CFA Program Level 1”

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CFA \({ }^{\text {® }}\) Program Level I

FORMULA SHEET (2025)

Table of Contents

• Setting Up the Texas BA II Plus Financial Calculator
• VOLUME 1: QUANTITATIVE METHODS
  • Learning Module 1: Rates and Returns
  • Learning Module 2: Time Value of Money in Finance
  • Learning Module 3: Statistical Measures of Asset Returns
  • Learning Module 4: Probability Trees and Conditional Expectations
  • Learning Module 5: Portfolio Mathematics
  • Learning Module 6: Simulation Methods
  • Learning Module 7: Estimation and Inference
  • Learning Module 8: Hypothesis Testing
  • Learning Module 9: Parametric and Non-Parametric Tests of Independence
  • Learning Module 10: Simple Linear Regression
  • Learning Module 11: Introduction to Big Data Techniques
• VOLUME 2: ECONOMICS
  • Learning Module 1: The Firm and Market Structures
  • Learning Module 2: Understanding Business Cycles
  • Learning Module 3: Fiscal Policy
  • Learning Module 4: Monetary Policy
  • Learning Module 5: Introduction to Geopolitics
  • Learning Module 6: International Trade
  • Learning Module 7: Capital Flows and the FX Market
  • Learning Module 8: Exchange Rate Calculations
• VOLUME 3: CORPORATE ISSUERS
  • Learning Module 1: Organizational Forms, Corporate Issuer Features, and Ownership
  • Learning Module 2: Investors and Other Stakeholders
  • Learning Module 3: Working Capital and Liquidity
  • Learning Module 4: Corporate Governance: Conflicts, Mechanisms, Risks, and Benefits
  • Learning Module 5: Capital Investments and Capital Allocation
  • Learning Module 6: Capital Structure
  • Learning Module 7: Business Models
• VOLUME 4: FINANCIAL STATEMENT ANALYSIS
  • Learning Module 1: Introduction to Financial Statement Analysis
  • Learning Module 2: Analyzing Income Statements
  • Learning Module 3: Analyzing Balance Sheets
  • Learning Module 4: Analyzing Statements of Cash Flows I
  • Learning Module 5: Analyzing Statements of Cash Flows II
  • Learning Module 6: Analysis of Inventories
  • Learning Module 7: Analysis of Long-Term Assets
  • Learning Module 8: Topics in Long-Term Liabilities and Equity
  • Learning Module 9: Analysis of Income Taxes
  • Learning Module 10: Financial Reporting Quality
  • Learning Module 11: Financial Analysis Techniques
  • Learning Module 12: Introduction to Financial Statement Modeling
• VOLUME 5: EQUITY INVESTMENTS
  • Learning Module 1: Market Organization and Structure
  • Learning Module 2: Security Market Indexes
  • Learning Module 3: Market Efficiency
  • Learning Module 4: Overview of Equity Securities
  • Learning Module 5: Company Analysis: Past and Present
  • Learning Module 6: Industry and Company Analysis
  • Learning Module 7: Company Analysis: Forecasting
  • Learning Module 8: Equity Valuation: Concepts and Basic Tools
• VOLUME 6: FIXED INCOME
  • Learning Module 1: Fixed-Income Instrument Features
  • Learning Module 2: Fixed-Income Cash Flows and Types
  • Learning Module 3: Fixed-Income Issuance and Trading
  • Learning Module 4: Fixed-Income Markets for Corporate Issuers
  • Learning Module 5: Fixed-Income Markets for Government Issuers
  • Learning Module 6: Fixed-Income Bond Valuation: Prices and Yields
  • Learning Module 7: Yield and Yield Spread Measures for Fixed Rate Bonds
  • Learning Module 8: Yield and Yield Spread Measures for Floating-Rate Instruments
  • Learning Module 9: The Term Structure of Interest Rates: Spot, Par, and Forward Curves
  • Learning Module 10: Interest Rate Risk and Return
  • Learning Module 11: Yield-Based Bond Duration Measures and Properties
  • Learning Module 12: Yield-Based Bond Convexity and Portfolio Properties
  • Learning Module 13: Curve-Based and Empirical Fixed-Income Risk Measures
  • Learning Module 14: Credit Risk
  • Learning Module 15: Credit Analysis for Government Issuers
  • Learning Module 16: Credit Analysis for Corporate Issuers
  • Learning Module 17: Fixed-Income Securitization
  • Learning Module 18: Asset-Backed Security (ABS) Instrument and Market Features
  • Learning Module 19: Mortgage-Backed Security (MBS) Instrument and Market Features
• VOLUME 7: DERIVATIVES
  • Learning Module 1: Derivative Instrument and Derivatives Market Features
  • Learning Module 2: Forward Commitments and Contingent Claim Features and Instruments
  • Learning Module 3: Derivative Benefits, Risks, and Issuer and Investor Uses
  • Learning Module 4: Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives
  • Learning Module 5: Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities
  • Learning Module 6: Pricing and Valuation of Futures Contracts
  • Learning Module 7: Pricing and Valuation of Interest Rates and Other Swaps
  • Learning Module 8: Pricing and Valuation of Options
  • Learning Module 9: Option Replication Using Put-Call Parity
  • Learning Module 10: Valuing a Derivative Using a One-Period Binomial Model
• VOLUME 8: ALTERNATIVE INVESTMENTS
  • Learning Module 1: Alternative Investment Features, Methods, and Structures
  • Learning Module 2: Alternative Investment Performance and Returns
  • Learning Module 3: Investments in Private Capital: Equity and Debt
  • Learning Module 4: Real Estate and Infrastructure
  • Learning Module 5: Natural Resources
  • Learning Module 6: Hedge Funds
  • Learning Module 7: Introduction to Digital Assets
• VOLUME 9: PORTFOLIO MANAGEMENT
  • Learning Module 1: Portfolio Risk and Return: Part I
  • Learning Module 2: Portfolio Risk and Return: Part II
  • Learning Module 3: Portfolio Management: An Overview
  • Learning Module 4: Basics of Portfolio Planning and Construction
  • Learning Module 5: The Behavioral Biases of Individuals
  • Learning Module 6: Introduction to Risk Management

CFA Level 1 - Formula Sheet (2025)

Setting Up the Texas BA II Plus Financial Calculator

Video: https://youtu.be/OMS8d8QOFmc

Using Texas BA II Plus Financial Calculator

Video: https://youtu.be/LWmTTiZz8BU

Video (Requires Login to Facebook): https://fb.watch/nci5V7Dwtj/

VOLUME 1: QUANTITATIVE METHODS

Learning Module 1: Rates and Returns

Determinants of Interest Rates

Interest rate, \(r=\) Real risk-free rate + Inflation premium + Default risk premium

• Liquidity premium + Maturity premium \((1+\) Nominal risk-free rate \()=(1+\) Real risk-free rate \() \times(1+\) Inflation premium \()\)

Nominal risk-free rate \(=\) Real risk-free rate + Inflation premium

Maturity premium = Interest rate on longer-maturity, liquid Treasury debt

• Interest rate on short-term Treasury debt

Holding Period Return

\[ R=\frac{P_{1}-P_{0}+I_{1}}{P_{0}} \]

where:

• \(P_{0}=\) Price at the beginning of the period
• \(P_{1}=\) Price at the end of the period
• \(I_{1}=\) Income

If given holding period returns \(R_{1}, R_{2}, \ldots, R_{T}\) over the holding period:

\[ R=\left(1+R_{1}\right) \times\left(1+R_{2}\right) \times \ldots \times\left(1+R_{T}\right)-1 \]

Arithmetic Return

\[ \bar{R}_{i}=\frac{1}{T} \sum_{t=1}^{T} R_{i t}=\frac{1}{T}\left(R_{i 1}+R_{i 2}+\cdots+R_{i T}\right) \]

Geometric Mean Return

\[ \bar{R}_{G i}=\sqrt[T]{\prod_{t=1}^{T}\left(1+R_{t}\right)-1}=\sqrt[T]{\left(1+R_{i 1}\right) \times\left(1+R_{i 2}\right) \times \ldots \times\left(1+R_{i T}\right)}-1 \]

Harmonic Mean

\[ \bar{X}_{H i}=\frac{n}{\sum_{i=1}^{n}\left(1 / X_{i}\right)} \quad \text { for } X_{i}>0 \]

Relationship between Arithmetic Mean, Geometric Mean, and Harmonic Mean

\[ (\text { Geometric mean })^{2}=\text { Arithmetic mean } \text { × } \text { Harmonic mean } \]

Money-Weighted Return (MWR)

\[ \sum_{t=0}^{T} \frac{C F_{t}}{(1+M W R)^{t}}=0 \]

Time-Weighted Return (TWR)

Given the holding period returns for each sub-period, \(R_{1}, R_{2}, \ldots, R_{T}\)

If \(\mathrm{T}>1\) year, then

\[ \text { Annualized TWR }=\left[\left(1+R_{1}\right) \times\left(1+R_{2}\right) \times \ldots \times\left(1+R_{T}\right)\right]^{1 / T}-1 \]

If \(T=1\) year, then

\[ \text { Annualized } T W R=\left(1+R_{1}\right) \times\left(1+R_{2}\right) \times \ldots \times\left(1+R_{T}\right)-1 \]

If \(T<1\) year, then

\[ \text { TWR for holding period }=\left(1+R_{1}\right) \times\left(1+R_{2}\right) \times \ldots \times\left(1+R_{T}\right)-1 \]

Non-Annual Compounding

\[ P V=F V_{N}\left(1+\frac{R_{S}}{m}\right)^{-m N} \]

where:

• \(m=\) Number of compounding periods per year
• \(R_{s}=\) Quoted annual interest rate
• \(N=\) Number of years

Annualizing Returns

\(R_{\text {annual }}=\left(1+R_{\text {weekly }}\right)^{52}-1\) \(R_{\text {annual }}=\left(1+R_{\text {monthly }}\right)^{12}-1\) \(R_{\text {annual }}=\left(1+R_{\text {daily }}\right)^{252}-1 \quad\) assuming 252 trading days per year \(R_{\text {weekly }}=\left(1+R_{\text {daily }}\right)^{5}-1 \quad\) assuming 5 trading days per week

Continuously Compounded Returns

\[ \begin{gathered} P_{t}=P_{0} e^{r_{0, T}} \\ r_{0, T}=\ln \left(\frac{P_{t}}{P_{0}}\right) \\ r_{0, T}=r_{0,1}+r_{1,2}+\cdots+r_{T-2, T-1}+r_{T-1, T} \end{gathered} \]

Real Returns

\((1+\) real return \()=(1+\) real risk-free rate \() \times(1+\) risk premium \()\)

Pre-Tax and After-Tax Nominal Return

After-tax nominal return \(=\) Pre-tax nominal return × ( 1 - Tax rate) After-tax real return \(=\frac{[1+\text { Pre-Tax nominal return } \times(1-\text { Tax rate })]}{1+\text { Inflation premium }}-1\)

Leveraged Return

Return on a leveraged portfolio

\[ R_{L}=R_{P}+\frac{V_{B}}{V_{E}}\left(R_{P}-r_{D}\right) \]

where:

• \(R_{P}=\) Return on the investment portfolio (unleveraged)
• \(r_{D}=\) Cost of debt
• \(V_{B}=\) Debt/borrowed funds
• \(V_{E}=\) Equity of the portfolio

Learning Module 2: Time Value of Money in Finance

\[ F V_{t}=P V(1+r)^{t} \quad P V=\frac{F V_{t}}{(1+r)^{t}} \]

where:

• \(F V_{t}=\) Future value at time \(t\)
• \(P V=\) Present value
• \(r=\) Discount rate per period
• \(t=\) Number of compounding periods

As compounding frequency becomes very large (i.e., continuous compounding)

\[ F V_{t}=P V e^{r t} \quad P V=F V_{t} e^{-r t} \]

Present Value of Zero-Coupon Bond

\[ P V(\text { Discount Bond })=\frac{F V}{(1+r)^{t}} \]

where:

• \(F V=\) Principal (or Face Value)
• \(r=\) Market discount rate per period
• \(t=\) Maturity of bond

\[ r=\left(\frac{F V_{t}}{P V}\right)^{1 / T}-1 \]

Present Value of Coupon Bond

\[ P V(\text { Coupon Bond })=\frac{P M T}{(1+r)^{1}}+\frac{P M T}{(1+r)^{2}}+\cdots+\frac{P M T+F V}{(1+r)^{N}} \]

where:

• \(P V=\) Bond’s price
• PMT = Periodic coupon payment
• \(F V=\) Face value
• \(N=\) Number of periods
• \(r=\) Market discount rate per period

Present Value of a Perpetual Bond (Perpetuity)

\[ P V(\text { Perpetual Bond })=\frac{P M T}{r} \]

Annuity Instruments (e.g., Mortgage)

\[ A=\frac{r P V}{1-(1+r)^{-t}} \]

where:

