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 €1 500 at the end of each year for 14 years. The interest rate is 5% per year. Find the present value PV of the annuity.

A = €1 500i = 5%n = 14

PV = €

2.Time value of moneyHard

You borrow €5 000 at 2.4% per year, compounded monthly, and repay €225 at the end of every month. How many months does it take until the loan is repaid (the last payment may be smaller)? Give the number of payments N.

P = €5 000A = €2252.4% / 12

N = payments

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays a 3% annual coupon in two half-yearly payments and matures in 6 years. The yield to maturity is 7% per year. Find the price P of the bond.

F = €1 000coupon = 3%YTM = 7%n = 6

P = €

4.Bond pricing & yieldHard

A bond with a face value of €1 000 and an annual coupon of 3% matures in 9 years and sells for €1 080. Find its yield to maturity YTM in percent.

F = €1 000coupon = 3%n = 9P = €1 080

YTM = %

5.CAPM & WACCMedium

A company has a debt-to-equity ratio (at market values) of 0.75. Its cost of equity is 9%, its pre-tax cost of debt is 3% and the tax rate is 21%. Find the WACC.

D/E = 0.75r_e = 9%r_d = 3%T = 21%

WACC = %

6.CAPM & WACCHard

A company has 1 000 000 shares trading at €25.50; the book value of its equity is €15 300 000. It just paid a dividend of €0.90 per share, and dividends grow by 4% per year. Its debt has a market value of €20 400 000 and a pre-tax cost of 6%. The tax rate is 21%. Find the WACC, using the dividend growth model for the cost of equity.

N = 1 000 000P₀ = €25.50book equity = €15 300 000D₀ = €0.90g = 4%D = €20 400 000r_d = 6%T = 21%

WACC = %

7.NPV & IRRMedium

A project costs €20 000 today and returns €6 000 at the end of each year for 6 years. The cost of capital is 6%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.

r = 6%CF₀ = −20 000CF₁ = 6 000CF₂ = 6 000CF₃ = 6 000CF₄ = 6 000CF₅ = 6 000CF₆ = 6 000

DPB = years

8.NPV & IRRHard

A project costs €200 000 today. Its cash flow is €36 500 at the end of year 1 and grows by 2% per year until the last cash flow at the end of year 6. The cost of capital is 8%. Find the net present value NPV.

C₀ = €200 000CF₁ = €36 500g = 2%n = 6r = 8%

NPV = €

3Solutions

1.Time value of money

Answer: PV = 14848 €

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 − 0.5051)/0.05 = 9.899

Present value

PV = 1 500·9.899 = €14 847.96

Answer

PV = €14 847.96

2.Time value of money

Answer: N = 23 payments

Rearrange the annuity formula

i = 0.002

(1 + i)^−N = 1 − 5 000·0.002/225 = 0.9556

Solve with logarithms

N = −ln(0.9556)/ln(1.002) = 22.75

Round up to whole payments

22 full payments leave a small balance, which the last, smaller payment clears.

N = 23

Answer

N = 23 payments

3.Bond pricing & yield

Answer: P = 806.7 €

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 = €15.00, i = 0.035, N = 12

Present value of the coupons

€15.00·(1 − 0.6618)/0.035 = €144.95

Present value of the face value

€1 000·0.6618 = €661.78

Price

P = €806.73

Answer

P = €806.73

The yield is above the coupon rate, so the bond sells at a discount.

4.Bond pricing & yield

Answer: YTM = 2.019 %

Equation for the yield

€1 080 = €30·(1 − (1 + y)⁻⁹)/y + €1 000/(1 + y)⁹

Solve numerically

P(2%) = €1 081.62, P(3%) = €1 000.00

YTM = 2.019%

Answer

YTM = 2.019%

5.CAPM & WACC

Answer: WACC = 6.159 %

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 + 0.75) = 0.5714

D/V = 0.75/(1 + 0.75) = 0.4286

WACC

WACC = 0.5714·9% + 0.4286·3%·(1 − 0.21) = 6.159%

Answer

WACC = 6.159%

6.CAPM & WACC

Answer: WACC = 6.368 %

Cost of equity

D₁ = 0.9·1.04 = €0.94

r_e = 0.936/25.5 + 0.04 = 7.671%

Market values

E = 1 000 000·€25.50 = €25 500 000

V = €45 900 000, E/V = 0.5556, D/V = 0.4444

WACC

WACC = 0.5556·7.671% + 0.4444·6%·(1 − 0.21) = 6.368%

Answer

WACC = 6.368%

7.NPV & IRR

Answer: DPB = 3.834 years

Payback period: the time until the cumulative cash flows repay the initial investment. Cash is assumed to come in evenly during each year, so the last part is a fraction of a year. The simple payback ignores the time value of money; the discounted payback period uses the discounted cash flows instead.

Discounted and cumulative cash flows

year 1: 6 000/1.06 = 5 660.38 → Σ = 5 660.38

year 2: 6 000/1.124 = 5 339.98 → Σ = 11 000.36

year 3: 6 000/1.191 = 5 037.72 → Σ = 16 038.07

year 4: 6 000/1.262 = 4 752.56 → Σ = 20 790.63

Fraction of the last year

DPB = 3 + 3 961.93/4 752.56 = 3.834

Answer

DPB = 3.834 years

The discounted payback is always longer than the simple payback (3.333 years).

8.NPV & IRR

Answer: NPV = −23385 €

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 growing annuity

((1.02)/(1.08))⁶ = 0.7097

PV = 36 500/(0.08 − 0.02)·(1 − 0.7097) = 176 615.38

NPV

NPV = 176 615.38 − 200 000 = −€23 384.62

Answer

NPV = −€23 384.62

NPV is negative, so the project earns less than the required return.

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