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 €15 000 in 8 years. An account pays 2% per year, compounded quarterly. How much must you deposit today? Find the present value PV.

FV = €15 0002% / 4n = 8

PV = €

2.Time value of moneyHard

A loan of €150 000 is repaid with equal monthly payments over 20 years. The interest rate is 6% per year, compounded monthly. Find the monthly payment A.

P = €150 0006% / 12N = 240

A = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 matures in 9 years and pays annual coupons. It sells for €760 when the yield to maturity is 9%. Find the coupon rate c in percent.

F = €1 000n = 9P = €760YTM = 9%

c = %

4.Bond pricing & yieldHard

A bond with a face value of €1 000 pays an annual coupon of 4% and matures in 5 years. The yield to maturity is 2.5%. Find the Macaulay duration D in years.

F = €1 000coupon = 4%n = 5YTM = 2.5%

D = years

5.CAPM & WACCMedium

A share with a beta of 1.25 has a required return of 7.88%. The risk-free rate is 1%. What expected market return r_m does CAPM imply?

β = 1.25r_e = 7.88%r_f = 1%

r_m = %

6.CAPM & WACCHard

A company has 5 000 000 shares at €13.00 each and 5 000 bonds trading at €1 020 each. The pre-tax cost of debt is 5% and the tax rate is 28%. The share’s beta is 0.8, the risk-free rate 2% and the market return 8.5%. Find the WACC.

β = 0.8r_f = 2%r_m = 8.5%r_d = 5%T = 28%

WACC = %

7.NPV & IRRMedium

A project costs €100 000 today. It generates €9 000 at the end of the first year, and the cash flow then grows by 4% per year forever. The cost of capital is 12%. Find the net present value NPV.

C₀ = €100 000CF₁ = €9 000g = 4%r = 12%

NPV = €

8.NPV & IRRHard

A project costs €25 000 and returns €11 000, €13 000, €17 500 at the end of years 1 to 3. Find its internal rate of return IRR in percent.

CF₀ = −25 000CF₁ = 11 000CF₂ = 13 000CF₃ = 17 500

IRR = %

3Solutions

1.Time value of money

Answer: PV = 12787 €

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 = 0.02/4 = 0.005

N = 4·8 = 32

Present value

PV = FV/(1 + i)^N = 15 000/1.173 = €12 787.25

Answer

PV = €12 787.25

2.Time value of money

Answer: A = 1075 €

Monthly rate and number of payments

i = 0.06/12 = 0.005

N = 12·20 = 240

Payment

The present value of N equal payments A is A·(1 − (1 + i)⁻ᴺ)/i. Set it equal to the loan and solve for A.

(1 + i)^−N = 0.3021

A = 150 000·0.005/(1 − 0.3021) = €1 074.65

Answer

A = €1 074.65

3.Bond pricing & yield

Answer: c = 4.997 %

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 + 0.09)⁻⁹ = €460.43

Coupon

C = (760 − 460.43)/5.995 = €49.97

Coupon rate

c = 49.97/1000 = 4.997%

Answer

c = 4.997%

4.Bond pricing & yield

Answer: D = 4.644 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: 40/1.025 = 39.02, t·PV = 39.02

t = 2: 40/1.051 = 38.07, t·PV = 76.15

t = 3: 40/1.077 = 37.14, t·PV = 111.43

t = 4: 40/1.104 = 36.24, t·PV = 144.95

t = 5: 1040/1.131 = 919.21, t·PV = 4 596.04

Duration

P = 1 069.69

D = 4 967.60/1 069.69 = 4.644

Answer

D = 4.644 years

5.CAPM & WACC

Answer: r_m = 6.504 %

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 = (7.88% − 1%)/1.25 = 5.504%

Market return

r_m = 1% + 5.504% = 6.504%

Answer

r_m = 6.504%

6.CAPM & WACC

Answer: WACC = 6.938 %

Market values

E = 5 000 000·€13.00 = €65 000 000

D = 5 000·€1 020 = €5 100 000

Cost of equity (CAPM)

r_e = 2% + 0.8·6.5% = 7.2%

Weights

V = €70 100 000, E/V = 0.9272, D/V = 0.07275

WACC

WACC = 0.9272·7.2% + 0.07275·5%·(1 − 0.28) = 6.938%

Answer

WACC = 6.938%

7.NPV & IRR

Answer: NPV = 12500 €

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.

Value of the cash flows

PV = 9 000/(0.12 − 0.04) = 112 500.00

NPV

NPV = 112 500.00 − 100 000 = €12 500.00

Answer

NPV = €12 500.00

NPV is positive, so the project earns more than the required return.

8.NPV & IRR

Answer: IRR = 27.67 %

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(25%) = 1 080.00

NPV(30%) = −880.75

Narrow down

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

IRR ≈ 25 + 5·1 080.00/(1 080.00 − (−880.75)) = 27.75%

exact: NPV(27.67%) = 0 → IRR = 27.67%

Answer

IRR = 27.67%

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