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

An annuity pays €5 000 at the end of each year for 17 years. The interest rate is 7% per year. Find the present value PV of the annuity.

A = €5 000i = 7%n = 17

PV = €

2.Time value of moneyHard

A mortgage of €300 000 is repaid with equal monthly payments over 25 years at 6% per year, compounded monthly. How much is still owed just after the payment at the end of year 2? Find the outstanding balance B.

P = €300 0006% / 12N = 300k = 24

B = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays a 6% annual coupon in two half-yearly payments and matures in 10 years. The yield to maturity is 2.5% per year. Find the price P of the bond.

F = €1 000coupon = 6%YTM = 2.5%n = 10

P = €

4.Bond pricing & yieldHard

A bond trades at €905 with a yield to maturity of 6% (annual compounding) and a Macaulay duration of 7.5 years. Use the modified duration to estimate the new price P if the yield rises by 25 basis points.

P₀ = €905y = 6%D = 7.5Δy = +25 bp

P = €

5.CAPM & WACCMedium

A company has a debt-to-equity ratio (at market values) of 1.5. Its cost of equity is 8%, its pre-tax cost of debt is 4.5% and the tax rate is 28%. Find the WACC.

D/E = 1.5r_e = 8%r_d = 4.5%T = 28%

WACC = %

6.CAPM & WACCHard

A company's business has an unlevered (asset) beta of 1.2. It plans a debt-to-equity ratio of 0.6. The tax rate is 20%, the risk-free rate is 4% and the market risk premium is 4.5%. Find the cost of equity r_e at the new capital structure.

β_U = 1.2D/E = 0.6T = 20%r_f = 4%r_m − r_f = 4.5%

r_e = %

7.NPV & IRRMedium

A machine costs €20 000 today, lasts 7 years and costs €1 500 per year to run (paid at the end of each year). The cost of capital is 5%. Find its equivalent annual cost EAC.

C₀ = €20 000M = €1 500n = 7r = 5%

EAC = €

8.NPV & IRRHard

A project costs €10 000 and returns €6 000, €4 000, €4 500 at the end of years 1 to 3. Cash flows can be reinvested at the cost of capital of 10%. Find the modified internal rate of return MIRR in percent.

r = 10%CF₀ = −10 000CF₁ = 6 000CF₂ = 4 000CF₃ = 4 500

MIRR = %

3Solutions

1.Time value of money

Answer: PV = 48816 €

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.

Annuity factor

(1 − (1 + i)⁻ⁿ)/i = (1 − 0.3166)/0.07 = 9.763

Present value

PV = 5 000·9.763 = €48 816.11

Answer

PV = €48 816.11

2.Time value of money

Answer: B = 288990 €

Monthly payment

i = 0.005, N = 300

A = 300 000·0.005/(1 − (1 + 0.005)⁻³⁰⁰) = €1 932.90

Present value of the remaining payments

Whatever is still owed must be exactly paid off by the remaining payments.

N − k = 300 − 24 = 276

B = 1933·(1 − (1 + 0.005)⁻²⁷⁶)/0.005 = €288 990.40

Answer

B = €288 990.40

3.Bond pricing & yield

Answer: P = 1308 €

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 = €30.00, i = 0.0125, N = 20

Present value of the coupons

€30.00·(1 − 0.78)/0.0125 = €527.98

Present value of the face value

€1 000·0.78 = €780.01

Price

P = €1 307.99

Answer

P = €1 307.99

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

4.Bond pricing & yield

Answer: P = 889 €

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 = 7.5/1.06 = 7.075

Estimated price change

%ΔP ≈ −7.075·(0.0025) = −1.769%

P ≈ 905·(1 − 0.01769) = €888.99

Answer

P = €888.99

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

5.CAPM & WACC

Answer: WACC = 5.144 %

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 + 1.5) = 0.4

D/V = 1.5/(1 + 1.5) = 0.6

WACC

WACC = 0.4·8% + 0.6·4.5%·(1 − 0.28) = 5.144%

Answer

WACC = 5.144%

6.CAPM & WACC

Answer: r_e = 11.99 %

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.

Levered beta

β_L = 1.2·(1 + 0.8·0.6) = 1.776

Cost of equity (CAPM)

r_e = 4% + 1.776·4.5% = 11.99%

Answer

r_e = 11.99%

7.NPV & IRR

Answer: EAC = 4956 €

Equivalent annual cost: spread the present value of all costs over the life as an equal yearly amount (an annuity): EAC = PV·r/(1 − (1 + r)⁻ⁿ). It lets you compare machines with different lives.

Annuity factor

(1 − 1.05⁻⁷)/0.05 = 5.786

Annualised purchase price

20 000/5.786 = 3 456.40

EAC

EAC = 3 456.40 + 1 500 = €4 956.40

Answer

EAC = €4 956.40

8.NPV & IRR

Answer: MIRR = 17.35 %

Modified IRR: compound every inflow forward to the end of the project at the reinvestment rate (usually the cost of capital) to get the terminal value TV. MIRR is the rate that turns the initial investment into TV: MIRR = (TV/C₀)^(1/n) − 1.

Terminal value of the inflows

6 000·1.1² = 7 260.00

4 000·1.1¹ = 4 400.00

4 500·1.1⁰ = 4 500.00

TV = 16 160.00

MIRR

MIRR = (16 160.00/10 000)^1/3 − 1 = 17.35%

Answer

MIRR = 17.35%

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