Investments and Valuation: cram pack

Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.

New problems Back to the course

In the print window, choose “Save as PDF” to keep a copy on your device.

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 invest €3 000 at a nominal rate of 4.8% per year, compounded monthly. Find the value FV of the investment after 9 years.

PV = €3 0004.8% / 12n = 9

FV = €

2.Time value of moneyHard

You deposit €6 000 at the start of every year for 21 years, the first deposit today. The account pays 4% per year. How much is in the account at the end of year 21? Find FV.

A = €6 000i = 4%n = 21

FV = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 matures in 8 years and pays annual coupons. It sells for €1 070 when the yield to maturity is 3%. Find the coupon rate c in percent.

F = €1 000n = 8P = €1 070YTM = 3%

c = %

4.Bond pricing & yieldHard

A bond with a face value of €1 000 pays an annual coupon of 5% and matures in 3 years. The yield to maturity is 4%. Find the Macaulay duration D in years.

F = €1 000coupon = 5%n = 3YTM = 4%

D = years

5.CAPM & WACCMedium

A company has equity worth €8 000 000 and debt worth €1 000 000. Its cost of equity is 9%, its pre-tax cost of debt is 5% and the tax rate is 25%. Find the WACC.

E = €8 000 000D = €1 000 000r_e = 9%r_d = 5%T = 25%

WACC = %

6.CAPM & WACCHard

A company has 2 000 000 shares trading at €23.50; the book value of its equity is €14 100 000. It just paid a dividend of €2.40 per share, and dividends grow by 4% per year. Its debt has a market value of €37 600 000 and a pre-tax cost of 4%. The tax rate is 28%. Find the WACC, using the dividend growth model for the cost of equity.

N = 2 000 000P₀ = €23.50book equity = €14 100 000D₀ = €2.40g = 4%D = €37 600 000r_d = 4%T = 28%

WACC = %

7.NPV & IRRMedium

An investment of €60 000 gives these cash flows at the end of years 1 to 3: €18 000, €17 000, €13 000. The cost of capital is 15%. Find the net present value NPV.

r = 15%CF₀ = −60 000CF₁ = 18 000CF₂ = 17 000CF₃ = 13 000

NPV = €

8.NPV & IRRHard

A project costs €10 000 and returns €7 000, €5 500, €4 000 at the end of years 1 to 3. Find its internal rate of return IRR in percent.

CF₀ = −10 000CF₁ = 7 000CF₂ = 5 500CF₃ = 4 000

IRR = %

3Solutions

1.Time value of money

Answer: FV = 4617 €

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 = r/m = 0.048/12 = 0.004

N = m·n = 12·9 = 108

Future value

FV = 3 000·(1 + 0.004)¹⁰⁸ = 3 000·1.539 = €4 617.03

Answer

FV = €4 617.03

2.Time value of money

Answer: FV = 199488 €

Ordinary annuity factor

((1 + 0.04)²¹ − 1)/0.04 = 31.97

Annuity due

Each deposit is invested one year longer.

FV = 6 000·31.97·1.04 = €199 487.82

Answer

FV = €199 487.82

3.Bond pricing & yield

Answer: c = 3.997 %

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.

Present value of the face value

1000·(1 + 0.03)⁻⁸ = €789.41

Coupon

C = (1070 − 789.41)/7.02 = €39.97

Coupon rate

c = 39.97/1000 = 3.997%

Answer

c = 3.997%

4.Bond pricing & yield

Answer: D = 2.861 years

Duration: the Macaulay duration is the weighted average time at which a bond pays its cash flows, with the present values of the cash flows as weights: D = Σ t·PVₜ/P. The modified duration D/(1 + y) estimates the percentage price change for a small change in yield: %ΔP ≈ −D_mod·Δy.

Present value of each cash flow

t = 1: 50/1.04 = 48.08, t·PV = 48.08

t = 2: 50/1.082 = 46.23, t·PV = 92.46

t = 3: 1050/1.125 = 933.45, t·PV = 2 800.34

Duration

P = 1 027.75

D = 2 940.87/1 027.75 = 2.861

Answer

D = 2.861 years

5.CAPM & WACC

Answer: WACC = 8.417 %

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 = €9 000 000, E/V = 0.8889, D/V = 0.1111

WACC

WACC = 0.8889·9% + 0.1111·5%·(1 − 0.25) = 8.417%

Answer

WACC = 8.417%

6.CAPM & WACC

Answer: WACC = 9.403 %

Cost of equity

D₁ = 2.4·1.04 = €2.50

r_e = 2.496/23.5 + 0.04 = 14.62%

Market values

E = 2 000 000·€23.50 = €47 000 000

V = €84 600 000, E/V = 0.5556, D/V = 0.4444

WACC

WACC = 0.5556·14.62% + 0.4444·4%·(1 − 0.28) = 9.403%

Answer

WACC = 9.403%

7.NPV & IRR

Answer: NPV = −22946 €

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 + 0.15)¹ = 18 000/1.15 = 15 652.17

CF₂/(1 + 0.15)² = 17 000/1.322 = 12 854.44

CF₃/(1 + 0.15)³ = 13 000/1.521 = 8 547.71

Add them up

NPV = €−22 945.67

Answer

NPV = €−22 945.67

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

8.NPV & IRR

Answer: IRR = 33.59 %

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(30%) = 459.72

NPV(35%) = −171.21

Narrow down

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

IRR ≈ 30 + 5·459.72/(459.72 − (−171.21)) = 33.64%

exact: NPV(33.59%) = 0 → IRR = 33.59%

Answer

IRR = 33.59%

ExamProblems · examproblems.com/courses/investments-and-valuation