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 €25 000 in 3 years. An account pays 2% per year, compounded monthly. How much must you deposit today? Find the present value PV.

FV = €25 0002% / 12n = 3

PV = €

2.Time value of moneyHard

You save €50 at the end of every month for 26 years. The account pays 3.6% per year, compounded monthly. How much have you saved at the end? Find FV.

A = €503.6% / 12N = 312

FV = €

3.NPV & IRRMedium

A project costs €20 000 today and returns €6 500, €7 500, €5 500, €7 000, €7 000 at the end of years 1 to 5. 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 = −20 000CF₁ = 6 500CF₂ = 7 500CF₃ = 5 500CF₄ = 7 000CF₅ = 7 000

DPB = years

4.NPV & IRRHard

Project A costs €10 000 and returns €6 000, €4 000, €2 500 at the end of years 1 to 3. Project B costs €15 000 and returns €4 500, €6 000, €7 500. Find the crossover rate r* at which both projects have the same NPV, in percent.

A: −10 000, 6 000, 4 000, 2 500B: −15 000, 4 500, 6 000, 7 500

r* = %

5.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays a 6% annual coupon in two half-yearly payments and matures in 9 years. The yield to maturity is 3.5% per year. Find the price P of the bond.

F = €1 000coupon = 6%YTM = 3.5%n = 9

P = €

6.Bond pricing & yieldHard

A bond with a face value of €1 000 and an annual coupon of 4% matures in 7 years and sells for €1 060. Find its yield to maturity YTM in percent.

F = €1 000coupon = 4%n = 7P = €1 060

YTM = %

7.CAPM & WACCMedium

A share with a beta of 0.9 has a required return of 6.95%. The risk-free rate is 2%. What expected market return r_m does CAPM imply?

β = 0.9r_e = 6.95%r_f = 2%

r_m = %

8.CAPM & WACCHard

A company's shares have a beta of 0.9 at its current debt-to-equity ratio of 0.4. It plans to raise its debt-to-equity ratio to 1. The tax rate is 21%, the risk-free rate 1% and the market risk premium 6%. Find the cost of equity r_e after the change.

β_L = 0.9D/E₁ = 0.4D/E₂ = 1T = 21%r_f = 1%r_m − r_f = 6%

r_e = %

9.Break-even analysisMedium

A product sells for €47 per unit with a variable cost of €45 per unit. Fixed costs are €3 000 per month. How many units Q must be sold per month to make a profit of €2 000?

F = €3 000p = €47v = €45profit = €2 000

Q = units

10.Break-even analysisHard

A product sells for €35 per unit, with a variable cost of €20 per unit and fixed costs of €37 500 per year. Budgeted sales are 3 130 units. Find the margin of safety MS as a percentage of budgeted sales.

F = €37 500p = €35v = €20budget = 3 130

MS = %

3Solutions

1.Time value of money

Answer: PV = 23545 €

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 = 0.02/12 = 0.001667

N = 12·3 = 36

Present value

PV = FV/(1 + i)^N = 25 000/1.062 = €23 545.29

Answer

PV = €23 545.29

2.Time value of money

Answer: FV = 25770 €

Monthly rate and number of deposits

i = 0.036000000000000004/12 = 0.003

N = 12·26 = 312

Future value of the deposits

FV = A·((1 + i)^N − 1)/i = 50·(2.546 − 1)/0.003 = €25 769.86

Answer

FV = €25 769.86

You paid in €15 600 yourself; the rest is interest.

3.NPV & IRR

Answer: DPB = 3.619 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: 6 500/1.08 = 6 018.52 → Σ = 6 018.52

year 2: 7 500/1.166 = 6 430.04 → Σ = 12 448.56

year 3: 5 500/1.26 = 4 366.08 → Σ = 16 814.64

year 4: 7 000/1.36 = 5 145.21 → Σ = 21 959.85

Fraction of the last year

DPB = 3 + 3 185.36/5 145.21 = 3.619

Answer

DPB = 3.619 years

The discounted payback is always longer than the simple payback (3.071 years).

4.NPV & IRR

Answer: r* = 3.029 %

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 = −15 000 − (−10 000) = −5 000

ΔCF₁ = 4 500 − 6 000 = −1 500

ΔCF₂ = 6 000 − 4 000 = 2 000

ΔCF₃ = 7 500 − 2 500 = 5 000

IRR of the difference

NPV_Δ(3%) = 4.59

NPV_Δ(4%) = −148.21

r* = 3.029%

Answer

r* = 3.029%

Check: at 3.029% both NPVs are €1 877.69.

5.Bond pricing & yield

Answer: P = 1192 €

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 = €30.00, i = 0.0175, N = 18

Present value of the coupons

€30.00·(1 − 0.7318)/0.0175 = €459.81

Present value of the face value

€1 000·0.7318 = €731.78

Price

P = €1 191.59

Answer

P = €1 191.59

The yield is below the coupon rate, so the bond sells at a premium.

6.Bond pricing & yield

Answer: YTM = 3.036 %

Equation for the yield

€1 060 = €40·(1 − (1 + y)⁻⁷)/y + €1 000/(1 + y)⁷

Solve numerically

P(3%) = €1 062.30, P(4%) = €1 000.00

YTM = 3.036%

Answer

YTM = 3.036%

7.CAPM & WACC

Answer: r_m = 7.5 %

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).

Market risk premium

r_m − r_f = (6.95% − 2%)/0.9 = 5.5%

Market return

r_m = 2% + 5.5% = 7.5%

Answer

r_m = 7.5%

8.CAPM & WACC

Answer: r_e = 8.345 %

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.

Unlever the beta

β_U = 0.9/(1 + 0.79·0.4) = 0.6839

Relever at the new D/E

β_L = 0.6839·(1 + 0.79·1) = 1.224

Cost of equity (CAPM)

r_e = 1% + 1.224·6% = 8.345%

Answer

r_e = 8.345%

9.Break-even analysis

Answer: Q = 2500 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).

Contribution margin per unit

p − v = 47 − 45 = €2

Units for the target profit

Q = (F + profit)/(p − v) = (3 000 + 2 000)/2 = 2 500

Answer

Q = 2 500 units

10.Break-even analysis

Answer: MS = 20.13 %

Break-even quantity

Q = F/(p − v) = 37 500/15 = 2 500

Margin of safety

MS = (3 130 − 2 500)/3 130·100 = 20.13%

Answer

MS = 20.13%

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