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Quantitative Methods
Future Value
Present Value
PV = FV / ( 1 + r ) n
Frequency of compounding
FV = PV ( 1 + r )
n
Perpetuity
PV =
PMT
(r/n)
Geometric Mean
[ ( 1 + r1 ) ( 1 + r ) ( 1 + rn ) ]
1/n
2
-1
Harmonic Mean
n
Position of an observation
at a given percentile
(n+1) x
y
100
Mean Abs. Deviation (MAD)
𝐢"𝟏
Population Variance
𝐍
!−
𝐢"𝟏
𝐍
!−
1 + 1
1
+
x2
xn
x1
Xi - x
n
Xi - 𝛍
2
n
Population stdev.
Pop. variance
Sample Variance
𝐍
!−
𝐢"𝟏
Xi - x
2
n-1
Population stdev.
Sample variance
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Quantitative Methods
Correlation of x & y
Coefficient of Variation
Standard Deviation
covariance (xy)
Mean
(σx) (σy)
Addition Rule
Multiplication Rule (Joint Probs.)
P(AB) = P( A I B ) x P(B)
P(A or B) = P(A) + P(B) - P(AB)
Total Probability Rule *
P(A) = P(A I B1 ) x P( B1 ) + P(A I B2 ) x P( B2 ) + … + P(A I Bn ) x P( Bn )
Expected Value
EV = (X1)P(X1) + (X2)P(X2) … + (Xn)P(Xn)
Variance of 2-Stock Portfolio
wA2 σA2 + wB2 σB2 + 2wA wB covarianceAB
FOR BINOMIAL RANDOM VARIABLE
Probability of “x” successes in “n” trials
p(x) =
n!
(n – x)! x!
x
n-x
p (1-p)
Expected
value
E(x) = (n)(p)
Variance
Var(x) = (n)(p)(1-p)
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Quantitative Methods
Normal Distributions
68% of observations fall within ± 1σ
90% fall within ± 1.65σ
95% fall within ± 1.96σ
99% fall within ± 2.58σ
Z-score (number of σ a given observation is from the population mean)
Z-score =
x-μ
=
σ
observed - population
value
mean
standard deviation
Roy’s Safety-First Ratio (SFR)
SFR =
E(Rp) - R L
σ
=
Expected Threshold
return –
level
Standard deviation
Continuously Compounded Rate Equalities
price relative
final price
initial price
Holding
Continuously
period return compounded rate
= ( 1 + HPR ) =
e RCC
Standard Error
Known population variance à
σ/ n
Unknown population variance à s / n
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Quantitative Methods
Common Z-values for confidence intervals
Zα/2 = 1.645 for 90% confidence intervals
(level of significance is 10%, 5% in each tail)
Zα/2 = 1.960 for 95% confidence intervals
(level of significance is 5%, 2.5% in each tail)
Zα/2 = 2.575 for 99% confidence intervals
(level of significance is 1%, 0.5% in each tail)
Formulas to calculate confidence intervals *
σ
– + Zα/2
x –
n
or
– + tα/2
x –
s
n
Linear Regression
Total Variation
=
SST
Explained variation
SSR
+ Unexplained variation
SSE
SSR
F-stat =
MSR
MSE
k
SSE
R2 =
n – ( k+1 )
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SSR
SST
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Quantitative Methods
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Economics
Elasticity Formulas
%Δ Qty
Own-price
=
Elasticity
%Δ Price
%Δ Qty
Cross-price =
Elasticity
%Δ Price (related good)
Income
Elasticity
=
%Δ Qty
%Δ Income
Marginal Product
Marginal
=
Product
N-firm Concentration Ratio
Δ output
Δ labor
Sum of mkt shares of N largest firms
HHI Ratio
Sum of SQUARED market shares of N largest firms
Gross Domestic Product
GDP = C + I + G + (X – M)
Saving, investment, Fiscal and Trade Balance *
S = I + (G – T) + (X – M)
GDP Deflator
Nominal GDP
Real GDP
x 100
Solow Growth Model
Growth in
Growth
Growth
Growth
=
+
W
L
+
W
c
potential GDP
in tech
in capital
in labor
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Economics
Unemployment Ratios
Unemployed
Unemployment
=
rate
Labor force
Participation
ratio
=
Labor force
Working age
population
Money Creation
Initial deposit
Money
=
created
Reserve requirement
Quantity Theory of Money
(M)
Money
supply
x
(V)
Velocity of $
in circulation
=
(P)
Price
level
x
(Y)
Real
output
Neutral Interest Rate
Neutral interest rate = trend growth rate + inflation
Balance of Payments
Current account = Capital account + Financial account
Real Exchange Rates
