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 need €50 000 in 9 years. An account pays 4.8% per year, compounded monthly. How much must you deposit today? Find the present value PV.

FV = €50 0004.8% / 12n = 9

PV = €

2.Time value of moneyHard

You borrow €5 000 at 3.6% per year, compounded monthly, and repay €125 at the end of every month. How many months does it take until the loan is repaid (the last payment may be smaller)? Give the number of payments N.

P = €5 000A = €1253.6% / 12

N = payments

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 matures in 6 years and pays annual coupons. It sells for €1 105 when the yield to maturity is 4%. Find the coupon rate c in percent.

F = €1 000n = 6P = €1 105YTM = 4%

c = %

4.Bond pricing & yieldHard

A bond with a face value of €1 000 pays a 5% annual coupon in two half-yearly payments and matures in 10 years. It sells for €1 220. Find its yield to maturity YTM as an annual rate (twice the half-yearly yield), in percent.

F = €1 000coupon = 5% (2×)n = 10P = €1 220

YTM = %

5.CAPM & WACCMedium

A company has a debt-to-equity ratio (at market values) of 1. Its cost of equity is 12%, its pre-tax cost of debt is 3% and the tax rate is 28%. Find the WACC.

D/E = 1r_e = 12%r_d = 3%T = 28%

WACC = %

6.CAPM & WACCHard

A company's shares have a beta of 1.2 at its current debt-to-equity ratio of 0.5. It plans to raise its debt-to-equity ratio to 0.75. The tax rate is 25%, the risk-free rate 2% and the market risk premium 6%. Find the cost of equity r_e after the change.

β_L = 1.2D/E₁ = 0.5D/E₂ = 0.75T = 25%r_f = 2%r_m − r_f = 6%

r_e = %

7.NPV & IRRMedium

A project costs €20 000 today and returns €6 000, €4 500, €4 500, €7 000, €7 000 at the end of years 1 to 5. The cost of capital is 10%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.

r = 10%CF₀ = −20 000CF₁ = 6 000CF₂ = 4 500CF₃ = 4 500CF₄ = 7 000CF₅ = 7 000

DPB = years

8.NPV & IRRHard

A project costs €25 000 and returns €13 500, €14 000 at the end of years 1 to 2. Find its internal rate of return IRR in percent.

CF₀ = −25 000CF₁ = 13 500CF₂ = 14 000

IRR = %

3Solutions

1.Time value of money

Answer: PV = 32488 €

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 = 0.048/12 = 0.004

N = 12·9 = 108

Present value

PV = FV/(1 + i)^N = 50 000/1.539 = €32 488.45

Answer

PV = €32 488.45

2.Time value of money

Answer: N = 43 payments

Rearrange the annuity formula

i = 0.003

(1 + i)^−N = 1 − 5 000·0.003/125 = 0.88

Solve with logarithms

N = −ln(0.88)/ln(1.003) = 42.68

Round up to whole payments

42 full payments leave a small balance, which the last, smaller payment clears.

N = 43

Answer

N = 43 payments

3.Bond pricing & yield

Answer: c = 6.003 %

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)⁻⁶ = €790.31

Coupon

C = (1105 − 790.31)/5.242 = €60.03

Coupon rate

c = 60.03/1000 = 6.003%

Answer

c = 6.003%

4.Bond pricing & yield

Answer: YTM = 2.5 %

Per period

C = €25.00, N = 20

Solve numerically

i = 1.25% per half-year

YTM

YTM = 2·1.25% = 2.5%

Answer

YTM = 2.5%

5.CAPM & WACC

Answer: WACC = 7.08 %

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 + 1) = 0.5

D/V = 1/(1 + 1) = 0.5

WACC

WACC = 0.5·12% + 0.5·3%·(1 − 0.28) = 7.08%

Answer

WACC = 7.08%

6.CAPM & WACC

Answer: r_e = 10.18 %

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 = 1.2/(1 + 0.75·0.5) = 0.8727

Relever at the new D/E

β_L = 0.8727·(1 + 0.75·0.75) = 1.364

Cost of equity (CAPM)

r_e = 2% + 1.364·6% = 10.18%

Answer

r_e = 10.18%

7.NPV & IRR

Answer: DPB = 4.613 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: 6 000/1.1 = 5 454.55 → Σ = 5 454.55

year 2: 4 500/1.21 = 3 719.01 → Σ = 9 173.55

year 3: 4 500/1.331 = 3 380.92 → Σ = 12 554.47

year 4: 7 000/1.464 = 4 781.09 → Σ = 17 335.56

year 5: 7 000/1.611 = 4 346.45 → Σ = 21 682.01

Fraction of the last year

DPB = 4 + 2 664.44/4 346.45 = 4.613

Answer

DPB = 4.613 years

The discounted payback is always longer than the simple payback (3.714 years).

8.NPV & IRR

Answer: IRR = 6.555 %

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(5%) = 555.56

NPV(10%) = −1 157.02

Narrow down

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

IRR ≈ 5 + 5·555.56/(555.56 − (−1 157.02)) = 6.622%

exact: NPV(6.555%) = 0 → IRR = 6.555%

Answer

IRR = 6.555%

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