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 problemsBack 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

  1. Draw a timeline with every cash flow and its timing.
  2. Match the rate to the period: a monthly rate for monthly payments.
  3. Use FV = PV·(1 + i)ⁿ for single amounts and the annuity formulas for equal payments.
  4. 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

  1. Find the coupon per period and the yield per period.
  2. Price = present value of the coupons (an annuity) + present value of the face value.
  3. For semi-annual coupons use half the coupon, half the yield and twice the periods.
  4. Check: if the yield is above the coupon rate, the price is below face value.

CAPM & WACC

CAPM
re = rf + β·(rm − rf)
WACC
E/V·re + D/V·rd·(1 − T)
Firm value
V = E + D

NPV & IRR

Net present value
NPV = −C0 + Σ CFt/(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

  1. List all cash flows with their year; the investment at year 0 is negative.
  2. Discount each cash flow: CFt/(1 + r)t.
  3. Add them up to get the NPV. Positive means the project earns more than r.
  4. For the IRR, find the rate that makes NPV zero by trial or interpolation.

2Practice problems

1.Time value of moneyMedium
periods0123411 + EAR = ?18% / 4

A credit card charges a nominal interest rate of 18% per year, compounded quarterly. Find the effective annual rate EAR in percent.

r = 18%m = 4
EAR = %
2.Time value of moneyHard
months0123…300€150 000A?AAA

A loan of €150 000 is repaid with equal monthly payments over 25 years. The interest rate is 3% per year, compounded monthly. Find the monthly payment A.

P = €150 0003% / 12N = 300
A = €
3.Bond pricing & yieldMedium
years0123456€1 053C = ?C = ?C = ? + €1 000

A bond with a face value of €1 000 matures in 6 years and pays annual coupons. It sells for €1 053 when the yield to maturity is 3.5%. Find the coupon rate c in percent.

F = €1 000n = 6P = €1 053YTM = 3.5%
c = %
4.Bond pricing & yieldHard

A bond trades at €900 with a yield to maturity of 5% (annual compounding) and a Macaulay duration of 7.2 years. Use the modified duration to estimate the new price P if the yield falls by 50 basis points.

P0 = €900y = 5%D = 7.2Δy = −50 bp
P = €
5.CAPM & WACCMedium
WACC = ?ordinary shares€10 000 000re​ = 14%preferred shares€1 500 000rp​ = 7.5%debt€2 000 000rd​ = 5%T = 25%

A company is financed with ordinary shares worth €10 000 000, preferred shares worth €1 500 000 and debt worth €2 000 000. The cost of equity is 14%, the cost of preferred stock 7.5% and the pre-tax cost of debt 5%. The tax rate is 25%. Find the WACC.

E = €10 000 000Ps = €1 500 000D = €2 000 000re = 14%rp = 7.5%rd = 5%T = 25%
WACC = %
6.CAPM & WACCHard
WACC = ?equity5 000 000 × €40.50β = 1.2rf​ = 3%, rm​ = 8.5%debt10 000 × €960rd​ = 3.5%T = 25.8%

A company has 5 000 000 shares at €40.50 each and 10 000 bonds trading at €960 each. The pre-tax cost of debt is 3.5% and the tax rate is 25.8%. The share’s beta is 1.2, the risk-free rate 3% and the market return 8.5%. Find the WACC.

β = 1.2rf = 3%rm = 8.5%rd = 3.5%T = 25.8%
WACC = %
7.NPV & IRRMedium
years0123−40 00015 50015 50019 500

A machine costs €40 000 today. It generates €15 500 at the end of each year for 3 years and can then be sold for €4 000. The cost of capital is 12%. Find the net present value NPV.

C0 = €40 000CF = €15 500S = €4 000n = 3r = 12%
NPV = €
8.NPV & IRRHard
years0123456−40 0007 5007 5007 5007 5007 5007 500

An investment of €40 000 returns €7 500 at the end of each year for 6 years. Find its internal rate of return IRR in percent.

C0 = €40 000CF = €7 500n = 6
IRR = %

3Solutions

1.Time value of money

Answer: EAR = 19.25 %

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 per period

r/m = 0.18/4 = 0.045

Compound over one year

EAR = (1 + 0.045)4 − 1 = 1.193 − 1 = 19.25%

Answer

EAR = 19.25%

The EAR is always above the nominal rate when interest compounds more than once a year.

2.Time value of money

Answer: A = 711.3 €

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.

Monthly rate and number of payments

i = 0.03/12 = 0.0025
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 = 0.4728
A = 150 000·0.0025/(1 − 0.4728) = €711.32

Answer

A = €711.32
3.Bond pricing & yield

Answer: c = 4.495 %

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 + 0.035)−6 = €813.50

Coupon

C = (1053 − 813.50)/5.329 = €44.95

Coupon rate

c = 44.95/1000 = 4.495%

Answer

c = 4.495%
4.Bond pricing & yield

Answer: P = 930.9 €

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 ≈ −Dmod·Δy.

Modified duration

Dmod = 7.2/1.05 = 6.857

Estimated price change

%ΔP ≈ −6.857·(−0.005) = 3.429%
P ≈ 900·(1 + 0.03429) = €930.86

Answer

P = €930.86

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

5.CAPM & WACC

Answer: WACC = 11.76 %

Cost of capital: CAPM gives the return shareholders require: re = rf + β·(rm − rf). The weighted average cost of capital mixes equity and debt by their market values: WACC = E/V·re + D/V·rd·(1 − T), with V = E + D. Interest is tax-deductible, hence the (1 − T).

Weights

V = 10 000 000 + 1 500 000 + 2 000 000 = €13 500 000
E/V = 0.7407, Ps/V = 0.1111, D/V = 0.1481

WACC

WACC = 0.7407·14% + 0.1111·7.5% + 0.1481·5%·(1 − 0.25) = 11.76%

Answer

WACC = 11.76%
6.CAPM & WACC

Answer: WACC = 9.283 %

Cost of capital: CAPM gives the return shareholders require: re = rf + β·(rm − rf). The weighted average cost of capital mixes equity and debt by their market values: WACC = E/V·re + D/V·rd·(1 − T), with V = E + D. Interest is tax-deductible, hence the (1 − T).

Market values

E = 5 000 000·€40.50 = €202 500 000
D = 10 000·€960 = €9 600 000

Cost of equity (CAPM)

re = 3% + 1.2·5.5% = 9.6%

Weights

V = €212 100 000, E/V = 0.9547, D/V = 0.04526

WACC

WACC = 0.9547·9.6% + 0.04526·3.5%·(1 − 0.258) = 9.283%

Answer

WACC = 9.283%
7.NPV & IRR

Answer: NPV = 75.51 €

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

15 500·2.402 = 37 228.38

Present value of the salvage value

4 000/1.405 = 2 847.12

NPV

NPV = −40 000 + 37 228.38 + 2 847.12 = €75.51

Answer

NPV = €75.51
8.NPV & IRR

Answer: IRR = 3.473 %

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.

Required annuity factor

(1 − (1 + i)−6)/i = 40 000/7 500 = 5.333

Solve for the rate

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

i = 3%: 5.417
i = 4%: 5.242
IRR = 3.473%

Answer

IRR = 3.473%

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