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

A credit card charges a nominal interest rate of 25% per year, compounded daily. Find the effective annual rate EAR in percent.

r = 25%m = 365

EAR = %

2.Time value of moneyHard

A loan of €25 000 is repaid with equal monthly payments over 3 years. The interest rate is 6% per year, compounded monthly. Find the monthly payment A.

P = €25 0006% / 12N = 36

A = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 matures in 12 years and pays annual coupons. It sells for €1 043 when the yield to maturity is 5.5%. Find the coupon rate c in percent.

F = €1 000n = 12P = €1 043YTM = 5.5%

c = %

4.Bond pricing & yieldHard

A bond with a face value of €1 000 and an annual coupon of 8% matures in 6 years and sells for €1 075. Find its yield to maturity YTM in percent.

F = €1 000coupon = 8%n = 6P = €1 075

YTM = %

5.CAPM & WACCMedium

A company has equity worth €8 000 000 and debt worth €3 000 000. Its WACC is 11.1%, its pre-tax cost of debt is 4.5% and the tax rate is 25%. Find the cost of equity r_e.

E = €8 000 000D = €3 000 000WACC = 11.1%r_d = 4.5%T = 25%

r_e = %

6.CAPM & WACCHard

A company has equity with a market value of €6 000 000 and a cost of equity of 13%. Its debt consists of 8 000 bonds (face value €1 000, annual coupon 4%, 8 years to maturity) that trade at €1 108 each. The tax rate is 21%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.

E = €6 000 000r_e = 13%bonds: 8 000 × €1 108coupon = 4%n = 8T = 21%

WACC = %

7.NPV & IRRMedium

A machine costs €60 000 today, lasts 4 years and costs €2 000 per year to run (paid at the end of each year). The cost of capital is 8%. Find its equivalent annual cost EAC.

C₀ = €60 000M = €2 000n = 4r = 8%

EAC = €

8.NPV & IRRHard

Project A costs €30 000 and returns €17 000, €12 000, €7 000 at the end of years 1 to 3. Project B costs €35 000 and returns €10 500, €13 500, €18 500. Find the crossover rate r* at which both projects have the same NPV, in percent.

A: −30 000, 17 000, 12 000, 7 000B: −35 000, 10 500, 13 500, 18 500

r* = %

3Solutions

1.Time value of money

Answer: EAR = 28.39 %

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 per period

r/m = 0.25/365 = 0.0006849

Compound over one year

EAR = (1 + 0.0006849)³⁶⁵ − 1 = 1.284 − 1 = 28.39%

Answer

EAR = 28.39%

The EAR is always above the nominal rate when interest compounds more than once a year.

2.Time value of money

Answer: A = 760.5 €

Monthly rate and number of payments

i = 0.06/12 = 0.005

N = 12·3 = 36

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 = 0.8356

A = 25 000·0.005/(1 − 0.8356) = €760.55

Answer

A = €760.55

3.Bond pricing & yield

Answer: c = 5.999 %

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 + 0.055)⁻¹² = €525.98

Coupon

C = (1043 − 525.98)/8.619 = €59.99

Coupon rate

c = 59.99/1000 = 5.999%

Answer

c = 5.999%

4.Bond pricing & yield

Answer: YTM = 6.453 %

Equation for the yield

€1 075 = €80·(1 − (1 + y)⁻⁶)/y + €1 000/(1 + y)⁶

Solve numerically

P(6%) = €1 098.35, P(7%) = €1 047.67

YTM = 6.453%

Answer

YTM = 6.453%

5.CAPM & WACC

Answer: r_e = 14 %

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 = €11 000 000, E/V = 0.7273, D/V = 0.2727

Debt part of the WACC

0.2727·4.5%·(1 − 0.25) = 0.9205%

Cost of equity

r_e = (11.1% − 0.9205%)/0.7273 = 14%

Answer

r_e = 14%

6.CAPM & WACC

Answer: WACC = 6.423 %

Cost of debt (YTM)

1108 = 40·(1 − (1 + y)⁻⁸)/y + 1000/(1 + y)⁸

y = 2.494%

Market values

D = 8 000·€1 108 = €8 864 000

V = €14 864 000, E/V = 0.4037, D/V = 0.5963

WACC

WACC = 0.4037·13% + 0.5963·2.494%·(1 − 0.21) = 6.423%

Answer

WACC = 6.423%

7.NPV & IRR

Answer: EAC = 20115 €

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 − 1.08⁻⁴)/0.08 = 3.312

Annualised purchase price

60 000/3.312 = 18 115.25

EAC

EAC = 18 115.25 + 2 000 = €20 115.25

Answer

EAC = €20 115.25

8.NPV & IRR

Answer: r* = 5.419 %

Crossover rate: the discount rate at which two projects have the same NPV. It is the IRR of the difference in their cash flows (B − A). Below it the project with the later, larger cash flows has the higher NPV; above it the other one wins.

Difference in cash flows

ΔCF₀ = −35 000 − (−30 000) = −5 000

ΔCF₁ = 10 500 − 17 000 = −6 500

ΔCF₂ = 13 500 − 12 000 = 1 500

ΔCF₃ = 18 500 − 7 000 = 11 500

IRR of the difference

NPV_Δ(5%) = 104.20

NPV_Δ(6%) = −141.46

r* = 5.419%

Answer

r* = 5.419%

Check: at 5.419% both NPVs are €2 899.12.

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