• \(A=\) Periodic cash flow
• \(r=\) Market interest rate per period
• \(P V=\) Present value or principal amount of loan/bond
• \(t=\) Number of payment periods

Price of a Preferred Share

\[ P V_{t}=\frac{D_{t}}{r} \]

where:

• \(D_{t}=\) Fixed periodic dividend
• \(r=\) Expected rate of return

Price of a Common Share

Constant Dividend Growth Rate into Perpetuity

\[ P V_{t}=\frac{D_{t}(1+g)}{r-g}=\frac{D_{t+1}}{r-g} \quad r>g \]

where:

• \(D_{t}=\) Common dividend at time \(t\)
• \(g=\) Constant growth rate
• \(r=\) Expected rate of return

\[ \begin{gathered} r=\frac{D_{t+1}}{P V_{t}}+g \\ \frac{P V_{t}}{E_{t}}=\frac{\frac{D_{t}}{E_{t}} \times(1+g)}{r-g} \\ \frac{P V_{t}}{E_{t+1}}=\frac{\frac{D_{t+1}}{E_{t+1}}}{r-g} \end{gathered} \]

where:

• \(E_{t}=\) Earnings per share for period \(t\)
• \(\frac{P V_{t}}{E_{t}}=\) Trailing price-to-earnings ratio
• \(\frac{P V_{t}}{E_{t+1}}=\) Forward price-to-earnings ratio

Two-stage Dividend Discount Model

\[ P V_{t}=\sum_{i=1}^{n} \frac{D_{t}\left(1+g_{s}\right)^{i}}{(1+r)^{i}}+\frac{E\left(S_{t+n}\right)}{(1+r)^{n}} \]

where:

• \(g_{s}=\) Higher short-term dividend growth rate
• \(g_{L}=\) Lower long-term dividend growth rate
• \(n=\) Initial growth phase
• \(E\left(S_{t+n}\right)=\) Stock value in \(n\) periods (Terminal value)

\[ =\frac{D_{t+n+1}}{r-g_{L}} \]

Forward Rate

\[ F_{1,1}=\frac{\left(1+r_{2}\right)^{2}}{\left(1+r_{1}\right)}-1 \]

where:

• \(F_{1,1}=\) One-year forward rate one year from now
• \(r_{1}=\) Discount rate on one-year risk-free discount bond
• \(r_{2}=\) Discount rate on two-year risk-free discount bond

Learning Module 3: Statistical Measures of Asset Returns

Measures of Central Tendency

\[ \text { Sample Mean, } \quad \bar{X}=\frac{1}{n} \sum_{i=1}^{n} X_{i} \]

where:

• \(X_{i}=\) Observation \(i(i=1,2,3, \ldots, n)\)

Median

\[ \text { Position of median }=\frac{\text { Number of observations }+1}{2} \]

Quantiles

\[ \text { Interquartile range }=Q_{3}-Q_{1} \]

where:

• \(Q_{1}=\) First quartile
• \(Q_{3}=\) Third quartile

Box and Whisker Plot

\[ \begin{aligned} & \text { Upper fence }=Q_{3}+1.5 \times I Q R \\ & \text { Lower fence }=Q_{1}-1.5 \times I Q R \end{aligned} \]

Measures of Dispersion

\[ \text { Range }=\text { Maximum value }- \text { Minimum value } \]

Mean Absolute Deviation (MAD)

\[ M A D=\frac{\sum_{i=1}^{n}\left|X_{i}-\bar{X}\right|}{n} \]

Sample Variance

\[ s^{2}=\frac{\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}}{n-1} \]

Sample Standard Deviation

\[ s=\sqrt{\frac{\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}}{n-1}} \]

Sample Target Semideviation

\[ s_{\text {Target }}=\sqrt{\frac{\sum_{X_{i} \leq B}^{n}\left(X_{i}-B\right)^{2}}{n-1}} \]

where:

• \(B=\) target
• n = total number of sample observations

Coefficient of Variation

\[ C V=\frac{s}{\bar{X}} \]

Sample Skewness

\[ \text { Skewness } \approx\left(\frac{1}{n}\right) \frac{\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{3}}{s^{3}} \]

Sample Excess Kurtosis

\[ K_{E} \approx\left(\frac{1}{n}\right) \frac{\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{4}}{s^{4}}-3 \]

Sample Covariance

\[ s_{X Y}=\frac{1}{n-1} \sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)\left(Y_{i}-\bar{Y}\right) \]

Sample Correlation Coefficient

\[ r_{X Y}=\frac{s_{X Y}}{s_{X} s_{Y}} \]

Learning Module 4: Probability Trees and Conditional Expectations

Expected Value of a Discrete Random Variable

\[ E(X)=\sum_{i=1}^{n} P\left(X_{i}\right) X_{i} \]

Variance of a Random Variable

\[ \begin{aligned} \sigma^{2}(X) & =E[X-E(X)]^{2} \\ & =\sum_{i=1}^{n} P\left(X_{i}\right)[X-E(X)]^{2} \end{aligned} \]

Conditional Expected Value of a Random Variable

\[ E(X \mid S)=P\left(X_{1} \mid S\right) X_{1}+P\left(X_{2} \mid S\right) X_{2}+\cdots+P\left(X_{n} \mid S\right) X_{n} \]

Conditional Variance of a Random Variable

\[ \begin{gathered} \sigma^{2}(X \mid S)=P\left(X_{1} \mid S\right)\left[X_{1}-E\left(X_{1} \mid S\right)\right]^{2}+P\left(X_{2} \mid S\right)\left[X_{2}-E\left(X_{2} \mid S\right)\right]^{2}+\cdots \\ +P\left(X_{n} \mid S\right)\left[X_{n}-E\left(X_{n} \mid S\right)\right]^{2} \end{gathered} \]

Total Probability Rule for Expected Value

\[ E(X)=E\left(X \mid S_{1}\right) P\left(S_{1}\right)+E\left(X \mid S_{2}\right) P\left(S_{2}\right)+\cdots+E\left(X \mid S_{n}\right) P\left(S_{n}\right) \]

where:

• \(S_{1}, S_{2}, \ldots, S_{n}\) are mutually exclusive and exhaustive events.

Bayes’ Formula

\[ \begin{gathered} P(A \mid B)=\frac{P(B \mid A)}{P(B)} \times P(A) \\ P(\text { Event } \mid \text { Information })=\frac{P(\text { Information } \mid \text { Event })}{P(\text { Information })} \times P(\text { Event }) \end{gathered} \]

Learning Module 5: Portfolio Mathematics

For \(n\) assets in a portfolio

Expected return on portfolio

\[ E\left(R_{P}\right)=w_{1} E\left(R_{1}\right)+w_{2} E\left(R_{2}\right)+\cdots+w_{n} E\left(R_{n}\right) \]

Variance on portfolio

\[ \sigma^{2}\left(R_{P}\right)=\sum_{i=1}^{n} \sum_{j=1}^{n} w_{i} w_{j} \operatorname{Cov}\left(R_{i}, R_{j}\right) \]

Requires \(n\) variances and \(\frac{n(n-1)}{2}\) distinct covariances to estimate portfolio variance.

Covariance

\[ \begin{aligned} \operatorname{Cov}\left(R_{i}, R_{j}\right) & =E\left[\left(R_{i}-E\left(R_{i}\right)\right)\left(R_{j}-E\left(R_{j}\right)\right)\right] \\ & =\frac{1}{n-1} \sum_{t=1}^{n}\left(R_{i, t}-\bar{R}_{i}\right)\left(R_{j, t}-\bar{R}_{j}\right) \end{aligned} \]

For a two-asset \((n=2)\) portfolio:

\[ \sigma^{2}\left(R_{P}\right)=w_{1}^{2} \sigma_{1}^{2}+w_{2}^{2} \sigma_{2}^{2}+2 w_{1} w_{2} \operatorname{Cov}\left(R_{1}, R_{2}\right) \]

where:

• \(\operatorname{Cov}\left(R_{1}, R_{2}\right)=\rho\left(R_{1}, R_{2}\right) \times \sigma\left(R_{1}\right) \times \sigma\left(R_{2}\right)\)

Video: https://youtu.be/IUwulZ9ONCO

For a three-asset ( \(n=3\) ) portfolio:

\[ \begin{aligned} \sigma^{2}\left(R_{P}\right)=w_{1}^{2} \sigma_{1}^{2}+ & w_{2}^{2} \sigma_{2}^{2}+w_{3}^{2} \sigma_{3}^{2}+2 w_{1} w_{2} \operatorname{Cov}\left(R_{1}, R_{2}\right) \\ & +2 w_{1} w_{3} \operatorname{Cov}\left(R_{1}, R_{3}\right)+2 w_{2} w_{3} \operatorname{Cov}\left(R_{2}, R_{3}\right) \end{aligned} \]

Covariance Given a Joint Probability Function

\[ \operatorname{Cov}\left(R_{A}, R_{B}\right)=\sum_{i=1} \sum_{j=1} P\left(R_{A, i}, R_{B, j}\right) \times\left[R_{A, i}-E\left(R_{A}\right)\right] \times\left[R_{B, j}-E\left(R_{B}\right)\right] \]

If \(X\) and \(Y\) are uncorrelated, then \(E(X Y)=E(X) E(Y)\)

If \(X\) and \(Y\) are independent, then \(P(X, Y)=P(X) P(Y)\)

Safety-First Optimal Portfolio

Safety-First Ratio

\[ \text { SFRatio }=\frac{E\left(R_{P}\right)-R_{L}}{\sigma_{P}} \]

\[ \text { Shortfall risk }=\operatorname{Pr}\left[E\left(R_{P}\right)<R_{L}\right]=\operatorname{Normal}(- \text { SFRatio }) \]

where:

• \(R_{L}=\) Investor’s threshold level
• \(E\left(R_{P}\right)=\) Expected portfolio return
• \(\sigma_{P}=\) Portfolio standard deviation

Video: https://youtu.be/S3x5JrGIOUA

Learning Module 6: Simulation Methods

Lognormal Distribution

Mean of a lognormal random variable

\[ \mu_{L}=\exp \left(\mu+0.50 \sigma^{2}\right) \]

Variance of a lognormal random variable

\[ \sigma_{L}^{2}=\exp \left(2 \mu+\sigma^{2}\right) \times\left[\exp \left(\sigma^{2}\right)-1\right] \]

where:

• \(\mu=\) Mean of the normal random variable
• \(\sigma^{2}=\) Variance of the normal random variable

Continuously Compounded Rates of Return

\[ P_{T}=P_{0} \exp \left(r_{0, T}\right) \]

where:

• \(P_{0}=\) Current asset price
• \(P_{T}=\) Asset price at time \(T\)
• \(r_{0, T}=\) Continuously compounded return from 0 to \(T\)

If returns are independently and identically distributed (i.i.d.), then

\[ r_{0, T}=r_{0,1}+r_{1,2}+\cdots+r_{T-2, T-1}+r_{T-1, T} \]

If the one-period continuously compounded returns are i.i.d. random variables with mean \(\mu\) and \(\sigma^{2}\), then

\[ \begin{aligned} E\left(r_{0, T}\right) & =\mu T \\ \sigma^{2}\left(r_{0, T}\right) & =\sigma^{2} T \\ \sigma\left(r_{0, T}\right) & =\sigma \sqrt{T} \end{aligned} \]

Learning Module 7: Estimation and Inference

\[ \text { Sharpe ratio }=\frac{R_{P}-R_{F}}{\sigma_{P}} \]

where:

• \(R_{P}=\) Portfolio return
• \(R_{F}=\) Risk-free rate
• \(\sigma_{P}=\) Portfolio standard deviation of return

\[ \begin{aligned} & \text { Variance of the sampling distribution }=\frac{\sigma^{2}}{n} \\ & \text { of the sample means } \\ & \text { Standard error of }=\frac{\sigma}{\sqrt{n}} \end{aligned} \]

where:

• \(\sigma=\) Population standard deviation
• \(n=\) Sample size

Note: If \(\sigma\) is not known, use \(s\), the sample standard deviation.

