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 €500 at the end of each year for 13 years. The interest rate is 3% per year. Find the present value PV of the annuity.

A = €500i = 3%n = 13

PV = €

2.Time value of moneyHard

A loan of €15 000 is repaid with equal monthly payments over 3 years. The interest rate is 3.6% per year, compounded monthly. Find the monthly payment A.

P = €15 0003.6% / 12N = 36

A = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays an annual coupon of 4.5% and matures in 12 years. Its yield to maturity changes from 9% to 8%. By how much ΔP does the bond price change?

F = €1 000coupon = 4.5%n = 12y₁ = 9%y₂ = 8%

ΔP = €

4.Bond pricing & yieldHard

A bond with a face value of €1 000 and an annual coupon of 7% matures in 7 years and sells for €925. Find its yield to maturity YTM in percent.

F = €1 000coupon = 7%n = 7P = €925

YTM = %

5.CAPM & WACCMedium

A company has a debt-to-equity ratio (at market values) of 0.25. Its cost of equity is 14%, its pre-tax cost of debt is 3.5% and the tax rate is 28%. Find the WACC.

D/E = 0.25r_e = 14%r_d = 3.5%T = 28%

WACC = %

6.CAPM & WACCHard

A company has 2 500 000 shares at €55.00 each and 40 000 bonds trading at €960 each. The pre-tax cost of debt is 3% and the tax rate is 25%. The share’s beta is 1.5, the risk-free rate 3.5% and the market return 7.5%. Find the WACC.

β = 1.5r_f = 3.5%r_m = 7.5%r_d = 3%T = 25%

WACC = %

7.NPV & IRRMedium

An investment of €60 000 gives these cash flows at the end of years 1 to 4: €28 000, €30 000, €22 000, €14 000. The cost of capital is 7.000000000000001%. Find the net present value NPV.

r = 7.000000000000001%CF₀ = −60 000CF₁ = 28 000CF₂ = 30 000CF₃ = 22 000CF₄ = 14 000

NPV = €

8.NPV & IRRHard

A project costs €50 000 and returns €27 000, €30 500, €36 500 at the end of years 1 to 3. Find its internal rate of return IRR in percent.

CF₀ = −50 000CF₁ = 27 000CF₂ = 30 500CF₃ = 36 500

IRR = %

3Solutions

1.Time value of money

Answer: PV = 5317 €

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 − 0.681)/0.03 = 10.63

Present value

PV = 500·10.63 = €5 317.48

Answer

PV = €5 317.48

2.Time value of money

Answer: A = 440.2 €

Monthly rate and number of payments

i = 0.036000000000000004/12 = 0.003

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 = 0.8978

A = 15 000·0.003/(1 − 0.8978) = €440.20

Answer

A = €440.20

3.Bond pricing & yield

Answer: ΔP = 58.47 €

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 = €45.00, i = 0.09, N = 12

€45.00·(1 − 0.3555)/0.09 + €1 000·0.3555 = €322.23 + €355.53 = €677.77

Price at the new yield

C = €45.00, i = 0.08, N = 12

€45.00·(1 − 0.3971)/0.08 + €1 000·0.3971 = €339.12 + €397.11 = €736.24

Change

ΔP = 736.24 − 677.77 = €58.47

Answer

ΔP = €58.47

4.Bond pricing & yield

Answer: YTM = 8.463 %

Equation for the yield

€925 = €70·(1 − (1 + y)⁻⁷)/y + €1 000/(1 + y)⁷

Solve numerically

P(8%) = €947.94, P(9%) = €899.34

YTM = 8.463%

Answer

YTM = 8.463%

5.CAPM & WACC

Answer: WACC = 11.7 %

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 + 0.25) = 0.8

D/V = 0.25/(1 + 0.25) = 0.2

WACC

WACC = 0.8·14% + 0.2·3.5%·(1 − 0.28) = 11.7%

Answer

WACC = 11.7%

6.CAPM & WACC

Answer: WACC = 7.917 %

Market values

E = 2 500 000·€55.00 = €137 500 000

D = 40 000·€960 = €38 400 000

Cost of equity (CAPM)

r_e = 3.5% + 1.5·4% = 9.5%

Weights

V = €175 900 000, E/V = 0.7817, D/V = 0.2183

WACC

WACC = 0.7817·9.5% + 0.2183·3%·(1 − 0.25) = 7.917%

Answer

WACC = 7.917%

7.NPV & IRR

Answer: NPV = 21010 €

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 = −60 000

CF₁/(1 + 0.07)¹ = 28 000/1.07 = 26 168.22

CF₂/(1 + 0.07)² = 30 000/1.145 = 26 203.16

CF₃/(1 + 0.07)³ = 22 000/1.225 = 17 958.55

CF₄/(1 + 0.07)⁴ = 14 000/1.311 = 10 680.53

Add them up

NPV = €21 010.47

Answer

NPV = €21 010.47

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

8.NPV & IRR

Answer: IRR = 37.22 %

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(35%) = 1 570.39

NPV(40%) = −1 851.31

Narrow down

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

IRR ≈ 35 + 5·1 570.39/(1 570.39 − (−1 851.31)) = 37.29%

exact: NPV(37.22%) = 0 → IRR = 37.22%

Answer

IRR = 37.22%

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