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 property is expected to earn net rent of €10 000 next year. The rent is expected to grow by 4% per year forever. The required return is 9% per year. Find the value PV of the property today.

C₁ = €10 000g = 4%r = 9%

PV = €

2.Time value of moneyHard

A loan of €25 000 is repaid with equal monthly payments over 7 years at 4.2% per year, compounded monthly. How much interest I is paid in total over the life of the loan?

P = €25 0004.2% / 12N = 84

I = €

3.Bond pricing & yieldMedium

You buy a bond with 8 years to maturity (face value €1 000, annual coupon 2%) when its yield to maturity is 8%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 9%. Find your holding period return HPR in percent.

F = €1 000coupon = 2%n = 8y₀ = 8%y₁ = 9%

HPR = %

4.Bond pricing & yieldHard

A bond with a face value of €1 000 and an annual coupon of 5% matures in 9 years and sells for €1 075. Find its yield to maturity YTM in percent.

F = €1 000coupon = 5%n = 9P = €1 075

YTM = %

5.CAPM & WACCMedium

A company has equity worth €2 000 000 and debt worth €2 000 000. Its cost of equity is 9%, its pre-tax cost of debt is 3.5% and the tax rate is 20%. Find the WACC.

E = €2 000 000D = €2 000 000r_e = 9%r_d = 3.5%T = 20%

WACC = %

6.CAPM & WACCHard

A company's business has an unlevered (asset) beta of 0.8. It plans a debt-to-equity ratio of 1. The tax rate is 20%, the risk-free rate is 4% and the market risk premium is 5.5%. Find the cost of equity r_e at the new capital structure.

β_U = 0.8D/E = 1T = 20%r_f = 4%r_m − r_f = 5.5%

r_e = %

7.NPV & IRRMedium

A project costs €60 000 today and returns €17 000 at the end of each year for 6 years. The cost of capital is 8%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.

r = 8%CF₀ = −60 000CF₁ = 17 000CF₂ = 17 000CF₃ = 17 000CF₄ = 17 000CF₅ = 17 000CF₆ = 17 000

DPB = years

8.NPV & IRRHard

An investment of €50 000 returns €8 500 at the end of each year for 9 years. Find its internal rate of return IRR in percent.

C₀ = €50 000CF = €8 500n = 9

IRR = %

3Solutions

1.Time value of money

Answer: PV = 200000 €

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.

Growing perpetuity

PV = C₁/(r − g) = 10 000/(0.09 − 0.04) = 10 000/0.05 = €200 000.00

Answer

PV = €200 000.00

2.Time value of money

Answer: I = 3898 €

Monthly payment

i = 0.0035, N = 84

A = 25 000·0.0035/(1 − (1 + 0.0035)⁻⁸⁴) = €344.03

Total interest

I = N·A − P = 84·344 − 25 000 = €3 898.23

Answer

I = €3 898.23

3.Bond pricing & yield

Answer: HPR = 1.907 %

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.

Purchase price

C = €20.00, i = 0.08, N = 8

€20.00·(1 − 0.5403)/0.08 + €1 000·0.5403 = €114.93 + €540.27 = €655.20

Selling price

C = €20.00, i = 0.09, N = 7

€20.00·(1 − 0.547)/0.09 + €1 000·0.547 = €100.66 + €547.03 = €647.69

Holding period return

HPR = (20 + 647.69 − 655.20)/655.20 = 1.907%

Answer

HPR = 1.907%

4.Bond pricing & yield

Answer: YTM = 3.992 %

Equation for the yield

€1 075 = €50·(1 − (1 + y)⁻⁹)/y + €1 000/(1 + y)⁹

Solve numerically

P(3%) = €1 155.72, P(4%) = €1 074.35

YTM = 3.992%

Answer

YTM = 3.992%

5.CAPM & WACC

Answer: WACC = 5.9 %

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 = €4 000 000, E/V = 0.5, D/V = 0.5

WACC

WACC = 0.5·9% + 0.5·3.5%·(1 − 0.2) = 5.9%

Answer

WACC = 5.9%

6.CAPM & WACC

Answer: r_e = 11.92 %

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 = 0.8·(1 + 0.8·1) = 1.44

Cost of equity (CAPM)

r_e = 4% + 1.44·5.5% = 11.92%

Answer

r_e = 11.92%

7.NPV & IRR

Answer: DPB = 4.319 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: 17 000/1.08 = 15 740.74 → Σ = 15 740.74

year 2: 17 000/1.166 = 14 574.76 → Σ = 30 315.50

year 3: 17 000/1.26 = 13 495.15 → Σ = 43 810.65

year 4: 17 000/1.36 = 12 495.51 → Σ = 56 306.16

year 5: 17 000/1.469 = 11 569.91 → Σ = 67 876.07

Fraction of the last year

DPB = 4 + 3 693.84/11 569.91 = 4.319

Answer

DPB = 4.319 years

The discounted payback is always longer than the simple payback (3.529 years).

8.NPV & IRR

Answer: IRR = 9.47 %

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.

Required annuity factor

(1 − (1 + i)⁻⁹)/i = 50 000/8 500 = 5.882

Solve for the rate

The annuity factor falls as the rate rises; bracket the rate and narrow down.

i = 9%: 5.995

i = 10%: 5.759

IRR = 9.47%

Answer

IRR = 9.47%

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