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## Cfa1 2 Quantitative Methods Notes This is an extract of our Cfa1 2 Quantitative Methods document, which we sell as part of our CFA Level 1 Notes collection written by the top tier of University Of London (examined By LSE) students.

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CFA Level 1

Quantitative Methods

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
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 1 CFA Level 1

Quantitative Methods

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
=NRP
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 2 CFA Level 1

Quantitative Methods

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)
3 CFA Level 1

Quantitative Methods

N = ln (PV / FV) / ln (1 + R)
Discounted Cash Flow
Discounting
PV = FV / (1 + R / M)M  N
Continuous Discounting
PV = FV e-R  N
Net Present Value (NPV) and Internal Rate of Return (IRR)
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)
4 CFA Level 1

Quantitative Methods

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 (1-year 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)

5 CFA Level 1

Quantitative Methods

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)
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)]
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
Interval Scale
Provide ranking like ordinal scales 6 CFA Level 1

Quantitative Methods

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
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 7

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