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 need €15 000 in 7 years. An account pays 2% per year, compounded monthly. How much must you deposit today? Find the present value PV.

FV = €15 0002% / 12n = 7

PV = €

2.Time value of moneyHard

A loan of €150 000 is repaid with equal monthly payments over 25 years. The interest rate is 3.6% per year, compounded monthly. Find the monthly payment A.

P = €150 0003.6% / 12N = 300

A = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays an annual coupon of 2% and matures in 8 years. Its yield to maturity changes from 8% to 6.5%. By how much ΔP does the bond price change?

F = €1 000coupon = 2%n = 8y₁ = 8%y₂ = 6.5%

ΔP = €

4.Bond pricing & yieldHard

A bond with 11 years to maturity (face value €1 000, annual coupon 9%) sells for €1 110. The issuer can call it in 5 years at a call price of €1 090. Find the yield to call YTC in percent.

F = €1 000coupon = 9%P = €1 110call price = €1 090call in 5 years

YTC = %

5.CAPM & WACCMedium

A company has equity worth €6 000 000 and debt worth €2 000 000. Its WACC is 9.72%, its pre-tax cost of debt is 4% and the tax rate is 28%. Find the cost of equity r_e.

E = €6 000 000D = €2 000 000WACC = 9.72%r_d = 4%T = 28%

r_e = %

6.CAPM & WACCHard

A company has 2 000 000 shares trading at €26.00; the book value of its equity is €31 200 000. It just paid a dividend of €0.80 per share, and dividends grow by 5% per year. Its debt has a market value of €41 600 000 and a pre-tax cost of 6%. The tax rate is 30%. Find the WACC, using the dividend growth model for the cost of equity.

N = 2 000 000P₀ = €26.00book equity = €31 200 000D₀ = €0.80g = 5%D = €41 600 000r_d = 6%T = 30%

WACC = %

7.NPV & IRRMedium

An investment of €60 000 gives these cash flows at the end of years 1 to 4: €17 000, €28 000, €22 000, €18 000. The cost of capital is 8%. Find the net present value NPV.

r = 8%CF₀ = −60 000CF₁ = 17 000CF₂ = 28 000CF₃ = 22 000CF₄ = 18 000

NPV = €

8.NPV & IRRHard

A project costs €150 000 today. Its cash flow is €30 000 at the end of year 1 and grows by 3% per year until the last cash flow at the end of year 6. The cost of capital is 10%. Find the net present value NPV.

C₀ = €150 000CF₁ = €30 000g = 3%n = 6r = 10%

NPV = €

3Solutions

1.Time value of money

Answer: PV = 13042 €

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 = 0.02/12 = 0.001667

N = 12·7 = 84

Present value

PV = FV/(1 + i)^N = 15 000/1.15 = €13 041.89

Answer

PV = €13 041.89

2.Time value of money

Answer: A = 759 €

Monthly rate and number of payments

i = 0.036000000000000004/12 = 0.003

N = 12·25 = 300

Payment

The present value of N equal payments A is A·(1 − (1 + i)⁻ᴺ)/i. Set it equal to the loan and solve for A.

(1 + i)^−N = 0.4071

A = 150 000·0.003/(1 − 0.4071) = €759.00

Answer

A = €759.00

3.Bond pricing & yield

Answer: ΔP = 70.8 €

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.

Price at the old yield

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

Price at the new yield

C = €20.00, i = 0.065, N = 8

€20.00·(1 − 0.6042)/0.065 + €1 000·0.6042 = €121.78 + €604.23 = €726.01

Change

ΔP = 726.01 − 655.20 = €70.80

Answer

ΔP = €70.80

4.Bond pricing & yield

Answer: YTC = 7.8 %

Equation for the yield

1110 = 90·(1 − (1 + y)⁻⁵)/y + 1090/(1 + y)⁵

Solve numerically

y = 7%: €1 146.17

y = 8%: €1 101.18

YTC = 7.8%

Answer

YTC = 7.8%

The yield to maturity is 7.497%. 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: r_e = 12 %

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 = 0.75, D/V = 0.25

Debt part of the WACC

0.25·4%·(1 − 0.28) = 0.72%

Cost of equity

r_e = (9.72% − 0.72%)/0.75 = 12%

Answer

r_e = 12%

6.CAPM & WACC

Answer: WACC = 6.439 %

Cost of equity

D₁ = 0.8·1.05 = €0.84

r_e = 0.84/26 + 0.05 = 8.231%

Market values

E = 2 000 000·€26.00 = €52 000 000

V = €93 600 000, E/V = 0.5556, D/V = 0.4444

WACC

WACC = 0.5556·8.231% + 0.4444·6%·(1 − 0.3) = 6.439%

Answer

WACC = 6.439%

7.NPV & IRR

Answer: NPV = 10441 €

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 + 0.08)¹ = 17 000/1.08 = 15 740.74

CF₂/(1 + 0.08)² = 28 000/1.166 = 24 005.49

CF₃/(1 + 0.08)³ = 22 000/1.26 = 17 464.31

CF₄/(1 + 0.08)⁴ = 18 000/1.36 = 13 230.54

Add them up

NPV = €10 441.07

Answer

NPV = €10 441.07

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

8.NPV & IRR

Answer: NPV = −10291 €

Present value of the growing annuity

((1.03)/(1.1))⁶ = 0.674

PV = 30 000/(0.1 − 0.03)·(1 − 0.674) = 139 709.40

NPV

NPV = 139 709.40 − 150 000 = −€10 290.60

Answer

NPV = −€10 290.60

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

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