Bootstrap Resampling

\[ s_{\bar{X}}=\sqrt{\frac{1}{B-1} \sum_{b=1}^{B}\left(\hat{\theta}_{b}-\bar{\theta}\right)^{2}} \]

where:

• \(s_{\bar{X}}=\) Estimate of the standard error of the sample mean
• \(B=\) Number of resamples drawn from the original sample
• \(\hat{\theta}_{b}=\) Mean of a resample
• \(\bar{\theta}=\) Mean across all the resample means

Learning Module 8: Hypothesis Testing

\[ \begin{aligned} & \text { Confidence level }=1-\alpha \\ & \text { Power of the test }=1-\beta \end{aligned} \]

where:

• \(\alpha=\) Significance level (Probability of Type I error)
• \(\beta=\) Probability of Type II error

Test of a Single Mean

Test statistic

\[ t=\frac{\bar{X}-\mu_{0}}{s / \sqrt{n}} \]

Degrees of freedom \(=n-1\) \((1-\alpha) \%\) Confidence Interval \(=\bar{X}+\) Critical value \(\times\left(\frac{s}{\sqrt{n}}\right)\)

Test of the Difference in Means

Test statistic

\[ t=\frac{\left(\bar{X}_{d 1}-\bar{X}_{d 2}\right)-\left(\mu_{d 1}-\mu_{d 2}\right)}{\sqrt{\frac{s_{p}^{2}}{n_{d 1}}+\frac{s_{p}^{2}}{n_{d 2}}}} \]

Degrees of freedom \(=n_{d 1}+n_{d 2}-2\)

\[ s_{p}^{2}=\frac{\left(n_{d 1}-1\right) s_{d 1}^{2}+\left(n_{d 2}-1\right) s_{d 2}^{2}}{n_{d 1}+n_{d 2}-2} \]

Test of the Mean of Differences

Test statistic

\[ t=\frac{\bar{d}-\mu_{d 0}}{s_{\bar{d}}} \]

Degrees of freedom \(=n-1\)

Test of a Single Variance

Test statistic

\[ \chi^{2}=\frac{(n-1) s^{2}}{\sigma_{0}^{2}} \]

Degrees of freedom \(=n-1\)

Test of the Difference in Variances

Test statistic

\[ F=\frac{s_{\text {Before }}^{2}}{s_{\text {After }}^{2}} \]

Degrees of freedom \(=n_{1}-1, n_{2}-1\)

Test of a Correlation

Test statistic

\[ t=\frac{r \sqrt{n-2}}{\sqrt{1-r^{2}}} \]

Degrees of freedom \(=n-2\)

Test of Independence (Categorical Data)

Test statistic

\[ \chi^{2}=\sum_{i=1}^{m} \frac{\left(O_{i j}-E_{i j}\right)^{2}}{E_{i j}} \]

Degrees of freedom \(=(r-1)(c-1)\)

where:

• \(m=\) Number of cells in the table
• \(O_{i j}=\) Number of observations in each cell of row \(i\) and column \(j\)
• \(E_{i j}=\) Expected number of observations in each cell of row \(i\) and column \(j\)

Learning Module 9: Parametric and Non-Parametric Tests of Independence

Test of a Correlation

Test statistic

\[ t=\frac{r \sqrt{n-2}}{\sqrt{1-r^{2}}} \]

Degrees of freedom \(=n-2\)

Pearson Correlation (or Bivariate Correlation)

\[ r_{X Y}=\frac{s_{X Y}}{s_{X} s_{Y}} \]

Spearman Rank Correlation Coefficient

\[ r_{S}=1-\frac{6 \sum_{i=1}^{n} d_{i}^{2}}{n\left(n^{2}-1\right)} \]

where:

• \(d=\) Difference in ranks

Test of Independence (Categorical Data)

Test statistic

\[ \chi^{2}=\sum_{i=1}^{m} \frac{\left(O_{i j}-E_{i j}\right)^{2}}{E_{i j}} \]

Degrees of freedom \(=(r-1)(c-1)\)

where:

• \(m=\) Number of cells in the table
• \(O_{i j}=\) Number of observations in each cell of row \(i\) and column \(j\)
• \(E_{i j}=\) Expected number of observations in each cell of row \(i\) and column \(j\)

\[ =\frac{(\text { Total row } i) \times(\text { Total column } j)}{\text { Overall total }} \]

Standardized Residual (or Pearson Residual)

\[ \text { Standardized Residual }=\frac{O_{i j}-E_{i j}}{\sqrt{E_{i j}}} \]

Learning Module 10: Simple Linear Regression

\[ Y_{i}=b_{0}+b_{1} X_{1}+\cdots+b_{n} X_{n}+\varepsilon_{i}, \quad i=1,2, \ldots, n \]

where:

• \(Y=\) Dependent variable
• \(X=\) Independent variable
• \(b_{0}=\) Intercept
• \(b_{i}=\) Slope coefficient, \(i=1,2, \ldots, n\)
• \(\varepsilon_{i}=\) Error term
• \(b_{0}, b_{1}, \ldots, b_{n}=\) Regression coefficients

\[ \hat{Y}_{i}=\hat{b}_{0}+\hat{b}_{1} X_{i}+e_{i} \]

where:

• \(\widehat{Y}_{i}=\) Estimated value on the regression line for the \(i\) th observation
• \(\hat{b}_{0}=\) Intercept
• \(\hat{b}_{1}=\) Slope
• \(e_{i}=\) Residual for the \(i\) th observation \(\hat{b}_{1}=\frac{\text { Covariance of } X \text { and } Y}{\text { Variance of } X}=\frac{\sum_{i=1}^{n}\left(Y_{i}-\bar{Y}\right)\left(X_{i}-\bar{X}\right)}{\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}}\) \(\hat{b}_{0}=\bar{Y}-\hat{b}_{1} \bar{X}\) Sum of Squares Total, \(S S T=\sum_{i=1}^{n}\left(Y_{i}-\bar{Y}\right)^{2}=S S R+S S E\) Sum of Squares Regression, \(S S R=\sum_{i=1}^{n}\left(\hat{Y}_{i}-\bar{Y}\right)^{2}\) Sum of Squares Error, \(S S E=\sum_{i=1}^{n}\left(Y_{i}-\widehat{Y}_{i}\right)^{2}=\sum_{i=1}^{n} e_{i}^{2}\) Coefficient of Determination

\[ R^{2}=\frac{S S R}{S S T}=1-\frac{S S E}{S S T} \]

Correlation coefficient

\[ r=\frac{\text { Covariance of } X \text { and } Y}{(\text { Standard deviation of } X)(\text { Standard deviation of } Y)} \]

Note: (Correlation coefficient) \({ }^{2}=\) Coefficient of determination

Sample standard deviation of X

\[ S_{X}=\sqrt{\frac{\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}}{n-1}} \]

Sample standard deviation of Y

\[ S_{Y}=\sqrt{\frac{\sum_{i=1}^{n}\left(Y_{i}-\bar{Y}\right)^{2}}{n-1}} \]

Homoskedasticity

\[ E\left(\varepsilon_{i}^{2}\right)=\sigma_{\varepsilon}^{2}, \quad i=1,2, \ldots, n \]

ANOVA F-Test

Mean square regression (MSR)

\[ M S R=\frac{S S R}{k} \]

Mean square error (MSE)

\[ M S E=\frac{S S E}{n-k-1} \]

F-distributed test statistic

\[ F=\frac{M S R}{M S E} \]

where:

• \(n=\) Number of observations
• \(k=\) Number of independent variables

Standard error of estimate

\[ s_{e}=\sqrt{M S E}=\sqrt{\frac{\sum_{i=1}^{n}\left(Y_{i}-\widehat{Y}_{i}\right)^{2}}{n-k-1}} \]

Hypothesis Test of the Slope Coefficient

\[ t=\frac{\hat{b}_{1}-B_{1}}{s_{\hat{b}_{1}}} \]

Degrees of freedom, \(d f=n-k-1\)

where:

• \(B_{1}=\) Hypothesized population slope
• \(s_{\hat{b}_{1}}=\) Standard error of the slope coefficient

\[ =\frac{s_{e}}{\sqrt{\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}}} \]

Hypothesis Test of the Intercept

\[ t_{\text {intercept }}=\frac{\hat{b}_{0}-B_{0}}{s_{\hat{b}_{0}}} \]

Standard error of the intercept, \(s_{\hat{b}_{0}}\)

\[ s_{\hat{b}_{0}}=\sqrt{\frac{1}{n}+\frac{\bar{X}^{2}}{\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}}} \]

Prediction Intervals

\[ \hat{Y}_{f} \pm t_{\alpha / 2} \times s_{f} \]

where: \(\widehat{Y}_{f}=\widehat{b}_{0}+\widehat{b}_{1} X_{f}\)

Variance of the prediction error of Y, given X

\[ s_{f}^{2}=s_{e}^{2}\left[1+\frac{1}{n}+\frac{\left(X_{f}-\bar{X}\right)^{2}}{(n-1) s_{X}^{2}}\right] \]

Standard error of the forecast

\[ s_{f}=s_{e} \sqrt{1+\frac{1}{n}+\frac{\left(X_{f}-\bar{X}\right)^{2}}{(n-1) s_{X}^{2}}} \]

The Log-Lin Model

\[ \ln Y_{i}=b_{0}+b_{1} X_{i} \]

The Lin-Log Model

\[ Y_{i}=b_{0}+b_{1} \ln X_{i} \]

The Log-Log Model

\[ \ln Y_{i}=b_{0}+b_{1} \ln X_{i} \]

Learning Module 11: Introduction to Big Data Techniques

No formula.

VOLUME 2: ECONOMICS

Learning Module 1: The Firm and Market Structures

Total profit \(=\) Total revenue - Total cost

Economic profit \(=\) Total revenue - Total economic costs

Accounting profit \(=\) Total revenue - Total accounting costs

Total revenue \(=\) Price × Quantity \(=P \times Q\)

Average revenue \(=\frac{\text { Total revenue }}{\text { Quantity }}\)

Marginal cost \(=\frac{\Delta T C}{\Delta Q}\)

Average variable cost \(=\frac{\text { Total variable cost }}{\text { Quantity }}\)

Average fixed cost \(=\frac{\text { Total fixed cost }}{\text { Quantity }}\)

Total cost = Total fixed cost + Total variable cost

Average total cost \(=\) Average fixed cost + Average variable cost

Concentration Ratio

\[ \text { Concentration ratio }=\sum_{i=1}^{n}(\text { Market share })_{i} \]

Herfindahl-Hirschman Index (HHI)

\[ H H I=\sum_{i=1}^{n}(\text { Market share })_{i}^{2} \]

Learning Module 2: Understanding Business Cycles

No formula

Learning Module 3: Fiscal Policy

\[ \text { Budget surplus/(deficit) }=G-T+B \]

where:

• \(G=\) Government spending
• \(T=\) Taxes
• \(B=\) Payments of transfer benefits

Disposable Income

\[ Y D=Y-N T=(1-t) Y \]

where:

• \(t=\) Net tax rate
• \(N T=\) Net taxes \(=\) Taxes - Transfers
• \(t Y=\) Total tax revenue

The Fiscal Multiplier

\[ \text { Fiscal multiplier }=\frac{1}{1-c(1-t)} \]

where:

• \(c=\) Marginal propensity to consume
• \(t=\) Tax rate

Learning Module 4: Monetary Policy

Neutral rate \(=\) Trend growth + Inflation target

Learning Module 5: Introduction to Geopolitics

No formula

Learning Module 6: International Trade

No formula

Learning Module 7: Capital Flows and the FX Market

Real exchange rate \(_{d / f}=S_{d / f} \times \frac{P_{f}}{P_{d}}\) \(\%\) Change in real exchange rate \(=\left(1+\% \Delta S_{d / f}\right) \times \frac{\left(1+\% \Delta P_{f}\right)}{\left(1+\% \Delta P_{d}\right)}-1\)

\[ \approx \% \Delta S_{d / f}+\% \Delta P_{f}-\% \Delta P_{d} \]

Percentage change in base currency \(f\) (vs currency \(d\) )

\[ \frac{E\left(S_{d / f}\right)-S_{d / f}}{S_{d / f}} \]

where:

• \(S_{d / f}=\) Spot exchange rate
• \(P_{f}=\) General price level of goods indexed in currency \(f\)
• \(P_{d}=\) General price level of goods indexed in currency \(d\)

Learning Module 8: Exchange Rate Calculations

Cross-Rate

\[ \frac{A}{B}=\frac{A}{C} \times \frac{C}{D} \]

Forward Rate

\[ F_{A / B}=S_{A / B} \times\left[\frac{1+r_{A} \times T}{1+r_{B} \times T}\right] \]

\[ \begin{aligned} \text { Forward points } & =F_{A / B}-S_{A / B} \\ & =S_{A / B}\left(\frac{r_{A}-r_{B}}{1+r_{B}}\right) T \end{aligned} \]

where:

• \(S_{A / B}=\) Spot exchange rate
• \(F_{A / B}=\) Forward exchange rate
• \(T=\) Time to maturity

Learning Module 1: Organizational Forms, Corporate Issuer Features, and Ownership

