Investments and Valuation: cram pack

Formula sheets, the key method steps for every topic and 8 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.

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

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.

2Practice problems

1.Time value of moneyMedium

You invest €10 000 at 7% per year with continuous compounding. Find the value FV after 17 years.

PV = €10 000r = 7%t = 17

FV = €

2.Time value of moneyHard

You save €150 at the end of every month for 15 years. The account pays 3% per year, compounded monthly. How much have you saved at the end? Find FV.

A = €1503% / 12N = 180

FV = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays a 7% annual coupon in two half-yearly payments and matures in 11 years. The yield to maturity is 3.5% per year. Find the price P of the bond.

F = €1 000coupon = 7%YTM = 3.5%n = 11

P = €

4.Bond pricing & yieldHard

A bond with a face value of €1 000 and an annual coupon of 3% matures in 8 years and sells for €900. Find its yield to maturity YTM in percent.

F = €1 000coupon = 3%n = 8P = €900

YTM = %

5.CAPM & WACCMedium

A company is financed with ordinary shares worth €4 000 000, preferred shares worth €1 000 000 and debt worth €4 000 000. The cost of equity is 10%, the cost of preferred stock 6% and the pre-tax cost of debt 4%. The tax rate is 25%. Find the WACC.

E = €4 000 000P_s = €1 000 000D = €4 000 000r_e = 10%r_p = 6%r_d = 4%T = 25%

WACC = %

6.CAPM & WACCHard

A company's shares have a beta of 0.9 at its current debt-to-equity ratio of 0.5. It plans to raise its debt-to-equity ratio to 1.5. The tax rate is 25%, the risk-free rate 3% and the market risk premium 5%. Find the cost of equity r_e after the change.

β_L = 0.9D/E₁ = 0.5D/E₂ = 1.5T = 25%r_f = 3%r_m − r_f = 5%

r_e = %

7.NPV & IRRMedium

A project costs €50 000 today and returns €13 500 at the end of each year for 5 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 = 13 500CF₂ = 13 500CF₃ = 13 500CF₄ = 13 500CF₅ = 13 500

DPB = years

8.NPV & IRRHard

A project costs €25 000 and returns €18 500, €13 000 at the end of years 1 to 2. Find its internal rate of return IRR in percent.

CF₀ = −25 000CF₁ = 18 500CF₂ = 13 000

IRR = %

3Solutions

1.Time value of money

Answer: FV = 32871 €

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.

Exponent

r·t = 0.07·17 = 1.19

e^1.19 = 3.287

Future value

FV = 10 000·3.287 = €32 870.81

Answer

FV = €32 870.81

2.Time value of money

Answer: FV = 34046 €

Monthly rate and number of deposits

i = 0.03/12 = 0.0025

N = 12·15 = 180

Future value of the deposits

FV = A·((1 + i)^N − 1)/i = 150·(1.567 − 1)/0.0025 = €34 045.90

Answer

FV = €34 045.90

You paid in €27 000 yourself; the rest is interest.

3.Bond pricing & yield

Answer: P = 1317 €

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 = €35.00, i = 0.0175, N = 22

Present value of the coupons

€35.00·(1 − 0.6827)/0.0175 = €634.56

Present value of the face value

€1 000·0.6827 = €682.72

Price

P = €1 317.28

Answer

P = €1 317.28

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

4.Bond pricing & yield

Answer: YTM = 4.517 %

Equation for the yield

€900 = €30·(1 − (1 + y)⁻⁸)/y + €1 000/(1 + y)⁸

Solve numerically

P(4%) = €932.67, P(5%) = €870.74

YTM = 4.517%

Answer

YTM = 4.517%

5.CAPM & WACC

Answer: WACC = 6.444 %

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 = 4 000 000 + 1 000 000 + 4 000 000 = €9 000 000

E/V = 0.4444, P_s/V = 0.1111, D/V = 0.4444

WACC

WACC = 0.4444·10% + 0.1111·6% + 0.4444·4%·(1 − 0.25) = 6.444%

Answer

WACC = 6.444%

6.CAPM & WACC

Answer: r_e = 9.955 %

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 + 0.75·0.5) = 0.6545

Relever at the new D/E

β_L = 0.6545·(1 + 0.75·1.5) = 1.391

Cost of equity (CAPM)

r_e = 3% + 1.391·5% = 9.955%

Answer

r_e = 9.955%

7.NPV & IRR

Answer: DPB = 4.575 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: 13 500/1.08 = 12 500.00 → Σ = 12 500.00

year 2: 13 500/1.166 = 11 574.07 → Σ = 24 074.07

year 3: 13 500/1.26 = 10 716.74 → Σ = 34 790.81

year 4: 13 500/1.36 = 9 922.90 → Σ = 44 713.71

year 5: 13 500/1.469 = 9 187.87 → Σ = 53 901.59

Fraction of the last year

DPB = 4 + 5 286.29/9 187.87 = 4.575

Answer

DPB = 4.575 years

The discounted payback is always longer than the simple payback (3.704 years).

8.NPV & IRR

Answer: IRR = 18.05 %

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.

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%) = 916.82

NPV(20%) = −555.56

Narrow down

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

IRR ≈ 15 + 5·916.82/(916.82 − (−555.56)) = 18.11%

exact: NPV(18.05%) = 0 → IRR = 18.05%

Answer

IRR = 18.05%

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