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 13 years. The interest rate is 10% per year. Find the present value PV of the annuity.

A = €5 000i = 10%n = 13

PV = €

2.Time value of moneyHard

You deposit €2 000 at the start of every year for 5 years, the first deposit today. The account pays 5% per year. How much is in the account at the end of year 5? Find FV.

A = €2 000i = 5%n = 5

FV = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays an annual coupon of 3% and matures in 13 years. Its yield to maturity changes from 5.5% to 6%. By how much ΔP does the bond price change?

F = €1 000coupon = 3%n = 13y₁ = 5.5%y₂ = 6%

ΔP = €

4.Bond pricing & yieldHard

A bond with 10 years to maturity (face value €1 000, annual coupon 6%) sells for €1 119. The issuer can call it in 4 years at a call price of €1 060. Find the yield to call YTC in percent.

F = €1 000coupon = 6%P = €1 119call price = €1 060call in 4 years

YTC = %

5.CAPM & WACCMedium

A share with a beta of 1.4 has a required return of 8.3%. The risk-free rate is 2%. What expected market return r_m does CAPM imply?

β = 1.4r_e = 8.3%r_f = 2%

r_m = %

6.CAPM & WACCHard

A company has 1 000 000 shares at €17.50 each and 20 000 bonds trading at €1 050 each. The pre-tax cost of debt is 3.5% and the tax rate is 20%. The share’s beta is 1.3, the risk-free rate 1.5% and the market return 7.5%. Find the WACC.

β = 1.3r_f = 1.5%r_m = 7.5%r_d = 3.5%T = 20%

WACC = %

7.NPV & IRRMedium

An investment of €75 000 gives these cash flows at the end of years 1 to 4: €30 000, €31 000, €33 000, €27 000. The cost of capital is 10%. Find the net present value NPV.

r = 10%CF₀ = −75 000CF₁ = 30 000CF₂ = 31 000CF₃ = 33 000CF₄ = 27 000

NPV = €

8.NPV & IRRHard

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

r = 6%CF₀ = −10 000CF₁ = 4 000CF₂ = 3 500CF₃ = 4 500

MIRR = %

3Solutions

1.Time value of money

Answer: PV = 35517 €

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 − 0.2897)/0.1 = 7.103

Present value

PV = 5 000·7.103 = €35 516.78

Answer

PV = €35 516.78

2.Time value of money

Answer: FV = 11604 €

Ordinary annuity factor

((1 + 0.05)⁵ − 1)/0.05 = 5.526

Annuity due

Each deposit is invested one year longer.

FV = 2 000·5.526·1.05 = €11 603.83

Answer

FV = €11 603.83

3.Bond pricing & yield

Answer: ΔP = −37.65 €

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 = €30.00, i = 0.055, N = 13

€30.00·(1 − 0.4986)/0.055 + €1 000·0.4986 = €273.51 + €498.56 = €772.07

Price at the new yield

C = €30.00, i = 0.06, N = 13

€30.00·(1 − 0.4688)/0.06 + €1 000·0.4688 = €265.58 + €468.84 = €734.42

Change

ΔP = 734.42 − 772.07 = −€37.65

Answer

ΔP = −€37.65

4.Bond pricing & yield

Answer: YTC = 4.123 %

Equation for the yield

1119 = 60·(1 − (1 + y)⁻⁴)/y + 1060/(1 + y)⁴

Solve numerically

y = 4%: €1 123.89

y = 5%: €1 084.82

YTC = 4.123%

Answer

YTC = 4.123%

The yield to maturity is 4.496%. Investors in a premium bond expect it to be called, so the lower yield to call is the more realistic one.

5.CAPM & WACC

Answer: r_m = 6.5 %

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 = (8.3% − 2%)/1.4 = 4.5%

Market return

r_m = 2% + 4.5% = 6.5%

Answer

r_m = 6.5%

6.CAPM & WACC

Answer: WACC = 5.755 %

Market values

E = 1 000 000·€17.50 = €17 500 000

D = 20 000·€1 050 = €21 000 000

Cost of equity (CAPM)

r_e = 1.5% + 1.3·6% = 9.3%

Weights

V = €38 500 000, E/V = 0.4545, D/V = 0.5455

WACC

WACC = 0.4545·9.3% + 0.5455·3.5%·(1 − 0.2) = 5.755%

Answer

WACC = 5.755%

7.NPV & IRR

Answer: NPV = 21127 €

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 each cash flow

CF₀ = −75 000

CF₁/(1 + 0.1)¹ = 30 000/1.1 = 27 272.73

CF₂/(1 + 0.1)² = 31 000/1.21 = 25 619.83

CF₃/(1 + 0.1)³ = 33 000/1.331 = 24 793.39

CF₄/(1 + 0.1)⁴ = 27 000/1.464 = 18 441.36

Add them up

NPV = €21 127.31

Answer

NPV = €21 127.31

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

8.NPV & IRR

Answer: MIRR = 8.306 %

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

4 000·1.06² = 4 494.40

3 500·1.06¹ = 3 710.00

4 500·1.06⁰ = 4 500.00

TV = 12 704.40

MIRR

MIRR = (12 704.40/10 000)^1/3 − 1 = 8.306%

Answer

MIRR = 8.306%

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