No formula

Learning Module 2: Investors and Other Stakeholders

No formula

Learning Module 3: Working Capital and Liquidity

\[ \begin{aligned} & \qquad \begin{array}{c} \text { Cash conversion } \\ \text { cycle } \end{array}=\begin{array}{c} \text { Days of inventory } \\ \text { on hand } \end{array}+\begin{array}{c} \text { Days sales } \\ \text { outstanding } \end{array}-\begin{array}{c} \text { Days payables } \\ \text { outstanding } \end{array} \\ & \quad \text { EAR of Supplier }_{\text {Financing }}^{\text {Days in Year }}=\left(1+\frac{\text { Discount } \%}{100 \%-\text { Discount } \%}\right)^{\frac{\text { Dayment Period-Discount Period }}{\text { Total working capital }}=\text { Current assets }- \text { Current Liabilities }}-1 \\ & \begin{array}{c} \text { Net working } \\ \text { capital }=\text { Current assets }(\text { excluding cash and marketable securities }) \\ - \text { Current Liabilities }(\text { excluding short-term and current debt }) \end{array} \end{aligned} \]

Cash flow from operations = Cash received from customers

• Interest and dividends received on financial investments
• Cash paid to employees and suppliers
• Taxes paid to governments
• Interest paid to lenders

Free cash flow \(=\) Cash flow from operations - Investments in long-term assets

\[ \text { Current ratio }=\frac{\text { Current assets }}{\text { Current liabilities }} \]

\[ \text { Quick ratio }=\frac{\text { Cash }+ \text { Short-term marketable instruments }+ \text { Receivables }}{\text { Current liabilities }} \]

\[ \text { Cash ratio }=\frac{\text { Cash }+ \text { Short-term marketable instruments }}{\text { Current liabilities }} \]

Learning Module 4: Corporate Governance: Conflicts, Mechanisms, Risks, and Benefits

No formula

Learning Module 5: Capital Investments and Capital Allocation

Net Present Value

\[ N P V=C F_{0}+\frac{C F_{1}}{(1+r)^{1}}+\frac{C F_{2}}{(1+r)^{2}}+\cdots+\frac{C F_{T}}{(1+r)^{T}}=\sum_{t=0}^{T} \frac{C F_{t}}{(1+r)^{t}} \]

where:

• \(C F_{t}=\) After-tax cash flow at time \(t\)
• \(r=\) Required rate of return
• \(C F_{0}=\) Initial outlay

Internal Rate of Return

\[ \sum_{t=0}^{T} \frac{C F_{t}}{(1+I R R)^{t}}=0 \]

Video: https://youtu.be/bzck7QLhICw

Return on Invested Capital

\[ \begin{aligned} \text { ROIC } & =\frac{\text { After-tax operating profit }}{\text { Average invested capital }} \\ & =\frac{\text { Operating profit } t_{t} \times(1-\text { Tax rate })}{\text { Average total long-term liabilities and equity }_{t-1, t}} \end{aligned} \]

\[ \text { ROIC }=\frac{\text { After-tax operating profit }}{\text { Sales }} \times \frac{\text { Sales }}{\text { Average invested capital }} \]

Real Options in Capital Budgeting

\[ \begin{aligned} & \text { Project NPV } \\ & (\text { with option }) \end{aligned}=\begin{gathered} \text { Project NPV } \\ (\text { without option }) \end{gathered}-\text { Option cost }+ \text { Option value } \]

Learning Module 6: Capital Structure

Weighted Average Cost of Capital

\[ W A C C=w_{d} r_{d}(1-t)+w_{e} r_{e} \]

where:

• \(w_{d}=\) Target weight of debt in capital structure \(=\frac{D}{D+E}\)
• \(w_{e}=\) Target weight of common stock in capital structure \(=\frac{E}{D+E}\)
• \(r_{d}=\) Before-tax marginal cost of debt
• \(t=\) Marginal tax rate
• \(r_{d}(1-t)=\) After-tax marginal cost of debt
• \(r_{e}=\) Marginal cost of common stock

Operating Leverage

\[ \text { Operating leverage }=\frac{\text { Fixed costs }}{\text { Total costs }} \]

Interest Coverage

\[ \text { Interest coverage }=\frac{\text { Profit before interest and taxes }}{\text { Interest expense }} \]

Modigliani-Miller Capital Structure Propositions

\[ \begin{gathered} V_{L}=V_{U}+t D \\ r_{e}=r_{0}+\left(r_{0}-r_{d}\right)(1-t) \frac{D}{E} \\ E=\frac{\left(C F_{e}-r_{d} D\right)(1-t)}{r_{e}} \\ V_{L}=\frac{C F_{e}(1-t)}{r_{\text {WACC }}} \end{gathered} \]

where:

• \(V_{L}=\) Value of levered firm
• \(V_{U}=\) Value of unlevered firm
• \(t=\) Marginal tax rate
• \(r_{e}=\) Cost of equity
• \(r_{d}=\) Cost of debt
• \(r_{0}=\) Cost of capital (for a \(100 \%\) equity-financed company)
• \(D=\) Market value of debt
• \(E=\) Market value of equity
• \(C F_{e}=\) After-tax cash flows to shareholders
• \(r_{d} D=\) Interest expense on debt

Static Trade-off Theory of Capital Structure

\[ V_{L}=V_{U}+t D-P V(\text { Costs of Financial Distress }) \]

Learning Module 7: Business Models

No formula

VOLUME 4: FINANCIAL STATEMENT ANALYSIS

Learning Module 1: Introduction to Financial Statement Analysis

No formula

Learning Module 2: Analyzing Income Statements

Gross profit \(=\) Revenue - Cost of Goods Sold

Operating income \(=\) Gross margin - Selling, General, and Administrative Expense

Taxable income \(=\) Operating income - Interest expense

Net income \(=\) Taxable income - Taxes

Ending shareholders’ equity \(=\) Beginning shareholders’ equity + Net income

• Other comprehensive income
• Dividends
• Net capital contributions from shareholders

Ending retained earnings \(=\) Beginning retained earnings + Net income - Dividends

Return on Equity

\[ R O E=\frac{\text { Net income }}{\text { Average shareholders' equity }} \]

Net Profit Margin

\[ \text { Net profit margin }=\frac{\text { Net income }}{\text { Revenue }} \]

Basic EPS

\[ \text { Basic EPS }=\frac{\text { Net income }- \text { Preferred dividends }}{\text { Weighted average number of shares outstanding }} \]

Diluted EPS (for convertible preferred stock)

\[ \text { Diluted EPS }=\frac{\text { Net income }}{\begin{array}{c} \text { Weighted average number } \\ \text { of shares outstanding } \end{array}+\begin{array}{c} \text { New common shares that would } \\ \text { have been issued at conversion } \end{array}} \]

Diluted EPS (for convertible debt)

\[ \text { Diluted EPS }=\frac{\begin{array}{c} \text { Net income }- \text { Preferred dividends }+ \\ \text { After tax interest expense } \\ \text { on convertible debt } \end{array}}{\begin{array}{c} \text { Weighted average number } \\ \text { of shares outstanding } \end{array}+\begin{array}{c} \text { New common shares that would } \\ \text { have been issued at conversion } \end{array}} \]

Diluted EPS (for options)

\[ \text { Diluted EPS }=\frac{\text { Net income }- \text { Preferred dividends }}{\begin{array}{c} \text { Weighted average number } \\ \text { of shares outstanding } \end{array} \begin{array}{c} \text { Additional common } \\ \text { shares issued upon } \\ \text { conversion } \end{array}} \]

Treasury stock method

\[ \begin{gathered} \text { Additional common } \\ \text { shares issued upon } \\ \text { conversion } \end{gathered}=\left(\begin{array}{ccc} \text { New shares } & \text { Shares repurchased } \\ \text { issued at } & - & \text { with cash received } \\ \text { option exercise } & \text { from option exercised } \end{array}\right) \times \begin{gathered} \text { Proportion of year } \\ \text { during which options } \\ \text { were outstanding } \end{gathered} \]

Video (Basic & Diluted EPS): https://youtu.be/2C-mwVqO2SQ

Learning Module 3: Analyzing Balance Sheets

Working capital \(=\) Current assets - Current liabilities

Liquidity Ratios

\[ \begin{gathered} \text { Current ratio }=\frac{\text { Current assets }}{\text { Current liabilities }} \\ \text { Quick }(\text { acid test }) \text { ratio }=\frac{\text { Cash }+ \text { Marketable securities }+ \text { Receivables }}{\text { Current liabilities }} \\ \text { Cash ratio }=\frac{\text { Cash }+ \text { Marketable securities }}{\text { Current liabilities }} \end{gathered} \]

Solvency Ratios

\[ \begin{gathered} \text { Long-term debt-to-equity }=\frac{\text { Long-term debt }}{\text { Total equity }} \\ \text { Debt-to-equity }=\frac{\text { Total debt }}{\text { Total equity }} \\ \text { Total debt }=\frac{\text { Total debt }}{\text { Total assets }} \\ \text { Financial leverage }=\frac{\text { Total assets }}{\text { Total equity }} \end{gathered} \]

Learning Module 4: Analyzing Statements of Cash Flows I

Cash flow Cash flow Cash flow \(\underset{\text { cash }}{\text { Ending }}=\underset{\text { cash }}{\text { Beginning }}+\underset{\text { activities }}{\text { from operating }}+\underset{\text { activities }}{\text { from investing }}+\underset{\text { activities }}{\text { from financing }}\)
Ending accounts \(=\) Beginning accounts receivable receivable	+ Revenue -
Ending inventory \(=\) inventory	Cost of goods sold
Ending accounts = Beginning accounts payable	+Purchases -	Cash paid to suppliers
Ending wages payable		Cash paid to employees
+ payable	Interest Cash paid expense \({ }^{-}\)for interest
\(\_\_\_\_\) taxpaple +	\(\_\_\_\_\)
Ending \(P P \& E=\) Beginning \(P P \& E+\) \(\_\_\_\_\)		Equipment sold
Ending accumulated Beginning accumulated Depreciation depreciation depreciation \(\_\_\_\_\) expense expense

Note:

Learning Module 5: Analyzing Statements of Cash Flows II

Free Cash Flow To Firm (FCFF)

\[ \begin{aligned} F C F F & =N I+N C C+\operatorname{Int}(1-\operatorname{Tax} \text { rate })-F C \operatorname{Inv}-W C \operatorname{Inv} \\ & =C F O+\operatorname{Int}(1-\text { Tax rate })-F C \operatorname{Inv} \end{aligned} \]

where:

• NI = Net income
• \(N C C=\) Non-cash charges (e.g., depreciation and amortization)
• Int = Interest expense
• FCInv = Capital expenditures
• WCInv = Working capital expenditures
• CFO = Cash flow from operating activities = NI + NCC - WCInv

Free Cash Flow to Equity (FCFE)

\(F C F E=C F O-F C I n v+N e t B o r r o w i n g\)

where:

• Net Borrowing \(=\) Debt issued - Debt repaid

Performance Ratios

Cash flow to revenue \(=\frac{C F O}{\text { Revenue }}\)

Cash return on assets \(=\frac{C F O}{\text { Average total assets }}\)

Cash return on equity \(=\frac{\text { CFO }}{\text { Average shareholders equity }}\)

Cash to income \(=\frac{C F O}{\text { Operating income }}\)

Cash flow per share \(=\frac{\text { CFO }- \text { Preferred dividends }}{\text { Number of common shares outstanding }}\)

Coverage Ratios

Debt coverage ratio \(=\frac{C F O}{\text { Total debt }}\)

Interest coverage ratio \(=\frac{\text { CFO }+ \text { Interest paid }+ \text { Taxes paid }}{\text { Interest paid }}\) Reinvestment ratio \(=\frac{\text { CFO }}{\text { Cash paid for long term assets }}\) Debt payment ratio \(=\frac{C F O}{\text { Cash paid for long term debt repayment }}\) Dividend payment ratio \(=\frac{\text { CFO }}{\text { Dividends paid }}\) Investing and financing ratio \(=\frac{\text { CFO }}{\begin{array}{c}\text { Cash flow for investing and } \\ \text { financing activities }\end{array}}\)

Learning Module 6: Analysis of Inventories

IFRS

Inventories \(=\) Lower of Cost and Net Realizable Value (NRV) \(N R V=\) Estimated selling price less estimated costs of completion and costs necessary to complete the sale

US GAAP

Inventories \(=\) Lower of Cost and NRV For last-in, first-out (LIFO) method or retail inventory methods Inventories \(=\) Lower of Cost and Market Value

Market value \(=\) Current replacement cost (subject to lower and upper limits) Lower limit \(=N R V-\) Normal profit margin Upper limit \(=N R V\) Video: https://youtu.be/V8C31msIBzs Inventory turnover ratio \(=\frac{\text { Cost of sales }}{\text { Average inventory }}\) Days of inventory on hand \(=\frac{\text { Number of days in period }}{\text { Inventory turnover ratio }}\)

Ending inventory \((\) FIFO \()=\) Ending inventory \((\) LIFO \()+\) LIFO reserve

