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CFA Level 1

Unit 2: Quantitative Methods

UOLLB CFA Top Notes

Level 1 Exam

Unit 2: Quantitative Methods

Table of Contents

Time Value of Money........................................1

Equal Cash Flows..............................................2

Discounted Cash Flow......................................2

Introduction to Statistics....................................3

Measurement Scales..........................................4

Graphic Representation of Data........................4

Measures of Central Tendency..........................4

Measures of Dispersion.....................................5

Measure of Downside Risk...............................5

Normal Distribution..........................................6

Probability.........................................................6

Probability Distribution.....................................7

Binomial Distribution........................................7

Independent and Identical Distribution (IID)....8

Sampling and Estimation..................................9

Standard Error (SE) of Sample Mean (S)..........9

Point Estimators................................................9

Hypothesis Testing..........................................10

Hypothesis Testing Concerning Variance........12

Technical Analysis...........................................12

Contrary-Opinion Rule (Contrarian)...............13

Smart Money Rule..........................................13

Momentum Indicators.....................................13

Stock Price and Volume Techniques...............14

Time Value of Money

Time Value of Money

Concern equivalence relationships between cash flows occurring on different dates

Roles of Interest Rate

Required rates of return

Discount rate

Opportunity cost

Premium

Required returns for bearing distinct types of risk

Composition of Interest Rate

Real risk-free interest rate

Inflation premium

Default risk premium

Liquidity premium

Maturity premium

Real Risk-free Interest Rate

Single-period interest rate for completely risk-free security

Reflect the time preference of individuals for current v. future consumption

Inflation Premium

Compensates investors for expected inflation

Reflect the average inflation rate expected over the maturity of the debt

Nominal Risk-free Interest Rate

= Real Risk-free Interest Rate + Inflation

Premium

Default Risk Premium

Compensate investors for the possibility that the borrower will default 1 CFA Level 1

Unit 2: Quantitative Methods

Liquidity Premium

Compensate investors for the risk of loss from quick conversion of investment into cash

Maturity Premium

Compensate investors for the increased sensitivity of the market value of debt to a change in market interest rates as maturity is extended

Stated Annual Interest Rate

Interest rate before taking into account the frequency of compounding in a period

Calculating Time Value of Money

Present value (PV)

Future value (FV)

Interest rate (R)

No. of periods from today (N)

Frequency of compounding in a period (M)

Principle (P)

Simple Interest

=NRP

Compounding

FV = PV (1 + R / M) M N

Continuous Compounding

FV = PV e R N

Periodic Interest Rate

Stated annual interest rate divided by the frequency of compounding in a period

Effective Annual Interest Rate (EAR)

Interest rate after taking into account the frequency of compounding in a period

EAR = (1 + Periodic Interest Rate)M - 1

Continuous EAR

= eR - 1

Equal Cash Flows

Annuity

A stream of equal cash flows that occurs at equal intervals over a given period

Types of Annuity

Ordinary annuity (cash flows occur at the

end of each period)

Annuity due (cash flows occur at the beginning of each period)

Future Value (FV) of Ordinary Annuity

A = Annuity payment

FV = A [1 + (1 + R)1 + (1 + R) 2 + … + (1 +

R)N]

FV = A {[(1 + R)N - 1] / R}

Present Value (PV) of Ordinary Annuity

PV = A / [1 + (1 + R)1 + (1 + R) 2 + … + (1 +

R)N]

PV = A {[1 - (1 + R)-N] / R}

Future Value (FV) of Annuity Due

FV = A {[(1 + R)N - 1] / R} (1 +R)

Present Value (PV) of Annuity Due

PV = A {[1 - (1 + R)-N] / R} (1 + R)

Perpetuity

A set of never-ending sequential cash flows with T = 1

A = Amount paid per period

PV = A / R

Perpetuity with First Payment after N

Period(s)

PV = (A / R) / (1 + R)N + 1

Present Value of Equal Cash Flows

C = Payment per period

PV = C1 / (1 + R) + C2 / (1 + R)1 + … CN / (1

+ R)N

Accumulative Growth Rates

Growth Rate per Year (G)

Value of First Year (V1)

Value of Final Year (VT)

No. of Period between First and Final Year

(N)

G = (VT / V1)1 / N - 1

Growth Rate between Two Consecutive

Periods

G = (V2 - V1) / V1

Solving for the Number of Periods (N)

PV = FV (1 + R)N

PV / FV = (1 + R)N

ln (PV / FV) = N ln (1 + R)

N = ln (PV / FV) / ln (1 + R)

2 CFA Level 1

Unit 2: Quantitative Methods

Discounted Cash Flow

Discounting

PV = FV / (1 + R / M)M N

Continuous Discounting

PV = FV e-R N

Applications of Discounting

Capital budgeting: Allocation of funds to relatively long-range projects or investments

