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 €20 000 at a nominal rate of 3.6% per year, compounded quarterly. Find the value FV of the investment after 7 years.

PV = €20 0003.6% / 4n = 7

FV = €

2.Time value of moneyHard

You deposit €2 000 at the start of every year for 27 years, the first deposit today. The account pays 8% per year. How much is in the account at the end of year 27? Find FV.

A = €2 000i = 8%n = 27

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 12 years. The yield to maturity is 5% per year. Find the price P of the bond.

F = €1 000coupon = 7%YTM = 5%n = 12

P = €

4.Bond pricing & yieldHard

A bond with a face value of €1 000 pays an annual coupon of 5% and matures in 4 years. The yield to maturity is 3.5%. Find the Macaulay duration D in years.

F = €1 000coupon = 5%n = 4YTM = 3.5%

D = years

5.CAPM & WACCMedium

A company has a debt-to-equity ratio (at market values) of 0.25. Its cost of equity is 11%, its pre-tax cost of debt is 4% and the tax rate is 25%. Find the WACC.

D/E = 0.25r_e = 11%r_d = 4%T = 25%

WACC = %

6.CAPM & WACCHard

A company has 4 000 000 shares at €56.00 each and 5 000 bonds trading at €1 000 each. The pre-tax cost of debt is 4.5% and the tax rate is 25.8%. The share’s beta is 1.2, the risk-free rate 1% and the market return 5%. Find the WACC.

β = 1.2r_f = 1%r_m = 5%r_d = 4.5%T = 25.8%

WACC = %

7.NPV & IRRMedium

A project costs €60 000 today and returns €15 000, €16 500, €22 500, €22 000, €15 500, €19 500 at the end of years 1 to 6. The cost of capital is 12%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.

r = 12%CF₀ = −60 000CF₁ = 15 000CF₂ = 16 500CF₃ = 22 500CF₄ = 22 000CF₅ = 15 500CF₆ = 19 500

DPB = years

8.NPV & IRRHard

A project costs €25 000 and returns €13 500, €13 000, €17 000 at the end of years 1 to 3. Find its internal rate of return IRR in percent.

CF₀ = −25 000CF₁ = 13 500CF₂ = 13 000CF₃ = 17 000

IRR = %

3Solutions

1.Time value of money

Answer: FV = 25703 €

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 = r/m = 0.036000000000000004/4 = 0.009

N = m·n = 4·7 = 28

Future value

FV = 20 000·(1 + 0.009)²⁸ = 20 000·1.285 = €25 702.93

Answer

FV = €25 702.93

2.Time value of money

Answer: FV = 188678 €

Ordinary annuity factor

((1 + 0.08)²⁷ − 1)/0.08 = 87.35

Annuity due

Each deposit is invested one year longer.

FV = 2 000·87.35·1.08 = €188 677.66

Answer

FV = €188 677.66

3.Bond pricing & yield

Answer: P = 1179 €

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 = 0.025, N = 24

Present value of the coupons

€35.00·(1 − 0.5529)/0.025 = €625.97

Present value of the face value

€1 000·0.5529 = €552.88

Price

P = €1 178.85

Answer

P = €1 178.85

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

4.Bond pricing & yield

Answer: D = 3.731 years

Duration: the Macaulay duration is the weighted average time at which a bond pays its cash flows, with the present values of the cash flows as weights: D = Σ t·PVₜ/P. The modified duration D/(1 + y) estimates the percentage price change for a small change in yield: %ΔP ≈ −D_mod·Δy.

Present value of each cash flow

t = 1: 50/1.035 = 48.31, t·PV = 48.31

t = 2: 50/1.071 = 46.68, t·PV = 93.35

t = 3: 50/1.109 = 45.10, t·PV = 135.29

t = 4: 1050/1.148 = 915.01, t·PV = 3 660.06

Duration

P = 1 055.10

D = 3 937.01/1 055.10 = 3.731

Answer

D = 3.731 years

5.CAPM & WACC

Answer: WACC = 9.4 %

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 from D/E

E/V = 1/(1 + 0.25) = 0.8

D/V = 0.25/(1 + 0.25) = 0.2

WACC

WACC = 0.8·11% + 0.2·4%·(1 − 0.25) = 9.4%

Answer

WACC = 9.4%

6.CAPM & WACC

Answer: WACC = 5.746 %

Market values

E = 4 000 000·€56.00 = €224 000 000

D = 5 000·€1 000 = €5 000 000

Cost of equity (CAPM)

r_e = 1% + 1.2·4% = 5.8%

Weights

V = €229 000 000, E/V = 0.9782, D/V = 0.02183

WACC

WACC = 0.9782·5.8% + 0.02183·4.5%·(1 − 0.258) = 5.746%

Answer

WACC = 5.746%

7.NPV & IRR

Answer: DPB = 4.393 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: 15 000/1.12 = 13 392.86 → Σ = 13 392.86

year 2: 16 500/1.254 = 13 153.70 → Σ = 26 546.56

year 3: 22 500/1.405 = 16 015.06 → Σ = 42 561.61

year 4: 22 000/1.574 = 13 981.40 → Σ = 56 543.01

year 5: 15 500/1.762 = 8 795.12 → Σ = 65 338.13

Fraction of the last year

DPB = 4 + 3 456.99/8 795.12 = 4.393

Answer

DPB = 4.393 years

The discounted payback is always longer than the simple payback (3.273 years).

8.NPV & IRR

Answer: IRR = 32.22 %

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(30%) = 814.75

NPV(35%) = −957.43

Narrow down

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

IRR ≈ 30 + 5·814.75/(814.75 − (−957.43)) = 32.3%

exact: NPV(32.22%) = 0 → IRR = 32.22%

Answer

IRR = 32.22%

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