COGS \((F I F O)=\) COGS \((\) LIFO \()\) - Change in LIFO reserve

Learning Module 7: Analysis of Long-Term Assets

Net book value \(=\) Historical cost - Accumulated depreciation

Gain on sale of asset \(=\) Sale proceeds - Net book value \(\begin{gathered}\text { Estimated total } \\ \text { useful life }\end{gathered}=\begin{gathered}\text { Estimated age } \\ \text { of equipment }\end{gathered}+\begin{gathered}\text { Estimated } \\ \text { remaining life }\end{gathered}\) \(\begin{gathered}\text { Estimated total } \\ \text { useful life }\end{gathered}=\frac{\text { Gross PP\&E }}{\text { Annual depreciation expense }}\) \(\begin{gathered}\text { Estimated age } \\ \text { of equipment }\end{gathered}=\frac{\text { Accumulated depreciation }}{\text { Annual depreciation expense }}\) \(\underset{\text { remaining life }}{\text { Estimated }}=\frac{\text { Net PP\&E }}{\text { Annual depreciation expense }}\)

Straight-line Depreciation

\[ \text { Annual depreciation expense }=\frac{\text { Historical cost }- \text { Salvage value }}{\text { Estimated useful life }} \]

Fixed Asset Turnover

\[ \text { Fixed asset turnover }=\frac{\text { Revenue }}{\text { Average net PP\&E }} \]

Impairment of Long-Lived Assets

IFRS

Impairment = Carrying amount - Recoverable amount

where:

• Recoverable amount \(=\) max(Fair value less costs to sell, Value in use)

US GAAP

If asset’s carrying amount > undiscounted expected future cash flows: Impairment = Carrying amount - Fair value

Learning Module 8: Topics in Long-Term Liabilities and Equity

Lessee Accounting - Finance Lease (IFRS)

\[ \begin{aligned} & \text { Interest expense }=\text { Implied interest rate × Beginning lease liability } \\ & \qquad \text { on lease } \\ & \text { Principal repayment }=\text { Lease payment }- \text { Interest expense } \\ & \begin{array}{l} \text { Ending lease } \\ \text { liability } \end{array}=\begin{array}{c} \text { Beginning lease } \\ \text { liability } \end{array}+\text { Interest expense }- \text { Lease payment } \end{aligned} \]

If ROU asset is amortized on a straight-line basis:

\[ \begin{gathered} \text { Amortization } \\ \text { expense } \end{gathered}=\frac{\text { Initial ROU asset value - Salvage value }}{\text { Lease term }} \]

\(\underset{\text { asset }}{\text { Ending } \text { ROU }}=\underset{\text { asset }}{\text { Beginning } \text { ROU }}-\underset{\text { expense }}{\text { Amortization }}\)

Lessee Accounting - Operating Lease (US GAAP)

$\underset{\text { expense }}{\text { Amortization }}=$ Lease payment - Interest expense

$\underset{\text { asset }}{\text { Ending } R O U}=\underset{\text { asset }}{\text { Beginning } R O U}-\underset{\text { expense }}{\text { Amortization }}$

$\underset{\text { liability }}{\text { Ending lease }}=\underset{\text { liability }}{\text { Beginning lease }}-\underset{\text { expense }}{\text { Amortization }}$

Stock Options

\[ \text { Compensation expense }=\frac{\text { Fair value of options granted }}{\text { Vesting period }} \]

Learning Module 9: Analysis of Income Taxes

Deferred Tax Asset/Liability

For Assets:

\[ \begin{gathered} \text { Deferred tax } \\ \text { liability } /(\text { asset }) \end{gathered}=\text { Tax rate } \times\left(\begin{array}{c} \text { Carrying amount } \\ \text { of asset } \end{array}-\begin{array}{c} \text { Tax base } \\ \text { of asset } \end{array}\right) \]

For Liabilities:

\[ \begin{gathered} \text { Deferred tax } \\ \text { liability } /(\text { asset }) \end{gathered}=\text { Tax rate } \times\left(\begin{array}{cc} \text { Tax base } & \text { Carrying amount } \\ \text { of liability } & \text { of liability } \end{array}\right) \]

Income tax expense \(=\) Income tax payable \(+\begin{gathered}\text { Changes in deferred tax } \\ \text { assets and liabilities }\end{gathered}\)

Effective tax rate \(=\frac{\text { Income tax expense }}{\text { Pre-tax income }}\)

Cash tax rate \(=\frac{\text { Cash tax }}{\text { Pre-tax income }}\)

Learning Module 10: Financial Reporting Quality

Adjusted EBITDA

\[ \underset{\text { EBITDA }}{\text { Adjusted }}=\underset{\text { EBIT }}{\text { Adjusted }}+\underset{\substack{\text { Software } \\ \text { and R\&D } \\ \text { amortization }}}{\text { Post-IPO }}+\underset{\substack{\text { Share-based } \\ \text { amortization }}}{\text { Depreciation }+\underset{\text { shartion }}{\text { amortion }}} \]

Straight-line method of depreciation

\[ \text { Depreciation expense }=\frac{\text { Cost }- \text { Salvage value }}{\text { Useful life }} \]

Double-Declining Balance method

\[ \text { Depreciaton expense }=\frac{2}{\text { Useful life }} \times(\text { Cost }- \text { Accumulated depreciation }) \]

Video: https://youtu.be/6RskYAxdAFk

Units-of-Production method

\[ \text { Depreciation expense }=\frac{\text { Units produced }}{\text { Total units over useful life }} \times(\text { Cost }- \text { Salvage value }) \]

Learning Module 11: Financial Analysis Techniques

Activity Ratios

\[ \begin{aligned} & \text { Inventory turnover }=\frac{\text { Cost of sales }}{\text { Average inventory }} \\ & \text { Days of inventory on hand }=\frac{\text { Number of days in the period }}{\text { Inventory turnover }} \\ & \text { Receivables turnover }=\frac{\text { Revenue }}{\text { Average receivables }} \\ & \text { Days of sales outstanding }=\frac{\text { Number of days in the period }}{\text { Receivables turnover }} \\ & \text { Payables turnover }=\frac{\text { Purchases }}{\text { Average payables }} \\ & \text { Number of days of payables }=\frac{\text { Number of days in the period }}{\text { Payables turnover }} \\ & \text { Working capital turnover }=\frac{\text { Revenue }}{\text { Average working capital }} \\ & \text { Fixed asset turnover }=\frac{\text { Revenue }}{\text { Average net fixed assets }} \\ & \text { Total asset turnover }=\frac{\text { Revenue }}{\text { Average total assets }} \\ & \text { Liquidity Ratios } \\ & \text { Current ratio }=\frac{\text { Current assets }}{\text { Current liabilities }} \\ & \text { Quick ratio }=\frac{\text { Cash }+ \text { Short term marketable investments }+ \text { Receivables }}{\text { Current liabilities }} \\ & \text { Cash ratio }=\frac{\text { Cash }+ \text { Short term marketable investments }}{\text { Current liabilities }} \\ & \text { Defensive interval }=\frac{\text { Cash }+ \text { Short term marketable investments }+ \text { Receivables }}{\text { Daily cash expenditures }} \end{aligned} \]

\(\begin{aligned} & \text { Cash conversion } \\ & \text { cycle }\end{aligned}=\begin{gathered}\text { Days of inventory } \\ \text { on hand }\end{gathered}+\begin{gathered}\text { Days of sales } \\ \text { outstanding }\end{gathered}-\begin{gathered}\text { Number of days } \\ \text { of payables }\end{gathered}\)

Solvency Ratios

\(\underset{(\text { "Total debt ratio" })}{\text { Debt-to-assets ratio }}=\frac{\text { Total debt }}{\text { Total assets }}\)

Debt-to-capital ratio \(=\frac{\text { Total debt }}{\text { Total debt }+ \text { Total equity }}\)

Debt-to-equity ratio \(=\frac{\text { Total debt }}{\text { Total equity }}\)

Financial leverage ratio \(=\frac{\text { Average total assets }}{\text { Average total equity }}\)

Debt-to-EBITDA ratio \(=\frac{\text { Total or net debt }}{\text { EBITDA }}\)

Coverage Ratios

Interest coverage ratio \(=\frac{\text { EBIT }}{\text { Interest payments }}\) Fixed charge coverage ratio \(=\frac{\text { EBIT }+ \text { Lease payments }}{\text { Interest payments }+ \text { Lease payments }}\)

Profitability Ratios

Gross profit margin \(=\frac{\text { Gross profit }}{\text { Revenue }}\) Operating profit margin \(=\frac{\text { Operating income }}{\text { Revenue }}\) Pretax margin \(=\frac{E B T}{\text { Revenue }}\) Net profit margin \(=\frac{\text { Net income }}{\text { Revenue }}\)

Operating ROA \(=\frac{\text { Operating income }}{\text { Average total assets }}\) ROA \(=\frac{\text { Net income }}{\text { Average total assets }}\) Return on invested capital \(=\frac{E B I T \times(1-E f f \text { ective tax rate })}{\text { Average total debt and equity }}\) ROE \(=\frac{\text { Net income }}{\text { Average total equity }}\) Return on common equity \(=\frac{\text { Net income }- \text { Preferred dividends }}{\text { Average common equity }}\)

DuPont Analysis

ROE \(=\) ROA × Financial Leverage

ROE \(=\) Net profit margin × Total asset turnover × Financial leverage ROE \(=\underset{\text { burden }}{\text { Tax }} \times \underset{\text { burden }}{\text { Interest }} \times \underset{\text { margin }}{\text { EBIT }} \times \underset{\text { turnover }}{\text { Total asset }} \times \underset{\text { leverage }}{\text { Financial }}\)

where:

• Tax burden \(=\frac{\text { Net income }}{\text { EBT }}\)
• Interest burden \(=\frac{E B T}{E B I T}\)

Business Risk

\(\begin{gathered}\text { Coefficient of variation } \\ \text { of operating income }\end{gathered}=\frac{\text { Standard deviation of operating income }}{\text { Average operating income }}\) \(\begin{gathered}\text { Coefficient of variation } \\ \text { of net income }\end{gathered}=\frac{\text { Standard deviation of net income }}{\text { Average net income }}\) \(\begin{gathered}\text { Coefficient of variation } \\ \text { of revenue }\end{gathered}=\frac{\text { Standard deviation of revenue }}{\text { Average revenue }}\)

Financial Sector Ratios

\(\begin{gathered}\text { Monetary reserve requirement } \\ (\text { Cash reserve ratio })\end{gathered}=\frac{\text { Reserves held at central bank }}{\text { Specified deposit liabilities }}\) Net interest margin \(=\frac{\text { Net interest income }}{\text { Total interest earning assets }}\) Liquid asset requirement \(=\frac{\text { Approved readily marketable securities }}{\text { Specified deposit liabilities }}\) Net interest margin \(=\frac{\text { Net interest income }}{\text { Total interest earning assets }}\)

Learning Module 12: Introduction to Financial Statement Modeling

Nothing new.

VOLUME 5: EQUITY INVESTMENTS

Learning Module 1: Market Organization and Structure

Maximum leverage ratio \(=\frac{1}{\text { Minimum margin requirement }}\) Total return on leveraged stock investment:

\[ \text { Total Return }=\frac{\begin{array}{c} \text { Sales } \\ \text { proceeds } \end{array}+\text { Dividends }- \text { Loan }-\frac{\text { Margin }}{\text { interest }}-\begin{array}{c} \text { Sales } \\ \text { commission } \end{array}}{\begin{array}{l} \text { Initial } \\ \text { equity } \end{array}+\begin{array}{c} \text { Purchase } \\ \text { commission } \end{array}}-1 \]

Initial equity \(=\begin{gathered}\text { Minimum margin } \\ \text { requirement }\end{gathered} \times\) Total purchase price Video (Return on Leveraged Stock Position): https://youtu.be/tZd4Xtvjill Margin Call Price \(=\frac{P_{0}(1-\text { Initial Margin })}{(1-\text { Maintenance Margin })}\)

Learning Module 2: Security Market Indexes

\[ \text { Price Return Index, } \quad V_{P R I}=\frac{\sum_{i=1}^{N} n_{i} P_{i}}{D} \]

where:

• \(n_{i}=\) the number of units of constituent security \(i\) held in the index portfolio
• \(N=\) the number of constituent securities in the index
• \(P_{i}=\) the unit price of constituent security \(i\)
• \(D=\) value of the divisor

\[ \begin{aligned} & \text { Price return of an index, } \quad P R_{I}=\frac{V_{P R I 1}-V_{P R I 0}}{V_{P R I 0}} \\ & \text { Total Return Index, } \quad T R_{I}=\frac{V_{P R I 1}-V_{P R I 0}+I n c_{I}}{V_{P R I 0}} \end{aligned} \]

where:

• \(V_{P R I 1}=\) value of the price return index at the end of the period
• \(V_{P R I 0}=\) value of the price return index at the beginning of the period
• \(I n c_{I}=\) total income (dividends and/or interest) from all securities in the index held over the period

Weighting Methods

\[ \text { Price weighting, } \quad w_{i}^{P}=\frac{P_{i}}{\sum_{j=1}^{N} P_{j}} \]

