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

You need €40 000 in 5 years. An account pays 3% per year, compounded monthly. How much must you deposit today? Find the present value PV.

FV = €40 0003% / 12n = 5

PV = €

2.Time value of moneyHard

A loan of €8 000 is repaid with equal monthly payments over 5 years. The interest rate is 3.6% per year, compounded monthly. Find the monthly payment A.

P = €8 0003.6% / 12N = 60

A = €

3.NPV & IRRMedium

An investment of €45 000 gives these cash flows at the end of years 1 to 3: €14 000, €15 000, €20 000. The cost of capital is 15%. Find the net present value NPV.

r = 15%CF₀ = −45 000CF₁ = 14 000CF₂ = 15 000CF₃ = 20 000

NPV = €

4.NPV & IRRHard

A project costs €25 000 and returns €14 500, €16 500 at the end of years 1 to 2. Find its internal rate of return IRR in percent.

CF₀ = −25 000CF₁ = 14 500CF₂ = 16 500

IRR = %

5.Bond pricing & yieldMedium

A bond with a face value of €1 000 matures in 14 years and pays annual coupons. It sells for €1 106 when the yield to maturity is 4%. Find the coupon rate c in percent.

F = €1 000n = 14P = €1 106YTM = 4%

c = %

6.Bond pricing & yieldHard

A bond with a face value of €1 000 and an annual coupon of 3% matures in 3 years and sells for €935. Find its yield to maturity YTM in percent.

F = €1 000coupon = 3%n = 3P = €935

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 25%. Find the WACC.

E = €12 000 000D = €2 000 000r_e = 10%r_d = 6%T = 25%

WACC = %

8.CAPM & WACCHard

A company's business has an unlevered (asset) beta of 1. It plans a debt-to-equity ratio of 0.4. The tax rate is 20%, the risk-free rate is 1% and the market risk premium is 6.5%. Find the cost of equity r_e at the new capital structure.

β_U = 1D/E = 0.4T = 20%r_f = 1%r_m − r_f = 6.5%

r_e = %

9.Break-even analysisMedium

A company sells 7 500 units at €35 each. The variable cost is €25 per unit and fixed costs are €25 000. Find the degree of operating leverage DOL at this sales level.

Q = 7 500p = €35v = €25F = €25 000

DOL =

10.Break-even analysisHard

A product sells for €47 per unit, with a variable cost of €45 per unit and fixed costs of €5 000 per year. Budgeted sales are 4 000 units. Find the margin of safety MS as a percentage of budgeted sales.

F = €5 000p = €47v = €45budget = 4 000

MS = %

3Solutions

1.Time value of money

Answer: PV = 34435 €

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.

Rate and number of periods

i = 0.03/12 = 0.0025

N = 12·5 = 60

Present value

PV = FV/(1 + i)^N = 40 000/1.162 = €34 434.76

Answer

PV = €34 434.76

2.Time value of money

Answer: A = 145.9 €

Monthly rate and number of payments

i = 0.036000000000000004/12 = 0.003

N = 12·5 = 60

Payment

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.

(1 + i)^−N = 0.8355

A = 8 000·0.003/(1 − 0.8355) = €145.89

Answer

A = €145.89

3.NPV & IRR

Answer: NPV = −8334 €

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.

Present value of each cash flow

CF₀ = −45 000

CF₁/(1 + 0.15)¹ = 14 000/1.15 = 12 173.91

CF₂/(1 + 0.15)² = 15 000/1.322 = 11 342.16

CF₃/(1 + 0.15)³ = 20 000/1.521 = 13 150.32

Add them up

NPV = €−8 333.61

Answer

NPV = €−8 333.61

NPV is negative, so the project earns less than the required return.

4.NPV & IRR

Answer: IRR = 15.26 %

Bracket the IRR

NPV falls as the rate rises, so the IRR lies between a rate with positive NPV and one with negative NPV.

NPV(15%) = 85.07

NPV(20%) = −1 458.33

Narrow down

Linear interpolation gives a first estimate; repeating it (or a calculator's IRR function) gives the exact rate.

IRR ≈ 15 + 5·85.07/(85.07 − (−1 458.33)) = 15.28%

exact: NPV(15.26%) = 0 → IRR = 15.26%

Answer

IRR = 15.26%

5.Bond pricing & yield

Answer: c = 5.003 %

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.

Present value of the face value

1000·(1 + 0.04)⁻¹⁴ = €577.48

Coupon

C = (1106 − 577.48)/10.56 = €50.03

Coupon rate

c = 50.03/1000 = 5.003%

Answer

c = 5.003%

6.Bond pricing & yield

Answer: YTM = 5.405 %

Equation for the yield

€935 = €30·(1 − (1 + y)⁻³)/y + €1 000/(1 + y)³

Solve numerically

P(5%) = €945.54, P(6%) = €919.81

YTM = 5.405%

Answer

YTM = 5.405%

7.CAPM & WACC

Answer: WACC = 9.214 %

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 − 0.25) = 9.214%

Answer

WACC = 9.214%

8.CAPM & WACC

Answer: r_e = 9.58 %

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 + 0.8·0.4) = 1.32

Cost of equity (CAPM)

r_e = 1% + 1.32·6.5% = 9.58%

Answer

r_e = 9.58%

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 = 7 500·(35 − 25) = €75 000

Operating profit

Π = 75 000 − 25 000 = €50 000

DOL

DOL = 75 000/50 000 = 1.5

Answer

DOL = 1.5

10.Break-even analysis

Answer: MS = 37.5 %

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).

Break-even quantity

Q = F/(p − v) = 5 000/2 = 2 500

Margin of safety

MS = (4 000 − 2 500)/4 000·100 = 37.5%

Answer

MS = 37.5%

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