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

You need €20 000 in 10 years. An account pays 6% per year, compounded quarterly. How much must you deposit today? Find the present value PV.

FV = €20 0006% / 4n = 10

PV = €

2.Time value of moneyHard

You want €30 000 in 11 years. You save the same amount at the end of every month in an account paying 3.6% per year, compounded monthly. Find the monthly deposit A.

FV = €30 0003.6% / 12N = 132

A = €

3.Bond pricing & yieldMedium

You buy a bond with 3 years to maturity (face value €1 000, annual coupon 2%) when its yield to maturity is 2.5%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 3.5%. Find your holding period return HPR in percent.

F = €1 000coupon = 2%n = 3y₀ = 2.5%y₁ = 3.5%

HPR = %

4.Bond pricing & yieldHard

A bond with 15 years to maturity (face value €1 000, annual coupon 7%) sells for €1 268. The issuer can call it in 4 years at a call price of €1 070. Find the yield to call YTC in percent.

F = €1 000coupon = 7%P = €1 268call price = €1 070call in 4 years

YTC = %

5.CAPM & WACCMedium

A company has equity worth €4 000 000 and debt worth €6 000 000. Its cost of equity is 11%, its pre-tax cost of debt is 6% and the tax rate is 28%. Find the WACC.

E = €4 000 000D = €6 000 000r_e = 11%r_d = 6%T = 28%

WACC = %

6.CAPM & WACCHard

A company has 2 000 000 shares at €18.00 each and 10 000 bonds trading at €1 020 each. The pre-tax cost of debt is 6% and the tax rate is 28%. The share’s beta is 1.3, the risk-free rate 4% and the market return 11%. Find the WACC.

β = 1.3r_f = 4%r_m = 11%r_d = 6%T = 28%

WACC = %

7.NPV & IRRMedium

A project costs €500 000 today. It generates €33 000 at the end of the first year, and the cash flow then grows by 4% per year forever. The cost of capital is 10%. Find the net present value NPV.

C₀ = €500 000CF₁ = €33 000g = 4%r = 10%

NPV = €

8.NPV & IRRHard

Project A costs €40 000 and returns €22 500, €16 000, €9 500 at the end of years 1 to 3. Project B costs €45 000 and returns €12 000, €19 000, €24 500. Find the crossover rate r* at which both projects have the same NPV, in percent.

A: −40 000, 22 500, 16 000, 9 500B: −45 000, 12 000, 19 000, 24 500

r* = %

3Solutions

1.Time value of money

Answer: PV = 11025 €

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 and number of periods

i = 0.06/4 = 0.015

N = 4·10 = 40

Present value

PV = FV/(1 + i)^N = 20 000/1.814 = €11 025.25

Answer

PV = €11 025.25

2.Time value of money

Answer: A = 185.6 €

Monthly rate and number of deposits

i = 0.036000000000000004/12 = 0.003

N = 12·11 = 132

Deposit

A = FV·i/((1 + i)^N − 1) = 30 000·0.003/(1.485 − 1) = €185.57

Answer

A = €185.57

3.Bond pricing & yield

Answer: HPR = 0.5869 %

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 = 0.025, N = 3

€20.00·(1 − 0.9286)/0.025 + €1 000·0.9286 = €57.12 + €928.60 = €985.72

Selling price

C = €20.00, i = 0.035, N = 2

€20.00·(1 − 0.9335)/0.035 + €1 000·0.9335 = €37.99 + €933.51 = €971.50

Holding period return

HPR = (20 + 971.50 − 985.72)/985.72 = 0.5869%

Answer

HPR = 0.5869%

4.Bond pricing & yield

Answer: YTC = 1.716 %

Equation for the yield

1268 = 70·(1 − (1 + y)⁻⁴)/y + 1070/(1 + y)⁴

Solve numerically

y = 1%: €1 301.39

y = 2%: €1 255.06

YTC = 1.716%

Answer

YTC = 1.716%

The yield to maturity is 4.504%. 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: WACC = 6.992 %

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 = €10 000 000, E/V = 0.4, D/V = 0.6

WACC

WACC = 0.4·11% + 0.6·6%·(1 − 0.28) = 6.992%

Answer

WACC = 6.992%

6.CAPM & WACC

Answer: WACC = 11.16 %

Market values

E = 2 000 000·€18.00 = €36 000 000

D = 10 000·€1 020 = €10 200 000

Cost of equity (CAPM)

r_e = 4% + 1.3·7% = 13.1%

Weights

V = €46 200 000, E/V = 0.7792, D/V = 0.2208

WACC

WACC = 0.7792·13.1% + 0.2208·6%·(1 − 0.28) = 11.16%

Answer

WACC = 11.16%

7.NPV & IRR

Answer: NPV = 50000 €

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.

Value of the cash flows

PV = 33 000/(0.1 − 0.04) = 550 000.00

NPV

NPV = 550 000.00 − 500 000 = €50 000.00

Answer

NPV = €50 000.00

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

8.NPV & IRR

Answer: r* = 7.173 %

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 = −45 000 − (−40 000) = −5 000

ΔCF₁ = 12 000 − 22 500 = −10 500

ΔCF₂ = 19 000 − 16 000 = 3 000

ΔCF₃ = 24 500 − 9 500 = 15 000

IRR of the difference

NPV_Δ(7%) = 51.70

NPV_Δ(8%) = −242.72

r* = 7.173%

Answer

r* = 7.173%

Check: at 7.173% both NPVs are €2 641.62.

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