Video (Recalculating the divisor of a price weighted index): https://youtu.be/eYiZNK-ETrg

\[ \begin{aligned} & \qquad \text { Equal weighting, } \quad w_{i}^{E}=\frac{1}{N} \\ & \text { Market-capitalization weighting, } \quad w_{i}^{M}=\frac{Q_{i} P_{i}}{\sum_{j=1}^{N} Q_{j} P_{j}} \\ & \text { Float-adjusted market capitalization weighting, } \quad w_{i}^{M}=\frac{f_{i} Q_{i} P_{i}}{\sum_{j=1}^{N} f_{j} Q_{j} P_{j}} \end{aligned} \]

where:

• \(f_{i}=\) fraction of shares outstanding in the market float
• \(Q_{i}=\) number of shares outstanding of security \(i\)
• \(P_{i}=\) share price of security \(i\)
• \(N=\) number of securities in the index

\[ \text { Fundamental weighting, } \quad w_{i}^{F}=\frac{F_{i}}{\sum_{j=1}^{N} F_{j}} \]

where \(F_{\mathrm{i}}\) denotes a fundamental size measure of company \(i\)

Learning Module 3: Market Efficiency

No formula

Learning Module 4: Overview of Equity Securities

Return on Equity (using average total book value of equity)

\[ R O E_{t}=\frac{N I_{t}}{\left(B V E_{t}+B V E_{t-1}\right) / 2} \]

Return on Equity (using beginning book value of equity)

\[ R O E_{t}=\frac{N I_{t}}{B V E_{t-1}} \]

where BVE = book value (Assets - Liabilities)

Learning Module 5: Company Analysis: Past and Present

Market share \(=\frac{\text { Revenue }}{\text { Market size }}\)

Sales potential \(=100 \%-\) Market share \(\%\)

Net sales \(=\) Average selling price × Quantity sold

Take rate \(=\frac{\text { Revenue earned from transactions }}{\text { Total transaction volume }} \times 100 \%\)

Operating income \(=Q \times(P-V C)-F C\)

where:

• \(Q=\) Units sold in a period
• \(P=\) Price per unit
• \(V C=\) Variable operating cost per unit
• \(F C=\) Fixed operating costs
• \(P-V C=\) Contribution margin per unit

Degree of operating leverage \((D O L)=\frac{\% \Delta \text { Operating income }}{\% \Delta \text { Sales }}\) Degree of financial leverage \((D F L)=\frac{\% \Delta \text { Net income }}{\% \Delta \text { Operating income }}\)

WACC \(=\begin{aligned} & \text { Weight } \\ & \text { of debt }\end{aligned} \times \begin{gathered}\text { Gross cost } \\ \text { of debt }\end{gathered} \times(1-\) tax rate \()+\underset{\text { of equity }}{\text { Weight }} \times \begin{gathered}\text { Cost of } \\ \text { equity }\end{gathered}\)

Learning Module 6: Industry and Company Analysis

Herfindahl-Hirschman Index (HHI)

\[ H H I=\sum_{i=1}^{\infty} s_{i}^{2} \]

where:

• \(s_{i}=\) Market share of participant \(i\) (stated as a whole number)

Learning Module 7: Company Analysis: Forecasting \(\%\) Variable cost \(\approx \frac{\% \Delta(\text { Cost of revenue }+ \text { Operating expense })}{\% \Delta \text { Revenue }}\) \(\%\) Fixed cost \(\approx 1-\%\) Variable cost \(\begin{gathered}\text { Number of units sold } \\ \text { post-cannibalization }\end{gathered}=\begin{gathered}\text { Number of units sold } \\ \text { pre-cannibalization }\end{gathered}-\begin{gathered}\text { Expected } \\ \text { cannibalization }\end{gathered}\) \(\underset{\text { cannibalization }}{\text { Expected }}=\underset{\text { pre-cannibalization }}{\text { Number of units sold }} \times \underset{\text { factor }}{\text { Cannibalization }}\)

Learning Module 8: Equity Valuation: Concepts and Basic Tools

Dividend Discount Model (DDM)

\[ V_{0}=\sum_{t=1}^{n} \frac{D_{t}}{(1+r)^{t}}+\frac{P_{n}}{(1+r)^{n}} \]

where:

• \(V_{0}=\) Intrinsic value of a share at \(t=0\)
• \(D_{t}=\) expected dividend in year \(t\)
• \(r=\) required rate of return on stock
• \(P_{n}=\) expected price per share at \(t=n\) (terminal value)

Free-cash-flow-to-equity (FCFE) Valuation Model

\[ V_{0}=\sum_{t=1}^{\infty} \frac{F C F E_{t}}{(1+r)^{t}} \]

where:

• \(F C F E=C F O-F C I n v+N e t B o r r o w i n g\)
• FCInv \(=\) Fixed capital investment
• Net Borrowing \(=\) Borrowings minus repayments Value of preferred stock (non-callable, non-convertible, perpetual)

\[ V_{0}=\frac{D_{0}}{r} \]

Value of preferred stock (non-callable, non-convertible, maturity at time \(n\) )

\[ V_{0}=\sum_{t=1}^{n} \frac{D_{t}}{(1+r)^{t}}+\frac{\text { Par value }}{(1+r)^{n}} \]

Gordon Growth Model

\[ P_{0}=\frac{D_{1}}{r-g}=\frac{D_{0}(1+g)}{r-g} \]

where:

• \(D_{0}=\) Most recent annual dividend
• \(D_{1}=\) Expected dividend in the next period
• \(g\) = Constant growth rate
• \(r=\) Required return on equity

Sustainable growth rate

\[ g=b \times R O E \]

where:

• \(b=\) earnings retention rate (= 1 - Dividend payout ratio)
• \(R O E=\) Return on equity

Video: https://youtu.be/MnfRRRhuGpA

Two-Stage Dividend Discount Model

\[ V_{0}=\sum_{t=1}^{n} \frac{D_{0}\left(1+g_{s}\right)^{t}}{(1+r)^{t}}+\frac{V_{n}}{(1+r)^{t}} \]

where:

• \(g_{L}=\) Long-term stable growth rate
• \(g_{s}=\) Short-term growth rate
• \(V_{n}=\frac{D_{n+1}}{r-g_{L}}=\frac{D_{0}\left(1+g_{s}\right)^{t}\left(1+g_{L}\right)}{r-g_{L}}\)

Justified forward P/E

\[ \frac{P_{0}}{E_{1}}=\frac{\text { Dividend payout ratio }}{r-g} \]

Enterprise Value

\[ E V=\begin{gathered} \text { Market value } \\ \text { of equity } \end{gathered}+\begin{gathered} \text { Market value } \\ \text { of preferred stock } \end{gathered}+\begin{gathered} \text { Market value } \\ \text { of debt } \end{gathered}-\begin{gathered} \text { Cash and } \\ \text { short term } \\ \text { investments } \end{gathered} \]

Asset-based Valuation

\[ \begin{gathered} \text { Adjusted } \\ \text { book value } \end{gathered}=\begin{gathered} \text { Market value } \\ \text { of assets } \end{gathered}-\begin{gathered} \text { Market value } \\ \text { of liabilities } \end{gathered} \]

VOLUME 6: FIXED INCOME

Learning Module 1: Fixed-Income Instrument Features Current yield \(=\frac{\text { Annual coupon }}{\text { Bond price }}\) Bond price \(=\frac{\text { Coupon }}{(1+r)^{1}}+\frac{\text { Coupon }}{(1+r)^{2}}+\cdots+\frac{\text { Coupon }+ \text { Face value }}{(1+r)^{n}}\)

where:

• Coupon per period \(=\) Coupon rate per period × Face value
• \(r=\) Yield to maturity per period
• \(n=\) Number of payments

Floating-rate Note (FRN) coupon rate \(=\) MRR + Spread

Learning Module 2: Fixed-Income Cash Flows and Types

Fully Amortizing Loan with Level Payment

\[ A=\frac{r \times \text { Principal }}{1-(1+r)^{-N}} \]

where:

• \(A=\) Periodic payment
• \(r=\) Market interest rate per period
• \(N=\) Number of payment periods

If the periodic payment is monthly:

\[ \begin{aligned} & \text { Monthly interest payment }=\text { Interest rate per month × Beginning principal of loan } \\ & \text { Monthly principal payment }=\text { Total monthly payment }- \text { Monthly interest payment } \\ & \text { Ending principal of loan }=\text { Beginning principal of loan }- \text { Monthly principal payment } \end{aligned} \]

Capital-Index Bond (e.g., TIPS)

Inflation-adjusted principal \(=\) Principal amount × \((1+\) Inflation adjustment \()\)

Coupon per period \(=\) Coupon rate per period × Inflation-adjusted principal

Deferred Coupon Bond

Video: https://youtu.be/erRbAUOGIyM

Convertible Bonds

\(\begin{gathered}\begin{array}{c}\text { Conversion } \\ \text { ratio }\end{array}=\frac{\text { Convertible bond par value }}{\text { Conversion price }} \\ \begin{array}{c}\text { Conversion } \\ \text { value }\end{array}\end{gathered}=\begin{gathered}\text { Conversion } \\ \text { ratio }\end{gathered} \times \begin{gathered}\text { Current share } \\ \text { price }\end{gathered}\)

Zero-Coupon Bond

Original issue discount = Bond par value - Issuance price

Learning Module 3: Fixed-Income Issuance and Trading

No formula

Learning Module 4: Fixed-Income Markets for Corporate Issuers

Repurchase Agreements

\[ \begin{gathered} \text { Repurchase price }=\text { Price of bond } \times\left[1+\text { Repo rate } \times \frac{\text { Repo term (in days) }}{\text { Number of days in a year }^{\text {Initial margin }}=\frac{\text { Security price }_{0}}{\text { Purchase price }_{0}}}\right. \\ \text { Haircut }=\frac{\text { Security price }_{0}-\text { Purchase price }_{0}}{\text { Security price }_{0}} \end{gathered} \]

\[ \text { Variation margin }=\left(\text { Initial margin } \times \text { Purchase price }_{t}\right)-\text { Security price }_{t} \]

Learning Module 5: Fixed-Income Markets for Government Issuers

No formula.

Learning Module 6: Fixed-Income Bond Valuation: Prices and Yields

\[ P V=\frac{P M T_{1}}{(1+r)^{1}}+\frac{P M T_{2}}{(1+r)^{2}}+\cdots+\frac{P M T_{N}+F V_{N}}{(1+r)^{N}} \]

where:

• \(P M T_{t}=\) Coupon that occurs in \(t\) periods
• \(r=\) Market discount rate per period
• \(N=\) Number of periods to maturity
• \(F V=\) Face value of bond

Full Price, Flat Price, and Accrued Interest (Video: https://youtu.be/I7G075JAu5w)

\[ \begin{aligned} \mathrm{PV}^{\text {Full }} & =\mathrm{PV}^{\text {Flat }}+\text { Accrued Interest } \\ & =P V_{B O P} \times(1+r)^{t / T} \end{aligned} \]

where:

• Accrued Interest \(=\frac{t}{T} \times P M T\)
• \(t=\) number of days from the last coupon payment to the settlement date
• \(T=\) number of days in the coupon period
• \(t / T=\) fraction of the coupon period that has gone by since the last payment
• \(P V_{B O P}=\) price on the previous coupon date (before the settlement date)

Matrix Pricing

\[ \text { Interpolated yield }=\text { Yield }_{S}+\left(\frac{\text { Tenor }_{\text {Target }}-\text { Tenor }_{S}}{\text { Tenor }_{L}-\text { Tenor }_{S}}\right) \times\left(\text { Yield }_{L}-\text { Yield }_{S}\right) \]

where:

• Yield \(_{S}=\) Yield of shorter-term bond
• Yield \(_{L}=\) Yield of longer-term bond
• Tenor \(_{S}\) = Tenor of shorter-term bond
• Tenor \(_{L}=\) Tenor of longer-term bond
• Tenor \(_{\text {Target }}=\) Tenor of the subject bond
• Tenor \(_{S}<\) Tenor \(_{\text {Target }}<\) Tenor \(_{L}\) Required yield spread \(=\) Bond YTM - Government Bond YTM (Similar maturity)

Learning Module 7: Yield and Yield Spread Measures for Fixed Rate Bonds

Periodicity Conversion

\[ \left(1+\frac{A P R_{m}}{m}\right)^{m}=\left(1+\frac{A P R_{n}}{n}\right)^{n} \]

where:

• \(A P R_{m}=\) Annual percentage rate for \(m\) periods per year
• \(A P R_{n}=\) Annual percentage rate for \(n\) periods per year Current yield \(_{t}=\frac{\text { Annual coupon }_{t}}{\text { Bond price }_{t}}\) Government equivalent yield, \(\quad\) Yield \(_{\text {ACT } / \text { ACT }}=\frac{365}{360} \times\) Yield \(_{30 / 360}\) Simple yield \(=\frac{\text { Coupon }+\left(\frac{F V-P V}{N}\right)}{\text { Flat price }}\)

