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

A property is expected to earn net rent of €12 000 next year. The rent is expected to grow by 1% per year forever. The required return is 7% per year. Find the value PV of the property today.

C₁ = €12 000g = 1%r = 7%

PV = €

2.Time value of moneyHard

You save €500 at the end of every month for 20 years. The account pays 7.2% per year, compounded monthly. How much have you saved at the end? Find FV.

A = €5007.2% / 12N = 240

FV = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays a 4.5% annual coupon in two half-yearly payments and matures in 8 years. The yield to maturity is 8% per year. Find the price P of the bond.

F = €1 000coupon = 4.5%YTM = 8%n = 8

P = €

4.Bond pricing & yieldHard

A bond trades at €1 015 with a yield to maturity of 3% (annual compounding) and a Macaulay duration of 4.3 years. Use the modified duration to estimate the new price P if the yield rises by 50 basis points.

P₀ = €1 015y = 3%D = 4.3Δy = +50 bp

P = €

5.CAPM & WACCMedium

A company has a debt-to-equity ratio (at market values) of 0.5. Its cost of equity is 8%, its pre-tax cost of debt is 3.5% and the tax rate is 21%. Find the WACC.

D/E = 0.5r_e = 8%r_d = 3.5%T = 21%

WACC = %

6.CAPM & WACCHard

A company has 2 500 000 shares at €10.00 each and 5 000 bonds trading at €1 050 each. The pre-tax cost of debt is 3.5% and the tax rate is 20%. The share’s beta is 1.1, the risk-free rate 4% and the market return 11%. Find the WACC.

β = 1.1r_f = 4%r_m = 11%r_d = 3.5%T = 20%

WACC = %

7.NPV & IRRMedium

An investment of €75 000 gives these cash flows at the end of years 1 to 4: €33 000, €34 000, €22 000, €33 000. The cost of capital is 6%. Find the net present value NPV.

r = 6%CF₀ = −75 000CF₁ = 33 000CF₂ = 34 000CF₃ = 22 000CF₄ = 33 000

NPV = €

8.NPV & IRRHard

Project A costs €40 000 and returns €23 000, €16 000, €7 000 at the end of years 1 to 3. Project B costs €45 000 and returns €12 500, €18 000, €23 000. Find the crossover rate r* at which both projects have the same NPV, in percent.

A: −40 000, 23 000, 16 000, 7 000B: −45 000, 12 500, 18 000, 23 000

r* = %

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) = 12 000/(0.07 − 0.01) = 12 000/0.06 = €200 000.00

Answer

PV = €200 000.00

2.Time value of money

Answer: FV = 266881 €

Monthly rate and number of deposits

i = 0.07200000000000001/12 = 0.006

N = 12·20 = 240

Future value of the deposits

FV = A·((1 + i)^N − 1)/i = 500·(4.203 − 1)/0.006 = €266 881.17

Answer

FV = €266 881.17

You paid in €120 000 yourself; the rest is interest.

3.Bond pricing & yield

Answer: P = 796.1 €

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.

Per period

C = €22.50, i = 0.04, N = 16

Present value of the coupons

€22.50·(1 − 0.5339)/0.04 = €262.18

Present value of the face value

€1 000·0.5339 = €533.91

Price

P = €796.08

Answer

P = €796.08

The yield is above the coupon rate, so the bond sells at a discount.

4.Bond pricing & yield

Answer: P = 993.8 €

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.

Modified duration

D_mod = 4.3/1.03 = 4.175

Estimated price change

%ΔP ≈ −4.175·(0.005) = −2.087%

P ≈ 1015·(1 − 0.02087) = €993.81

Answer

P = €993.81

Duration is a linear estimate; for large yield changes convexity makes the true price a little higher.

5.CAPM & WACC

Answer: WACC = 6.255 %

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 from D/E

E/V = 1/(1 + 0.5) = 0.6667

D/V = 0.5/(1 + 0.5) = 0.3333

WACC

WACC = 0.6667·8% + 0.3333·3.5%·(1 − 0.21) = 6.255%

Answer

WACC = 6.255%

6.CAPM & WACC

Answer: WACC = 10.16 %

Market values

E = 2 500 000·€10.00 = €25 000 000

D = 5 000·€1 050 = €5 250 000

Cost of equity (CAPM)

r_e = 4% + 1.1·7% = 11.7%

Weights

V = €30 250 000, E/V = 0.8264, D/V = 0.1736

WACC

WACC = 0.8264·11.7% + 0.1736·3.5%·(1 − 0.2) = 10.16%

Answer

WACC = 10.16%

7.NPV & IRR

Answer: NPV = 31003 €

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 = −75 000

CF₁/(1 + 0.06)¹ = 33 000/1.06 = 31 132.08

CF₂/(1 + 0.06)² = 34 000/1.124 = 30 259.88

CF₃/(1 + 0.06)³ = 22 000/1.191 = 18 471.62

CF₄/(1 + 0.06)⁴ = 33 000/1.262 = 26 139.09

Add them up

NPV = €31 002.67

Answer

NPV = €31 002.67

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

8.NPV & IRR

Answer: r* = 6.982 %

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 = 12 500 − 23 000 = −10 500

ΔCF₂ = 18 000 − 16 000 = 2 000

ΔCF₃ = 23 000 − 7 000 = 16 000

IRR of the difference

NPV_Δ(6%) = 308.24

NPV_Δ(7%) = −5.44

r* = 6.982%

Answer

r* = 6.982%

Check: at 6.982% both NPVs are €1 195.45.

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