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

An annuity pays €1 000 at the end of each year for 5 years. The interest rate is 10% per year. Find the present value PV of the annuity.

A = €1 000i = 10%n = 5

PV = €

2.Time value of moneyHard

A loan of €20 000 is repaid with equal monthly payments over 6 years. The interest rate is 4.2% per year, compounded monthly. Find the monthly payment A.

P = €20 0004.2% / 12N = 72

A = €

3.Bond pricing & yieldMedium

A bond with a face value of €1 000 pays a 4% annual coupon in two half-yearly payments and matures in 3 years. The yield to maturity is 3.5% per year. Find the price P of the bond.

F = €1 000coupon = 4%YTM = 3.5%n = 3

P = €

4.Bond pricing & yieldHard

A bond with a face value of €1 000 and an annual coupon of 8% matures in 7 years and sells for €950. Find its yield to maturity YTM in percent.

F = €1 000coupon = 8%n = 7P = €950

YTM = %

5.CAPM & WACCMedium

A share with a beta of 2 has a required return of 12.5%. The risk-free rate is 3.5%. What expected market return r_m does CAPM imply?

β = 2r_e = 12.5%r_f = 3.5%

r_m = %

6.CAPM & WACCHard

A company has equity with a market value of €5 000 000 and a cost of equity of 11%. Its debt consists of 8 000 bonds (face value €1 000, annual coupon 4%, 11 years to maturity) that trade at €879 each. The tax rate is 28%. Find the WACC, using the yield to maturity of the bonds as the pre-tax cost of debt.

E = €5 000 000r_e = 11%bonds: 8 000 × €879coupon = 4%n = 11T = 28%

WACC = %

7.NPV & IRRMedium

A project costs €50 000 today and returns €17 500 at the end of each year for 6 years. The cost of capital is 5%. Assume cash flows arrive evenly through each year. Find the discounted payback period DPB in years.

r = 5%CF₀ = −50 000CF₁ = 17 500CF₂ = 17 500CF₃ = 17 500CF₄ = 17 500CF₅ = 17 500CF₆ = 17 500

DPB = years

8.NPV & IRRHard

A project costs €25 000 and returns €10 000, €10 000, €13 000 at the end of years 1 to 3. Cash flows can be reinvested at the cost of capital of 8%. Find the modified internal rate of return MIRR in percent.

r = 8%CF₀ = −25 000CF₁ = 10 000CF₂ = 10 000CF₃ = 13 000

MIRR = %

3Solutions

1.Time value of money

Answer: PV = 3791 €

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.

Annuity factor

(1 − (1 + i)⁻ⁿ)/i = (1 − 0.6209)/0.1 = 3.791

Present value

PV = 1 000·3.791 = €3 790.79

Answer

PV = €3 790.79

2.Time value of money

Answer: A = 314.7 €

Monthly rate and number of payments

i = 0.042/12 = 0.0035

N = 12·6 = 72

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 = 0.7776

A = 20 000·0.0035/(1 − 0.7776) = €314.73

Answer

A = €314.73

3.Bond pricing & yield

Answer: P = 1014 €

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 = €20.00, i = 0.0175, N = 6

Present value of the coupons

€20.00·(1 − 0.9011)/0.0175 = €112.98

Present value of the face value

€1 000·0.9011 = €901.14

Price

P = €1 014.12

Answer

P = €1 014.12

The yield is below the coupon rate, so the bond sells at a premium.

4.Bond pricing & yield

Answer: YTM = 8.993 %

Equation for the yield

€950 = €80·(1 − (1 + y)⁻⁷)/y + €1 000/(1 + y)⁷

Solve numerically

P(8%) = €1 000.00, P(9%) = €949.67

YTM = 8.993%

Answer

YTM = 8.993%

5.CAPM & WACC

Answer: r_m = 8 %

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).

Market risk premium

r_m − r_f = (12.5% − 3.5%)/2 = 4.5%

Market return

r_m = 3.5% + 4.5% = 8%

Answer

r_m = 8%

6.CAPM & WACC

Answer: WACC = 6.883 %

Cost of debt (YTM)

879 = 40·(1 − (1 + y)⁻¹¹)/y + 1000/(1 + y)¹¹

y = 5.495%

Market values

D = 8 000·€879 = €7 032 000

V = €12 032 000, E/V = 0.4156, D/V = 0.5844

WACC

WACC = 0.4156·11% + 0.5844·5.495%·(1 − 0.28) = 6.883%

Answer

WACC = 6.883%

7.NPV & IRR

Answer: DPB = 3.163 years

Payback period: the time until the cumulative cash flows repay the initial investment. Cash is assumed to come in evenly during each year, so the last part is a fraction of a year. The simple payback ignores the time value of money; the discounted payback period uses the discounted cash flows instead.

Discounted and cumulative cash flows

year 1: 17 500/1.05 = 16 666.67 → Σ = 16 666.67

year 2: 17 500/1.103 = 15 873.02 → Σ = 32 539.68

year 3: 17 500/1.158 = 15 117.16 → Σ = 47 656.84

year 4: 17 500/1.216 = 14 397.29 → Σ = 62 054.13

Fraction of the last year

DPB = 3 + 2 343.16/14 397.29 = 3.163

Answer

DPB = 3.163 years

The discounted payback is always longer than the simple payback (2.857 years).

8.NPV & IRR

Answer: MIRR = 12.36 %

Modified IRR: compound every inflow forward to the end of the project at the reinvestment rate (usually the cost of capital) to get the terminal value TV. MIRR is the rate that turns the initial investment into TV: MIRR = (TV/C₀)^(1/n) − 1.

Terminal value of the inflows

10 000·1.08² = 11 664.00

10 000·1.08¹ = 10 800.00

13 000·1.08⁰ = 13 000.00

TV = 35 464.00

MIRR

MIRR = (35 464.00/25 000)^1/3 − 1 = 12.36%

Answer

MIRR = 12.36%

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