Callable Bonds

\[ P V=\frac{P M T}{(1+Y T C)^{1}}+\frac{P M T}{(1+Y T C)^{2}}+\cdots+\frac{P M T+\text { Call price }}{(1+Y T C)^{N}} \]

where:

• \(P V=\) Price of the callable bond
• PMT = Coupon payment per period
• YTC = Yield to call per period
• \(N=\) Number of periods to when the bond can be called at the call price

Option-adjusted price = Flat price of bond + Value of embedded call option

Value of call option = Price of option-free bond - Price of callable bond

G-spread \(=\) Bond YTM - Interpolated sovereign bond YTM

I-spread = Bond YTM - Swap rate

Z-Spread

\[ P V=\frac{P M T}{\left(1+z_{1}+Z\right)^{1}}+\frac{P M T}{\left(1+z_{2}+Z\right)^{2}}+\cdots+\frac{P M T+F V}{\left(1+z_{N}+Z\right)^{N}} \]

where:

• \(Z=Z\)-spread
• \(z_{N}=\) Spot rate for \(N\) periods

OAS \(=\) Z-spread - Option value (in basis points per year)

Learning Module 8: Yield and Yield Spread Measures for Floating-Rate Instruments

Value of Floating Rate Note (FRN)

\[ P V=\frac{\left(\frac{M R R+Q M}{m}\right) \times F V}{\left(1+\frac{M R R+D M}{m}\right)^{1}}+\frac{\left(\frac{M R R+Q M}{m}\right) \times F V}{\left(1+\frac{M R R+D M}{m}\right)^{2}}+\cdots+\frac{\left(\frac{M R R+Q M}{m}\right) \times F V+F V}{\left(1+\frac{M R R+D M}{m}\right)^{n}} \]

where:

• \(Q M=\) Quoted Margin
• DM = Discount Margin
• \(M R R=\) Market reference rate
• \(m=\) Periodicity (i.e., number of payment periods per year)
• FV = Face Value of FRN
• \(N=\) Number of evenly spaced periods to maturity

Video: https://youtu.be/zqYOtVLkYR8

Yield Measures for Money Market Instruments

Discount Rate Basis

\[ \begin{gathered} P V=F V \times\left(1-\frac{D a y s}{Y e a r} \times D R\right) \\ D R=\frac{Y e a r}{D a y s} \times\left(\frac{F V-P V}{F V}\right) \end{gathered} \]

where:

• \(P V=\) present value of money market instrument
• \(F V=\) future value paid at maturity
• Days = number of days between settlement and maturity
• Year \(=\) number of days in the year
• \(D R=\) discount rate (stated as annual percentage rate)

Add-on Rate Basis

\[ \begin{gathered} P V=\frac{F V}{\left(1+\frac{\text { Days }}{\text { Year }} \times A O R\right)} \\ A O R=\frac{Y e a r}{\text { Days }} \times\left(\frac{F V-P V}{P V}\right) \\ \text { Bond equivalent yield }=\frac{365}{D a y s} \times\left(\frac{F V-P V}{P V}\right) \end{gathered} \]

Learning Module 9: The Term Structure of Interest Rates: Spot, Par, and Forward Curves

Calculation of Bond Price Using Spot Rates

\[ P V=\frac{P M T}{\left(1+Z_{1}\right)^{1}}+\frac{P M T}{\left(1+Z_{2}\right)^{2}}+\cdots+\frac{P M T+F V}{\left(1+Z_{N}\right)^{N}} \]

where:

• \(P V=\) Price of bond
• PMT = Bond coupon payment
• \(Z_{N}=\) Spot rate (or zero-coupon yield or zero rate) for period \(N\)
• \(F V=\) Face value of bond

Given a Par Rate, \(F V=P V\) and \(P M T=\operatorname{Par}\) Rate (%) \(\times F V\)

\[ 100=\frac{P M T}{\left(1+Z_{1}\right)^{1}}+\frac{P M T}{\left(1+Z_{2}\right)^{2}}+\cdots+\frac{P M T+100}{\left(1+Z_{N}\right)^{N}} \]

Forward Rates, IFR

\[ \left(1+z_{A}\right)^{A} \times\left(1+I F R_{A, B-A}\right)^{B-A}=\left(1+z_{B}\right)^{B} \]

where:

• \(I F R_{A, B-A}=\) Forward rate for ( \(B-A\) ) periods that starts in period \(A\)

Learning Module 10: Interest Rate Risk and Return

Duration Gap

Duration gap = Macaulay duration - Investment horizon

Macaulay Duration

Macaulay duration \(=\left(1-\frac{t}{T}\right)\left[\frac{\frac{P M T}{(1+r)^{1-t / T}}}{P V^{\text {Full }}}\right]+\left(2-\frac{t}{T}\right)\left[\frac{P M T}{\frac{(1+r)^{2-t / T}}{P V^{\text {Full }}}}\right]+\cdots\)

\[ +\left(N-\frac{t}{T}\right)\left[\frac{\frac{P M T+F V}{(1+r)^{N-t / T}}}{P V^{F u l l}}\right] \]

Macaulay duration \(=\left\{\frac{1+r}{r}-\frac{1+r+[N \times(c-r)]}{c \times\left[(1+r)^{N}-1\right]+r}\right\}-\frac{t}{T}\)

where:

• \(r=\) Yield per period
• \(c=\) Coupon rate per period
• \(N=\) Number of evenly spaced periods to maturity as of the beginning of the current period
• \(t=\) Number of days from the last coupon payment to the settlement date
• \(T=\) Number of days in the coupon period

Video: https://youtu.be/USgjcdCk7Fs

Learning Module 11: Yield-Based Bond Duration Measures and Properties

Modified Duration

\[ \text { Modified Duration }=\frac{\text { Macaulay Duration }}{1+r} \]

Approximate Modified Duration

\[ \begin{aligned} & \text { AnnModDur } \approx \frac{\left(P V_{-}\right)-\left(P V_{+}\right)}{2 \times(\Delta \text { Yield }) \times\left(P V_{0}\right)} \\ & \% \Delta P V^{\text {Full }} \approx-\text { AnnModDur } \times \Delta \text { Yield } \end{aligned} \]

Money Duration

Money duration \(=\) AnnModDur × \(P V^{\text {full }}\)

\[ \Delta P V^{\text {Full }} \approx-\text { MoneyDur } \times \Delta \text { Yield } \]

Duration of Zero-Coupon Bond

\[ \begin{aligned} & \text { MacDur }=\text { Time to maturity } \\ & \text { ModDur }=\frac{\text { Time to maturity }}{1+r} \end{aligned} \]

Duration of Perpetual Bond

\[ \begin{gathered} \text { MacDur }=\frac{1+r}{r} \\ \text { ModDur }=\frac{1}{r} \end{gathered} \]

Duration of Floating-Rate Notes

\[ \text { MacDur }=\frac{T-t}{T}=\begin{gathered} \text { Fraction of period remaining until } \\ \text { the next reset date } \end{gathered} \]

Learning Module 12: Yield-Based Bond Convexity and Portfolio Properties

Convexity

\[ \text { Convexity }=\sum_{t=1}^{N} \frac{t(t+1) \times \frac{P V_{t}}{P V^{F u l l}}}{(1+Y T M)^{2}} \]

Approximate Annualized Convexity

\[ \begin{gathered} \text { ApproxConv } \approx \frac{\left(P V_{-}\right)+\left(P V_{+}\right)-2\left(P V_{0}\right)}{(\Delta \text { Yield })^{2} \times\left(P V_{0}\right)} \\ \% \Delta P V^{\text {Full }} \approx-\text { AnnModDur } \times \Delta \text { Yield }+\frac{1}{2} \times \text { AnnConvexity } \times(\Delta \text { Yield })^{2} \end{gathered} \]

Money Convexity

\[ \begin{gathered} \text { MoneyCon }=\text { AnnConvexity } \times P V^{\text {Full }} \\ \Delta P V^{\text {Full }} \approx-(\text { MoneyDur } \times \Delta \text { Yield })+\left[\frac{1}{2} \times \text { MoneyCon } \times(\Delta \text { Yield })^{2}\right] \end{gathered} \]

Portfolio Duration and Convexity

\[ \begin{gathered} \text { Portfolio Modified Duration }=\sum_{i=1}^{N} w_{i} \times \text { ModDur }_{i} \\ \text { Portfolio Convexity }=\sum_{i=1}^{N} w_{i} \times \text { Convexity }_{i} \end{gathered} \]

where:

• \(w_{i}=\) Weight of bond \(i\), measured in market value

Learning Module 13: Curve-Based and Empirical Fixed-Income Risk Measures

Effective Duration

\[ \text { EffDur }=\frac{\left(P V_{-}\right)-\left(P V_{+}\right)}{2 \times(\Delta \text { Curve }) \times P V_{0}} \]

Effective Convexity

\[ \begin{gathered} \text { EffCon }=\frac{\left(P V_{-}\right)+\left(P V_{+}\right)-2 \times P V_{0}}{(\Delta \text { Curve })^{2} \times P V_{0}} \\ \% \Delta P V^{\text {Full }} \approx-\text { EffDur } \times \Delta \text { Curve }+\frac{1}{2} \times \text { EffCon } \times(\Delta \text { Curve })^{2} \end{gathered} \]

Key Rate Duration

\[ \begin{gathered} \operatorname{KeyRateDur}_{k}=-\frac{1}{P V} \times \frac{\Delta P V}{\Delta r_{k}} \\ \% \Delta P V=-\operatorname{KeyRateDur}_{k} \times \Delta r_{k} \\ \sum_{k=1}^{n} \operatorname{KeyRateDur}_{k}=\text { EffDur } \end{gathered} \]

where:

• \(r_{k}=k\) th key rate

Learning Module 14: Credit Risk

Expected Loss

\[ \begin{gathered} E L=L G D \times P O D \\ L G D=E E \times(1-R R) \end{gathered} \]

where:

• POD = Probability of default
• LGD = Loss given default
• \(E E=\) Expected exposure
• \(R R=\) Recovery rate
• \(1-R R=\) Loss severity

Credit spread \(\approx\) POD × LGD

Decomposing Bond Yields

Yield spread \(=\) Bond YTM - Government bond YTM (Similar maturity)

Liquidity spread \(=\) Bond YTM \((\) Bid \()-\) Bond YTM \((\) Offer \()\)

Credit spread \(=\) Yield spread - Liquidity spread

Price Impact Given a Change in Yield Spread

\[ \% \Delta P V^{\text {Full }} \approx-\text { AnnModDur } \times \Delta \text { Spread }+\frac{1}{2} \times \text { AnnConvexity } \times(\Delta \text { Spread })^{2} \]

where:

• AnnModDur \(\approx \frac{\left(P V_{-}\right)-\left(P V_{+}\right)}{2 \times(\Delta \text { Yield }) \times\left(P V_{0}\right)}\)
• AnnConvexity \(\approx \frac{\left(P V_{-}\right)+\left(P V_{+}\right)-2\left(P V_{0}\right)}{(\Delta \text { Yield })^{2} \times\left(P V_{0}\right)}\)

Learning Module 15: Credit Analysis for Government Issuers

No formula.

Learning Module 16: Credit Analysis for Corporate Issuers

\[ \begin{gathered} \text { EBIT margin }=\frac{\text { Operating income }}{\text { Revenue }} \\ \text { EBIT to interest expense }=\frac{\text { Operating income }}{\text { Interest expense }} \\ \text { Debt to EBITDA }=\frac{\text { Debt }}{\text { EBITDA }} \\ \text { RCF to net debt }=\frac{\text { Retained cash flow }}{\text { Debt - Cash and marketable securities }} \\ \text { FFO to debt }=\frac{F F O}{\text { Debt }} \end{gathered} \]

where:

• FFO = Net income from continuing operations + Depreciation & amortization

• Deferred income taxes + Other non-cash items

Learning Module 17: Fixed-Income Securitization

No formula.

Learning Module 18: Asset-Backed Security (ABS) Instrument and Market Features

No formula.

