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 8% per year with continuous compounding. Find the value FV after 18 years.

PV = €20 000r = 8%t = 18

FV = €

2.Time value of moneyHard

You deposit €1 000 at the start of every year for 7 years, the first deposit today. The account pays 4% per year. How much is in the account at the end of year 7? Find FV.

A = €1 000i = 4%n = 7

FV = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 matures in 12 years and pays annual coupons. It sells for €1 154 when the yield to maturity is 2.5%. Find the coupon rate c in percent.

F = €1 000n = 12P = €1 154YTM = 2.5%

c = %

4.Bond pricing & yieldHard

A bond trades at €980 with a yield to maturity of 4% (annual compounding) and a Macaulay duration of 3.9 years. Use the modified duration to estimate the new price P if the yield rises by 25 basis points.

P₀ = €980y = 4%D = 3.9Δy = +25 bp

P = €

5.CAPM & WACCMedium

A company is financed with ordinary shares worth €10 000 000, preferred shares worth €500 000 and debt worth €4 000 000. The cost of equity is 12%, the cost of preferred stock 7.5% and the pre-tax cost of debt 3%. The tax rate is 28%. Find the WACC.

E = €10 000 000P_s = €500 000D = €4 000 000r_e = 12%r_p = 7.5%r_d = 3%T = 28%

WACC = %

6.CAPM & WACCHard

A company has 1 000 000 shares at €47.00 each and 10 000 bonds trading at €1 000 each. The pre-tax cost of debt is 4.5% and the tax rate is 30%. The share’s beta is 1.3, the risk-free rate 2.5% and the market return 6.5%. Find the WACC.

β = 1.3r_f = 2.5%r_m = 6.5%r_d = 4.5%T = 30%

WACC = %

7.NPV & IRRMedium

A machine costs €100 000 today. It generates €31 000 at the end of each year for 4 years and can then be sold for €15 000. The cost of capital is 8%. Find the net present value NPV.

C₀ = €100 000CF = €31 000S = €15 000n = 4r = 8%

NPV = €

8.NPV & IRRHard

A project costs €20 000 and returns €7 500, €7 500, €10 000 at the end of years 1 to 3. Find its internal rate of return IRR in percent.

CF₀ = −20 000CF₁ = 7 500CF₂ = 7 500CF₃ = 10 000

IRR = %

3Solutions

1.Time value of money

Answer: FV = 84414 €

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 = 0.08·18 = 1.44

e^1.44 = 4.221

Future value

FV = 20 000·4.221 = €84 413.92

Answer

FV = €84 413.92

2.Time value of money

Answer: FV = 8214 €

Ordinary annuity factor

((1 + 0.04)⁷ − 1)/0.04 = 7.898

Annuity due

Each deposit is invested one year longer.

FV = 1 000·7.898·1.04 = €8 214.23

Answer

FV = €8 214.23

3.Bond pricing & yield

Answer: c = 4.001 %

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 + 0.025)⁻¹² = €743.56

Coupon

C = (1154 − 743.56)/10.26 = €40.01

Coupon rate

c = 40.01/1000 = 4.001%

Answer

c = 4.001%

4.Bond pricing & yield

Answer: P = 970.8 €

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 = 3.9/1.04 = 3.75

Estimated price change

%ΔP ≈ −3.75·(0.0025) = −0.9375%

P ≈ 980·(1 − 0.009375) = €970.81

Answer

P = €970.81

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

5.CAPM & WACC

Answer: WACC = 9.13 %

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 = 10 000 000 + 500 000 + 4 000 000 = €14 500 000

E/V = 0.6897, P_s/V = 0.03448, D/V = 0.2759

WACC

WACC = 0.6897·12% + 0.03448·7.5% + 0.2759·3%·(1 − 0.28) = 9.13%

Answer

WACC = 9.13%

6.CAPM & WACC

Answer: WACC = 6.902 %

Market values

E = 1 000 000·€47.00 = €47 000 000

D = 10 000·€1 000 = €10 000 000

Cost of equity (CAPM)

r_e = 2.5% + 1.3·4% = 7.7%

Weights

V = €57 000 000, E/V = 0.8246, D/V = 0.1754

WACC

WACC = 0.8246·7.7% + 0.1754·4.5%·(1 − 0.3) = 6.902%

Answer

WACC = 6.902%

7.NPV & IRR

Answer: NPV = 13701 €

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.

Present value of the yearly cash flows

31 000·3.312 = 102 675.93

Present value of the salvage value

15 000/1.36 = 11 025.45

NPV

NPV = −100 000 + 102 675.93 + 11 025.45 = €13 701.38

Answer

NPV = €13 701.38

8.NPV & IRR

Answer: IRR = 11.43 %

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(10%) = 529.68

NPV(15%) = −1 232.02

Narrow down

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

IRR ≈ 10 + 5·529.68/(529.68 − (−1 232.02)) = 11.5%

exact: NPV(11.43%) = 0 → IRR = 11.43%

Answer

IRR = 11.43%

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