Real
Nominal exchange rate x CPI base currency
exchange rate =
(spot rate)
CPI price currency
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Economics
No Arbitrage Forward Exchange Rates *
Forward
price
= Spot
price
base
base
x
(1 + r price)
(1 + r base)
Rate of Change
Rate of
=
change
final value
initial value
-1
Marshall-Lerner Condition
(Wx)(Ex) + (WM)(EM – 1) > 0
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Financial Statement Analysis
Basic EPS
Comprehensive Income
Net income
-
Preferred
dividends
Net
Other comprehensive
+
Income
income
Weighted avg # of
common shares
Diluted EPS
Calculating Inventory
Beginning inventory
+ Purchases (cash paid to suppliers)
- Cogs (cost inventory sold)
Ending inventory
Free Cash Flows *
FCFF = NI + NCC + Interest (1-t) - FC Inv - WC Inv
FCFF = CFO + Interest (1-t) - FC Inv
FCFE = CFO - FC Inv + Net borrowing
FCFE = FCFF - Interest (1-t) + Net borrowing
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Financial Statement Analysis
Coverage Ratios
Performance Ratios
Activity Turnover Ratios
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Financial Statement Analysis
“Days of” Ratios *
Liquidity Ratios
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Financial Statement Analysis
Solvency Ratios
Profitability Ratios
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Financial Statement Analysis
Straight-line Depreciation
Original cost – salvage value
Useful life
Double-declining Balance Depreciation
2
x
useful life
Cost – Accumulated
depreciation
Deferred Tax Liability *
DTL = (CV – TB) x Tax rate
• If taxable income < pre-tax income, deferred tax liability
• If taxable income > pre-tax income, deferred tax asset
Effective Interest Method
Interest
CV of bond liability
=
@ beg. of yr
expense
Mkt rate
at issuance
Amount = Int. expense – Int. payment
(coupon)
amortized
Retirement Plans
Fair value of
Funded
=
status
fund’s asset
_
PV of estimated
pension liabilities
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Financial Statement Analysis
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Corporate Issuers
Weighted Average Cost of Capital
WACC = (Wd)(rd)(1-t) + (Wp)(rp) + (We)(re)
Cost of Preferred Stock
rp =
CAPM
Dividend
re = rf + B(rm – rf)
Current
share price
MRP
Levered Beta
B’e = Bu
Unlevered Beta
1
D’
1 + (1-t)
E’
Bu = Be
Degree of Operating Leverage (DOL)
1 + (1-t)
D
E
Degree of Financial Leverage (DFL)
Q (P - V)
Q (P - V) - F
Q (P - V) - F
Q (P - V) – F - I
Degree of Total Leverage
Q (P – V)
Q (P - V) – F - I
Break-even Quantity *
Fixed
Fixed operating
+ interests (I)
costs (F)
QBE =
Contribution margin
(P – V)
Operating Break-even Quantity
QoBE =
P = price per unit
V = variable cost per unit
Fixed operating costs (F)
(P – V)
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Equity Investments
Leverage Ratio
Value of asset
Investor’s equity position
or
Margin Call Price
P0
Market Cap
(1 - initial margin)
(1 – maintenance margin)
Price-to-book Ratio
Market cap
Book value
1
initial margin requirement
(# of shares)(price per share)
Free Cash Flow to Equity *
NI + depr – ΔinWC – FCInv + net borrowing
or
CFO – FCInv + net borrowing
Price at t=0
P0 =
D1
r-g
D1 = (D0)(1+g)
g = (ROE)(RR)
RR = ( 1 – div. payout ratio)
Enterprise Value
mkt value of common mkt value of
cash and short
+
and preferred equity
debt
term investments
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Fixed Income
Full Price of a Bond
PV x
Flat Price of a Bond
Full price – Accrued interest
YTM t T
1+
n
Full price – (coupon)( t T )
Periodic Rate
Effective Yield
n
EFF = ( 1 + r ) - 1
r =
YTM
n
Money Market Instruments *
Money
market
yield
=
FV - PV
PV
x
Bond
FV - PV
equivalent =
PV
yield
Discount
yield
=
FV - PV
FV
x
x
360
n
365
n
360
n
Z-spread
Yield or Bond with an embedded option
Yield on
+
government bonds
