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 €12 000 at a nominal rate of 2% per year, compounded monthly. Find the value FV of the investment after 9 years.

PV = €12 0002% / 12n = 9

FV = €

2.Time value of moneyHard

A loan of €10 000 is repaid with equal monthly payments over 4 years at 4.2% per year, compounded monthly. How much interest I is paid in total over the life of the loan?

P = €10 0004.2% / 12N = 48

I = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays an annual coupon of 6% and matures in 9 years. Its yield to maturity changes from 8% to 6%. By how much ΔP does the bond price change?

F = €1 000coupon = 6%n = 9y₁ = 8%y₂ = 6%

ΔP = €

4.Bond pricing & yieldHard

A bond trades at €950 with a yield to maturity of 4% (annual compounding) and a Macaulay duration of 5.4 years. Use the modified duration to estimate the new price P if the yield falls by 25 basis points.

P₀ = €950y = 4%D = 5.4Δy = −25 bp

P = €

5.CAPM & WACCMedium

A company has equity worth €8 000 000 and debt worth €4 000 000. Its cost of equity is 10%, its pre-tax cost of debt is 4.5% and the tax rate is 21%. Find the WACC.

E = €8 000 000D = €4 000 000r_e = 10%r_d = 4.5%T = 21%

WACC = %

6.CAPM & WACCHard

A company has 5 000 000 shares at €23.00 each and 10 000 bonds trading at €980 each. The pre-tax cost of debt is 6% and the tax rate is 30%. The share’s beta is 1.2, the risk-free rate 1.5% and the market return 7%. Find the WACC.

β = 1.2r_f = 1.5%r_m = 7%r_d = 6%T = 30%

WACC = %

7.NPV & IRRMedium

A project costs €50 000 today and returns €18 500, €16 500, €10 000, €19 500, €20 000, €21 000 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 = −50 000CF₁ = 18 500CF₂ = 16 500CF₃ = 10 000CF₄ = 19 500CF₅ = 20 000CF₆ = 21 000

DPB = years

8.NPV & IRRHard

An investment of €40 000 returns €8 500 at the end of each year for 9 years. Find its internal rate of return IRR in percent.

C₀ = €40 000CF = €8 500n = 9

IRR = %

3Solutions

1.Time value of money

Answer: FV = 14364 €

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 = 0.02/12 = 0.001667

N = m·n = 12·9 = 108

Future value

FV = 12 000·(1 + 0.001667)¹⁰⁸ = 12 000·1.197 = €14 364.46

Answer

FV = €14 364.46

2.Time value of money

Answer: I = 881 €

Monthly payment

i = 0.0035, N = 48

A = 10 000·0.0035/(1 − (1 + 0.0035)⁻⁴⁸) = €226.69

Total interest

I = N·A − P = 48·226.7 − 10 000 = €880.96

Answer

I = €880.96

3.Bond pricing & yield

Answer: ΔP = 124.9 €

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.

Price at the old yield

C = €60.00, i = 0.08, N = 9

€60.00·(1 − 0.5002)/0.08 + €1 000·0.5002 = €374.81 + €500.25 = €875.06

Price at the new yield

C = €60.00, i = 0.06, N = 9

€60.00·(1 − 0.5919)/0.06 + €1 000·0.5919 = €408.10 + €591.90 = €1 000.00

Change

ΔP = 1 000.00 − 875.06 = €124.94

Answer

ΔP = €124.94

4.Bond pricing & yield

Answer: P = 962.3 €

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.

Modified duration

D_mod = 5.4/1.04 = 5.192

Estimated price change

%ΔP ≈ −5.192·(−0.0025) = 1.298%

P ≈ 950·(1 + 0.01298) = €962.33

Answer

P = €962.33

Duration is a linear estimate; for large yield changes convexity makes the true price a little higher.

5.CAPM & WACC

Answer: WACC = 7.852 %

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 = €12 000 000, E/V = 0.6667, D/V = 0.3333

WACC

WACC = 0.6667·10% + 0.3333·4.5%·(1 − 0.21) = 7.852%

Answer

WACC = 7.852%

6.CAPM & WACC

Answer: WACC = 7.794 %

Market values

E = 5 000 000·€23.00 = €115 000 000

D = 10 000·€980 = €9 800 000

Cost of equity (CAPM)

r_e = 1.5% + 1.2·5.5% = 8.1%

Weights

V = €124 800 000, E/V = 0.9215, D/V = 0.07853

WACC

WACC = 0.9215·8.1% + 0.07853·6%·(1 − 0.3) = 7.794%

Answer

WACC = 7.794%

7.NPV & IRR

Answer: DPB = 4.072 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: 18 500/1.12 = 16 517.86 → Σ = 16 517.86

year 2: 16 500/1.254 = 13 153.70 → Σ = 29 671.56

year 3: 10 000/1.405 = 7 117.80 → Σ = 36 789.36

year 4: 19 500/1.574 = 12 392.60 → Σ = 49 181.96

year 5: 20 000/1.762 = 11 348.54 → Σ = 60 530.50

Fraction of the last year

DPB = 4 + 818.04/11 348.54 = 4.072

Answer

DPB = 4.072 years

The discounted payback is always longer than the simple payback (3.256 years).

8.NPV & IRR

Answer: IRR = 15.39 %

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.

Required annuity factor

(1 − (1 + i)⁻⁹)/i = 40 000/8 500 = 4.706

Solve for the rate

The annuity factor falls as the rate rises; bracket the rate and narrow down.

i = 15%: 4.772

i = 16%: 4.607

IRR = 15.39%

Answer

IRR = 15.39%

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