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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
 Required returns for bearing distinct types of risk
Composition of Interest Rate
 Real risk-free interest rate
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
 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
 Compensate investors for the possibility that the borrower will default
 Compensate investors for the risk of loss from quick conversion of investment into cash
 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)
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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