 814.81 → Σ = 14 814.81

year 2: 16 000/1.166 = 13 717.42 → Σ = 28 532.24

year 3: 16 000/1.26 = 12 701.32 → Σ = 41 233.55

year 4: 16 000/1.36 = 11 760.48 → Σ = 52 994.03

Fraction of the last year

DPB = 3 + 8 766.45/11 760.48 = 3.745

Answer

DPB = 3.745 years

The discounted payback is always longer than the simple payback (3.125 years).

4.NPV & IRR

Answer: r* = 4.517 %

Crossover rate: the discount rate at which two projects have the same NPV. It is the IRR of the difference in their cash flows (B − A). Below it the project with the later, larger cash flows has the higher NPV; above it the other one wins.

Difference in cash flows

ΔCF₀ = −32 000 − (−30 000) = −2 000

ΔCF₁ = 8 000 − 16 000 = −8 000

ΔCF₂ = 12 500 − 12 000 = 500

ΔCF₃ = 17 500 − 7 000 = 10 500

IRR of the difference

NPV_Δ(4%) = 104.43

NPV_Δ(5%) = −95.24

r* = 4.517%

Answer

r* = 4.517%

Check: at 4.517% both NPVs are €2 424.56.

5.Bond pricing & yield

Answer: P = 961 €

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.

Per period

C = €22.50, i = 0.025, N = 20

Present value of the coupons

€22.50·(1 − 0.6103)/0.025 = €350.76

Present value of the face value

€1 000·0.6103 = €610.27

Price

P = €961.03

Answer

P = €961.03

The yield is above the coupon rate, so the bond sells at a discount.

6.Bond pricing & yield

Answer: YTM = 2.019 %

Equation for the yield

€1 145 = €40·(1 − (1 + y)⁻⁸)/y + €1 000/(1 + y)⁸

Solve numerically

P(2%) = €1 146.51, P(3%) = €1 070.20

YTM = 2.019%

Answer

YTM = 2.019%

7.CAPM & WACC

Answer: WACC = 9.249 %

Cost of capital: CAPM gives the return shareholders require: r_e = r_f + β·(r_m − r_f). The weighted average cost of capital mixes equity and debt by their market values: WACC = E/V·r_e + D/V·r_d·(1 − T), with V = E + D. Interest is tax-deductible, hence the (1 − T).

Weights

V = €14 000 000, E/V = 0.8571, D/V = 0.1429

WACC

WACC = 0.8571·10% + 0.1429·6%·(1 − 0.21) = 9.249%

Answer

WACC = 9.249%

8.CAPM & WACC

Answer: r_e = 11.88 %

Levered beta (Hamada): debt makes equity riskier. β_L = β_U·(1 + (1 − T)·D/E), where β_U 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.

Levered beta

β_L = 1·(1 + 0.79·0.25) = 1.198

Cost of equity (CAPM)

r_e = 3.5% + 1.198·7% = 11.88%

Answer

r_e = 11.88%

9.Break-even analysis

Answer: DOL = 1.5

Operating leverage: fixed costs make profit react more strongly than sales. The degree of operating leverage DOL = contribution margin/operating profit tells you that a 1% change in sales changes profit by DOL %. It is very high just above break-even and falls as sales grow.

Total contribution margin

CM = 12 000·(45 − 30) = €180 000

Operating profit

Π = 180 000 − 60 000 = €120 000

DOL

DOL = 180 000/120 000 = 1.5

Answer

DOL = 1.5

10.Break-even analysis

Answer: Q = 1000 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.

Contribution margins

cm_A = €5, cm_B = €10

Weighted contribution margin

(3·5 + 2·10)/5 = €7.00

Break-even units

Q = 7 000/7 = 1 000

A: 600, B: 400

Answer

Q = 1 000 units

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