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 invest €2 000 at a nominal rate of 3.6% per year, compounded monthly. Find the value FV of the investment after 5 years.

PV = €2 0003.6% / 12n = 5

FV = €

2.Time value of moneyHard

You borrow €8 000 at 3.6% per year, compounded monthly, and repay €200 at the end of every month. How many months does it take until the loan is repaid (the last payment may be smaller)? Give the number of payments N.

P = €8 000A = €2003.6% / 12

N = payments

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays a 7% annual coupon in two half-yearly payments and matures in 5 years. The yield to maturity is 8% per year. Find the price P of the bond.

F = €1 000coupon = 7%YTM = 8%n = 5

P = €

4.Bond pricing & yieldHard

A bond trades at €1 040 with a yield to maturity of 6% (annual compounding) and a Macaulay duration of 7.9 years. Use the modified duration to estimate the new price P if the yield rises by 150 basis points.

P₀ = €1 040y = 6%D = 7.9Δy = +150 bp

P = €

5.CAPM & WACCMedium

A company has equity worth €3 000 000 and debt worth €6 000 000. Its cost of equity is 13%, its pre-tax cost of debt is 4% and the tax rate is 21%. Find the WACC.

E = €3 000 000D = €6 000 000r_e = 13%r_d = 4%T = 21%

WACC = %

6.CAPM & WACCHard

A company has 2 000 000 shares at €16.00 each and 20 000 bonds trading at €1 020 each. The pre-tax cost of debt is 5% and the tax rate is 20%. The share’s beta is 1.5, the risk-free rate 1.5% and the market return 5.5%. Find the WACC.

β = 1.5r_f = 1.5%r_m = 5.5%r_d = 5%T = 20%

WACC = %

7.NPV & IRRMedium

A machine costs €50 000 today. It generates €18 500 at the end of each year for 3 years and can then be sold for €5 000. The cost of capital is 12%. Find the net present value NPV.

C₀ = €50 000CF = €18 500S = €5 000n = 3r = 12%

NPV = €

8.NPV & IRRHard

An investment of €40 000 returns €5 000 at the end of each year for 9 years. Find its internal rate of return IRR in percent.

C₀ = €40 000CF = €5 000n = 9

IRR = %

3Solutions

1.Time value of money

Answer: FV = 2394 €

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 = 0.036000000000000004/12 = 0.003

N = m·n = 12·5 = 60

Future value

FV = 2 000·(1 + 0.003)⁶⁰ = 2 000·1.197 = €2 393.79

Answer

FV = €2 393.79

2.Time value of money

Answer: N = 43 payments

Rearrange the annuity formula

i = 0.003

(1 + i)^−N = 1 − 8 000·0.003/200 = 0.88

Solve with logarithms

N = −ln(0.88)/ln(1.003) = 42.68

Round up to whole payments

42 full payments leave a small balance, which the last, smaller payment clears.

N = 43

Answer

N = 43 payments

3.Bond pricing & yield

Answer: P = 959.4 €

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 = €35.00, i = 0.04, N = 10

Present value of the coupons

€35.00·(1 − 0.6756)/0.04 = €283.88

Present value of the face value

€1 000·0.6756 = €675.56

Price

P = €959.45

Answer

P = €959.45

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

4.Bond pricing & yield

Answer: P = 923.7 €

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 = 7.9/1.06 = 7.453

Estimated price change

%ΔP ≈ −7.453·(0.015) = −11.18%

P ≈ 1040·(1 − 0.1118) = €923.74

Answer

P = €923.74

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

5.CAPM & WACC

Answer: WACC = 6.44 %

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 = 0.3333, D/V = 0.6667

WACC

WACC = 0.3333·13% + 0.6667·4%·(1 − 0.21) = 6.44%

Answer

WACC = 6.44%

6.CAPM & WACC

Answer: WACC = 6.137 %

Market values

E = 2 000 000·€16.00 = €32 000 000

D = 20 000·€1 020 = €20 400 000

Cost of equity (CAPM)

r_e = 1.5% + 1.5·4% = 7.5%

Weights

V = €52 400 000, E/V = 0.6107, D/V = 0.3893

WACC

WACC = 0.6107·7.5% + 0.3893·5%·(1 − 0.2) = 6.137%

Answer

WACC = 6.137%

7.NPV & IRR

Answer: NPV = −2007 €

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 the yearly cash flows

18 500·2.402 = 44 433.88

Present value of the salvage value

5 000/1.405 = 3 558.90

NPV

NPV = −50 000 + 44 433.88 + 3 558.90 = −€2 007.22

Answer

NPV = −€2 007.22

8.NPV & IRR

Answer: IRR = 2.423 %

Required annuity factor

(1 − (1 + i)⁻⁹)/i = 40 000/5 000 = 8

Solve for the rate

The annuity factor falls as the rate rises; bracket the rate and narrow down.

i = 2%: 8.162

i = 3%: 7.786

IRR = 2.423%

Answer

IRR = 2.423%

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