OAS +
Option value
OAS = z-spread – option value
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Fixed Income
Single Monthly Mortality (SMM) rate
Prepayment for the month
Mortgage balance
Scheduled principal
_
at the beginning of month
repayments that month
Loan-to-value (LTV) Ratio
*
Debt-Service coverage (DSCR) ratio
Loan amount
LTV =
Market value
of collateral
Modified Duration
(1 + YTM)
n
Approximate Modified Duration
_
Debt service
% Change in Bond Price
Macaulay Duration
V_
DSCR =
Net operating
income
V+
2 x V 0 x ΔYTM
Money Duration
Annual
Full price of
Modified x
a bond
Duration
_ Annual modified
duration
Effective Duration
V_
_
V+
2 x V 0 x ΔCurve
Price Value of a Basis Point
PVBP =
V_
_
V+
2
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x
ΔYTM
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Fixed Income
Approximate Convexity
Effective Convexity
V_ + V + _ 2V0
(ΔYTM) 2 V0
V_ + V + _ 2V0
(Δcurve) 2 V0
Change in Bond’s Full Price *
_ Annual modified
x ΔYTM
duration
Duration
+
Annual
2 convexity
1
2
ΔYTM
Convexity
Duration Gap
Macaulay _ Investment
Duration
Horizon
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Derivatives
Spot Price
S0 =
Contract Value at time = t
F0(T)
( 1 + rf ) T
Vt = St _ PVt [F0(T)]
Spot price at initiation with cost of carry
S0 =
F0(T)
( 1 + rf ) T
+ PV(Benefits) - PV(Costs)
“Cost of carry”
Contract Value at time = t with Costs and Benefits
St
_
PVt [F0(T)] + PVt (Benefits) – PVt (Costs)
Put-Call parity *
+
S
Underlying
asset
P
=
Put
C
+
X
(1 + r)T
Call
Risk-free
bond
C
X
(1 + r)T
Forward Put-Call parity
Fwd price
(1 + r)T
Forward
contract
+
P
Put
=
Call
+
Risk-free
bond
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Portfolio Management
Diversification Ratio
Risk of equally weighted portfolio of “n” securities
Risk of a single random security
Holding Period Return (HPR)
Final price
-1
Initial price
Portfolio Variance
wA2 σA2 + wB2 σB2 + 2wA wB covarianceAB
Covariance of x and y
covariance (xy) = (σy) (σx) (correlation of x & y)
For any point on the Capital Allocation Line (CAL) *
E(r)p = ( W Rf ) ( Rf ) + ( W Rp ) [E(Rp)]
σp = ( W Rp ) ( σRp )
For any point on the Capital Market Line (CML)
E(r)p = ( W Rf ) ( Rf ) + ( Wm ) [E(Rm)]
σp = ( W m
Beta of Stock “i”
Beta i =
covariance i,mkt
σ 2 mkt
Market Model
or
ρ i,mkt σi
σ mkt
CAPM
Ri = B(Rm) + αi + ei
E(r) = Rf + B [E(Rm) – Rf]
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) ( σm )
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Portfolio Management
Sharpe Ratio
(Rp – Rf)
σp
Treynor Ratio
(Rp – Rf)
B
For any point on the Capital Market Line (CML)
E(Rp) =
[E(Rm) – Rf ]
σm
σp + Rf
Security Market Line (SML)
E(r) = [E(Rm) – Rf] B + Rf
M2 ratio
Jensen’s Alpha
(Rp – Rf)
σp
σm - (Rm – Rf)
Rp – [ Rf + B (Rm – Rf)
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Ethical and Professional
Standards
1: Professionalism
5: Investment analysis,
recommendations, and actions
1A. Knowledge of the law
5A. Diligence & reasonable basis
1B. Independence & Objectivity
5B. Communication with clients
and prospective clients
1C. Misrepresentation
1D. Misconduct
2: Integrity of capital markets
2A. Material nonpublic information
2B. Market manipulation
3: Duties to clients
3A. Loyalty, prudence and care
3B. Fair dealing
3C. Suitability
3D. Performance presentation
3E. Preservation of confidentiality
5C. Record retention
6: Conflicts of interest *
6A. Disclosure of conflicts
6B. Priority of transactions
6C. Referral fees
7: Responsibilities as a CFA®
institute member / candidate
7A. Conduct as participants in
CFA® programs
7B. Reference to CFA® institute,
the CFA® designation, and the
CFA® program
4. Duties to employers
4A. Loyalty
4B. Additional compensation
arrangements
4C. Responsibilities of supervisors
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CFA ®