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 €5 000 in 10 years. An account pays 2% per year, compounded quarterly. How much must you deposit today? Find the present value PV.

FV = €5 0002% / 4n = 10

PV = €

2.Time value of moneyHard

A mortgage of €250 000 is repaid with equal monthly payments over 20 years at 6% per year, compounded monthly. How much is still owed just after the payment at the end of year 13? Find the outstanding balance B.

P = €250 0006% / 12N = 240k = 156

B = €

3.Bond pricing & yieldMedium

You buy a bond with 11 years to maturity (face value €1 000, annual coupon 5%) when its yield to maturity is 3.5%. One year later, just after receiving the first coupon, you sell it when the yield to maturity is 4%. Find your holding period return HPR in percent.

F = €1 000coupon = 5%n = 11y₀ = 3.5%y₁ = 4%

HPR = %

4.Bond pricing & yieldHard

A bond with a face value of €1 000 and an annual coupon of 6% matures in 3 years and sells for €975. Find its yield to maturity YTM in percent.

F = €1 000coupon = 6%n = 3P = €975

YTM = %

5.CAPM & WACCMedium

A company has equity worth €3 000 000 and debt worth €5 000 000. Its WACC is 6%, its pre-tax cost of debt is 5% and the tax rate is 28%. Find the cost of equity r_e.

E = €3 000 000D = €5 000 000WACC = 6%r_d = 5%T = 28%

r_e = %

6.CAPM & WACCHard

A company has 1 000 000 shares at €45.00 each and 10 000 bonds trading at €1 020 each. The pre-tax cost of debt is 3.5% and the tax rate is 20%. The share’s beta is 0.9, the risk-free rate 4% and the market return 11%. Find the WACC.

β = 0.9r_f = 4%r_m = 11%r_d = 3.5%T = 20%

WACC = %

7.NPV & IRRMedium

A project costs €100 000 today. It generates €9 000 at the end of the first year, and the cash flow then grows by 1% per year forever. The cost of capital is 12%. Find the net present value NPV.

C₀ = €100 000CF₁ = €9 000g = 1%r = 12%

NPV = €

8.NPV & IRRHard

Project A costs €10 000 and returns €6 000, €4 000, €2 000 at the end of years 1 to 3. Project B costs €15 000 and returns €4 000, €6 000, €7 500. Find the crossover rate r* at which both projects have the same NPV, in percent.

A: −10 000, 6 000, 4 000, 2 000B: −15 000, 4 000, 6 000, 7 500

r* = %

3Solutions

1.Time value of money

Answer: PV = 4096 €

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 = 0.02/4 = 0.005

N = 4·10 = 40

Present value

PV = FV/(1 + i)^N = 5 000/1.221 = €4 095.69

Answer

PV = €4 095.69

2.Time value of money

Answer: B = 122605 €

Monthly payment

i = 0.005, N = 240

A = 250 000·0.005/(1 − (1 + 0.005)⁻²⁴⁰) = €1 791.08

Present value of the remaining payments

Whatever is still owed must be exactly paid off by the remaining payments.

N − k = 240 − 156 = 84

B = 1791·(1 − (1 + 0.005)⁻⁸⁴)/0.005 = €122 604.71

Answer

B = €122 604.71

3.Bond pricing & yield

Answer: HPR = −0.3449 %

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 = €50.00, i = 0.035, N = 11

€50.00·(1 − 0.6849)/0.035 + €1 000·0.6849 = €450.08 + €684.95 = €1 135.02

Selling price

C = €50.00, i = 0.04, N = 10

€50.00·(1 − 0.6756)/0.04 + €1 000·0.6756 = €405.54 + €675.56 = €1 081.11

Holding period return

HPR = (50 + 1 081.11 − 1 135.02)/1 135.02 = −0.3449%

Answer

HPR = −0.3449%

4.Bond pricing & yield

Answer: YTM = 6.952 %

Equation for the yield

€975 = €60·(1 − (1 + y)⁻³)/y + €1 000/(1 + y)³

Solve numerically

P(6%) = €1 000.00, P(7%) = €973.76

YTM = 6.952%

Answer

YTM = 6.952%

5.CAPM & WACC

Answer: r_e = 10 %

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 = €8 000 000, E/V = 0.375, D/V = 0.625

Debt part of the WACC

0.625·5%·(1 − 0.28) = 2.25%

Cost of equity

r_e = (6% − 2.25%)/0.375 = 10%

Answer

r_e = 10%

6.CAPM & WACC

Answer: WACC = 8.914 %

Market values

E = 1 000 000·€45.00 = €45 000 000

D = 10 000·€1 020 = €10 200 000

Cost of equity (CAPM)

r_e = 4% + 0.9·7% = 10.3%

Weights

V = €55 200 000, E/V = 0.8152, D/V = 0.1848

WACC

WACC = 0.8152·10.3% + 0.1848·3.5%·(1 − 0.2) = 8.914%

Answer

WACC = 8.914%

7.NPV & IRR

Answer: NPV = −18182 €

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 = 9 000/(0.12 − 0.01) = 81 818.18

NPV

NPV = 81 818.18 − 100 000 = −€18 181.82

Answer

NPV = −€18 181.82

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

8.NPV & IRR

Answer: r* = 2.859 %

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 = −15 000 − (−10 000) = −5 000

ΔCF₁ = 4 000 − 6 000 = −2 000

ΔCF₂ = 6 000 − 4 000 = 2 000

ΔCF₃ = 7 500 − 2 000 = 5 500

IRR of the difference

NPV_Δ(2%) = 144.33

NPV_Δ(3%) = −23.28

r* = 2.859%

Answer

r* = 2.859%

Check: at 2.859% both NPVs are €1 451.82.

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