Capital structure: Choice of long-term financing for investments

Working capital management: Management of short-term assets (e.g. inventory)

Net Present Value (NPV)

Net Cash Flow (NCF) = Cash Inflows -

Cash Outflows

NPVN = NCF0 + NCF1 / (1 + R) + NFC2 / (1

+ R)2 + … + NCFN / (1 + R)N

Internal Rate of Return (IRR)

Set NPV = 0

Accept the project if IRR > Opportunity cost of capital (hurdle rate)

Reject the project if IRR < Opportunity cost of capital (hurdle rate)

0 = NPV

0 = NCF0 + NCF1 / (1 + IRR) + NCF2 / (1 +

IRR)2 + … + NCFN / (1 + IRR)N

NCF0 = Initial investment

Initial investment = NCF1 / (1 + IRR) +

NCF2 / (1 + IRR)2 + … + NCFN / (1 + IRR)N

Problems with IRR

Give conflicting recommendations to NPV

Cannot accommodate changes in interest rates over the life of a project

May lead to choices that do not maximize shareholders' wealth

May give multiple IRRs

Portfolio Return Measurement

Holding Period Return (HPR)

Initial Investment (P0)

Price Received (P1)

Interest payment (accrued) at the end of the holding period (D1)

HPR = (P1 - P0 + D1) / P0

Money-Weighted Rate of Return

Measure the compound growth rate in the value of all funds invested

= Internal Rate of Return

= YTM

Time-Weighted Rate of Return (RTW)

Measure the compound rate of return for one unit of money

= Average Returns over Each Sub-period

RTW = [(1 + R1) (1 + R2) … (1 + RN)]1 / N

-1

Rate of Return at each Sub-period (RT)

Market Value at the Beginning (MVB)

Market Value at the End (MVE)

= MVET - MVBT / MVBT

Short-Term Money Market Yield

Holding Period Yield (HPY) (= Holding

Period Return)

Money Market Yield

Bank Discount Yield

Effective Annual Yield

Money Market

Market for short-term debt instruments (1year maturity or less)

Pure Discount Instrument (e.g. T-Bill)

An instrument that pays no income until maturity

US Treasury Bill (T-Bill)

Investors pay the face value less the discount amount (Bank Discount Basis)

Investors receive the face value (P1) at maturity

Accrued Interest

Interest that has been declared but has not yet paid

The purchase and sale prices of a T-bill must include any accrued interest

The price with accrued interest is called the full price (dirty price)

The price without accrued interest is called the trade price (clean price)

Bank Discount Yield (BDY)

Purchase price paid at the beginning (P0)

Face value received at the end (P1)

No. of remaining day to maturity (T)

No. of days in a year (365)

3 CFA Level 1

Unit 2: Quantitative Methods

BDY = [(P1 - P0) / P1] (365 / T)

Effective Annual Yield (EAY)

= (1 + HPY)365 / T - 1

Money Market Yield (MMY) (also CD

Equivalent Yield)

= (HPY) (365 / T)

= (BDY) [(P1 / P0]

= (365 BDY) / [365 - (T) (BDY)]

Introduction to Statistics

Population

All members of a specified group

Sample

A subset of a population

Sample Statistic

A quantity computed from or used to describe a sample

Measurement Scales

Nominal Scales

Weakest level of measurement

Categorize data but do not rank them

Statistical methods: Mode, Chi Square

Ordinal Scale

Stronger level of measurement

Sort data into categories that are ordered with respect to some characteristics

Statistical methods: Median, Percentile

A tabular display of data summarized into a relatively small number of intervals

Interval

A set of values with which an observation falls

Absolute Frequency

Simply the frequency

Relative Frequency

Absolute frequency of each interval divided by the total no. of observation (%)

Cumulative Absolute Frequency

Add up the absolute frequencies from the first to the last interval (max 100%)

Cumulative Relative Frequency

Add up the relative frequencies from the first to the last interval (max 100%)

Graphic Representation of Data

Histogram

A bar chart showing the absolute frequency of each interval

Frequency Polygon

A straight line connecting the absolute frequency of each interval

Cumulative Frequency Distribution

A curve connecting the frequency of each interval from the lowest to the highest

Measures of Central Tendency

Interval Scale

Provide ranking like ordinal scales

Assure an equal interval value between scales

Statistical methods: Mean, Standard

Deviation, Correlation, Regression

Ratio Scale

Strongest level of measurement

Provide rankings like ordinal scales

Assure an equal interval value between scales

Have a true zero point as the origin

Statistical methods: Geometric Mean,

Harmonic Mean, Coefficient

Frequency Distribution

Arithmetic Mean

Sum of the observations divided by the number of observations (N)

Use all info about the size and magnitude of the observations

Sensitive to extreme values

Population Mean ()

N = Entire Population

Sample Mean (X')

N < Entire Population

Deviation

Distance between the mean and each outcome (X - X')

4

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