Learning Module 19: Mortgage-Backed Security (MBS) Instrument and Market Features

Loan-to-value (LTV) ratio

\[ L T V=\frac{\text { Loan amount }}{\text { House price }} \]

Debt-to-income (DTI) ratio

\[ D T I=\frac{\text { Monthly debt payment }}{\text { Monthly pre-tax gross income }} \]

Mortgage Pass-Through Securities

\[ \begin{aligned} W A C & =\sum_{i=1}^{N} c_{i}\left(\frac{C B_{i}}{C B}\right) \\ W A M & =\sum_{i=1}^{N} M M_{i}\left(\frac{C B_{i}}{C B}\right) \end{aligned} \]

where:

• WAC = Weighted average coupon
• \(W A M=\) Weighted average maturity
• \(c_{i}=\) Coupon rate on mortgage \(i\)
• \(M M_{i}=\) Number of months to maturity for mortgage \(i\)
• \(N=\) Number of mortgages in MBS
• \(C B_{i}=\) Current balance on mortgage \(i\)
• \(C B=\) Total current balance of mortgages in MBS

Commercial Mortgage-Backed Securities (CMBS)

Debt Service Coverage Ratio (DSCR)

\[ \text { DSCR }=\frac{\text { Net operating income }}{\text { Debt service }} \]

Net Operating Income (NOI)

NOI \(=(\) Rental income - Cash operating expenses \()-\) Replacement reserves

VOLUME 7: DERIVATIVES

Learning Module 1: Derivative Instrument and Derivatives Market Features

No formula.

Learning Module 2: Forward Commitments and Contingent Claim Features and Instruments

Forward Contract

\[ \begin{gathered} \text { Buyer (Long) payoff }=S_{T}-F_{0}(T) \\ \text { Seller (Short) payoff }=-\left[S_{T}-F_{0}(T)\right] \end{gathered} \]

where:

• \(S_{T}=\) Spot price on contract’s maturity
• \(F_{0}(T)=\) Forward price with maturity of \(T\)

Futures Contract

For one futures contract:

Long Futures daily mark-to-market \(=f_{t}(T)-f_{t-1}(T)\) Short Futures daily mark-to-market \(=-\left[f_{t}(T)-f_{t-1}(T)\right]\)

where:

• \(f_{t}(T)=\) Closing price of futures contract on day \(t\)
• \(f_{t-1}(T)=\) Closing price of futures contract on day \(t-1\) \(T=\) Maturity of futures contract

If margin balance < maintenance margin:

Variation Margin \(=\) Initial margin - Margin balance

Options Contract

LONG Call option

Payoff or Value at expiration, \(c_{T}=\max \left(0, S_{T}-X\right)\)

Profit at expiration, \(\Pi=\max \left(0, S_{T}-X\right)-c_{0}\)

where:

• \(c_{0}=\) Call premium
• \(X=\) Exercise/Strike price \(S_{T}=\) Spot price at expiration

SHORT Call option

Payoff or Value at expiration, \(c_{T}=-\max \left(0, S_{T}-X\right)\)

Profit at expiration, \(\Pi=-\left[\max \left(0, S_{T}-X\right)-c_{0}\right]\)

LONG Put option

Payoff or Value at expiration, \(p_{T}=\max \left(0, X-S_{T}\right)\)

Profit at expiration, \(\Pi=\max \left(0, X-S_{T}\right)-p_{0}\)

SHORT Put option

Payoff or Value at expiration, \(p_{T}=-\max \left(0, X-S_{T}\right)\)

Profit at expiration, \(\Pi=-\left[\max \left(0, X-S_{T}\right)-p_{0}\right]\)

Credit Default Swap (CDS)

CDS MTM Change \(=\triangle\) CDS Spread × CDS Notional × EffDur \(_{\text {CDS }}\)

In a credit event, payment from CDS seller to CDS buyer \(\approx L G D(\%) \times\) Notional

Learning Module 3: Derivative Benefits, Risks, and Issuer and Investor Uses

No formula.

Learning Module 4: Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

If there are no underlying costs or benefits:

\[ \text { Forward price, } F_{0}(T)=S_{0}(1+r)^{T} \]

If there are underlying costs or benefits in present value terms:

\[ \text { Forward price, } F_{0}(T)=\left[S_{0}-P V_{0}(\text { Income })+P V_{0}(\text { Cost })\right](1+r)^{T} \]

where:

• \(S_{0}=\) Current spot price
• \(r=\) Risk-free rate
• \(T=\) Tenor of forward contract

Under continuous compounding, \(F_{0}(T)=S_{0} e^{r T}\) Under continuous compounding, with income (i) and cost (c) expressed in %:

\[ F_{0}(T)=S_{0} e^{(r+c-i) T} \]

Foreign Exchange Forward Contract

\[ F_{0, f / d}(T)=S_{0, f / d}(T) e^{\left(r_{f}-r_{d}\right) T} \]

where:

• \(F_{0, f / d}=\) Forward exchange rate
• \(S_{0, f / d}=\) Spot exchange rate
• \(r_{f}=\) Continuously compounded risk-free rate (for price/quote currency)
• \(r_{d}=\) Continuously compounded risk-free rate (for base currency)
• T = Maturity of forward contract

Learning Module 5: Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities

Value of LONG Forward Prior to Expiration

\[ \begin{gathered} V_{0}(T)=0 \\ V_{t}(T)=S_{t}-\frac{F_{0}(T)}{(1+r)^{T-t}}=S_{t}-F_{0}(T) \times(1+r)^{-(T-t)} \\ V_{T}(T)=S_{0}-F_{0}(T) \end{gathered} \]

If the asset incurs cost or generates income from time \(t\) through maturity,

\[ V_{t}(T)=\left[S_{t}-P V_{t}(\text { Income })+P V_{t}(\text { Cost })\right]-F_{0}(T) \times(1+r)^{-(T-t)} \]

For foreign exchange forward contract,

\[ V_{t}(T)=S_{t, f / d}-F_{0, f / d}(T) \times e^{-\left(r_{f}-r_{d}\right)(T-t)} \]

Value of SHORT Forward Prior to Expiration

\[ \begin{gathered} V_{0}(T)=0 \\ V_{t}(T)=-\left[S_{t}-\frac{F_{0}(T)}{(1+r)^{T-t}}\right] \\ V_{T}(T)=-\left[S_{0}-F_{0}(T)\right] \end{gathered} \]

Interest Rate Forward Contracts (Forward Rate Agreements (FRA))

\[ \left(1+z_{A}\right)^{A} \times\left(1+I F R_{A, B-A}\right)^{B-A}=\left(1+z_{B}\right)^{B} \]

where:

• \(z_{A}=\) Spot rate for \(A\) periods
• \(z_{B}=\) Spot rate for \(B\) periods
• \(I F R_{A, B-A}=\) Implied forward rate for \((B-A)\) periods, starting in \(A\) periods

Payoff for a Long FRA \(=\left(M R R_{B-A}-I F R_{A, B-A}\right) \times\) Notional principal × Period

Payoff for a Short FRA \(=-\left(M R R_{B-A}-I F R_{A, B-A}\right) \times\) Notional principal × Period

Learning Module 6: Pricing and Valuation of Futures Contracts

If there are no underlying costs or benefits:

\[ \text { Futures price, } f_{0}(T)=S_{0}(1+r)^{T} \]

If there are underlying costs or benefits in present value terms:

\[ f_{0}(T)=\left[S_{0}-P V_{0}(\text { Income })+P V_{0}(\text { Cost })\right](1+r)^{T} \]

Under continuous compounding, \(f_{0}(T)=S_{0} e^{r T}\)

Under continuous compounding, with income ( \(i\) ) and cost ( \(c\) ) expressed in %:

\[ f_{0}(T)=S_{0} e^{(r+c-i) T} \]

Foreign Exchange Forward Contract

\[ f_{0, f / d}(T)=S_{0, f / d}(T) e^{\left(r_{f}-r_{d}\right) T} \]

Interest Rate Futures Contract

\[ f_{A, B-A}=100-\left(100 \times M R R_{A, B-A}\right) \]

where:

• \(f_{A, B-A}=\) Futures price for a market reference rate for ( \(B-A\) ) periods that begins in \(A\) periods

Futures contract basis point value, \(B P V=\) Notional principal \(\times 0.01 \% \times\) Period

Learning Module 7: Pricing and Valuation of Interest Rates and Other Swaps

For a fixed-rate payer in an interest rate swap:

\[ \text { Periodic settlement value }=\left(M R R-s_{N}\right) \times \text { Swap Notional × Period } \]

For a fixed-rate receiver in an interest rate swap:

\[ \text { Periodic settlement value }=\left(s_{N}-M R R\right) \times \text { Swap Notional × Period } \]

where:

• \(s_{N}=\) Fixed swap rate
• \(M R R=\) Market reference rate

Calculating Par Swap Rate

\[ \sum_{i=1}^{N} \frac{I F R}{\left(1+z_{i}\right)^{i}}=\sum_{i=1}^{N} \frac{s_{N}}{\left(1+z_{i}\right)^{i}} \]

where:

• \(I F R=\) Implied forward rates
• \(s_{N}=\) Fixed swap rate
• \(N=\) Tenor of swap contract

Valuation of Interest Rate Swap

Value of a pay-fixed interest rate swap on a settlement date after inception \(=\begin{gathered}\text { Current settlement } \\ \text { value }\end{gathered}+\Sigma(\) Floating payments \()-\Sigma(\) Fixed payments \()\)

Value of a receive-fixed interest rate swap on a settlement date after inception \(=\begin{gathered}\text { Current settlement } \\ \text { value }\end{gathered}+\Sigma(\) Fixed payments \()-\Sigma(\) Floating payments \()\)

Learning Module 8: Pricing and Valuation of Options

Option value = Exercise value + Time value

At time \(t\) (prior to option expiration):

Call option exercise value \(=\operatorname{Max}\left[0, S_{t}-X(1+r)^{-(T-t)}\right]\)

Call option time value \(=c_{t}-\operatorname{Max}\left[0, S_{t}-X(1+r)^{-(T-t)}\right]\)

Put option exercise value \(=\operatorname{Max}\left[0, X(1+r)^{-(T-t)}-S_{t}\right]\)

Put option time value \(=p_{t}-\operatorname{Max}\left[0, X(1+r)^{-(T-t)}-S_{t}\right]\)

\[ \begin{aligned} & \text { Lower bound of call option value }=\operatorname{Max}\left[0, S_{t}-X(1+r)^{-(T-t)}\right] \\ & \text { Upper bound of call option value }=S_{t} \\ & \text { Lower bound of put option value }=\operatorname{Max}\left[0, X(1+r)^{-(T-t)}-S_{t}\right] \\ & \text { Upper bound of put option value }=X \end{aligned} \]

where:

• \(S_{t}=\) Spot price at time \(t\)
• \(X=\) Exercise price (or strike price)
• \(T=\) Maturity of option
• \(r=\) Risk-free rate

Learning Module 9: Option Replication Using Put-Call Parity

Put-Call Parity

\[ S_{0}+p_{0}=c_{0}+X(1+r)^{-T} \]

Put-Call Forward Parity

\[ F_{0}(T)(1+r)^{-T}+p_{0}=c_{0}+X(1+r)^{-T} \]

Value of the Firm

\[ V_{0}=c_{0}+P V(\text { Debt })-p_{0} \]

Value of debt \(=P V(\) Debt \()-p_{0}\) Value of equity \(=c_{0}\)

Learning Module 10: Valuing a Derivative Using a One-Period Binomial Model

Risk-neutral probability of a price increase in underlying

\[ \pi=\frac{1+r-R^{d}}{R^{u}-R^{d}} \]

where:

• \(R^{u}=U p\) factor \(=\frac{S_{1}^{u}}{S_{0}}>1\)
• \(R^{d}=\) Down factor \(=\frac{S_{1}^{d}}{S_{0}}<1\)
• \(S_{0}=\) Current asset price
• \(S_{1}^{u}=\) One-period asset price when price moves up
• \(S_{1}^{d}=\) One-period asset price when price moves down

Video: https://youtu.be/ymUIKgz-rAw

Hedge ratio

\[ h^{*}=\frac{c_{1}^{u}-c_{1}^{d}}{S_{1}^{u}-S_{1}^{d}} \]

where:

• \(c_{1}^{u}=\max \left(0, S_{1}^{u}-X\right)\)
• \(c_{1}^{d}=\max \left(0, S_{1}^{d}-X\right)\)

Riskless portfolio with a Call: \(\boldsymbol{h}\) of the underlying, \(\boldsymbol{S}\), and short call position, \(\boldsymbol{c}\) \(V_{0}=h S_{0}-c_{0}\) \(V_{1}^{u}=h S_{1}^{u}-c_{1}^{u}\) \(V_{1}^{d}=h S_{1}^{d}-c_{1}^{d}\)

Riskless portfolio with a Put: \(\boldsymbol{h}\) of the underlying, \(\boldsymbol{S}\), and long put position, \(\boldsymbol{p}\)

\(V_{0}=h S_{0}+p_{0}\) \(V_{1}^{u}=h S_{1}^{u}+p_{1}^{u}\) \(V_{1}^{d}=h S_{1}^{d}+p_{1}^{d}\)

Value of a one-period call option

\[ c_{0}=\frac{\pi c_{1}^{u}+(1-\pi) c_{1}^{d}}{1+r} \]

Value of a one-period put option

\[ p_{0}=\frac{\pi p_{1}^{u}+(1-\pi) p_{1}^{d}}{1+r} \]

where:

• \(p_{1}^{u}=\max \left(0, X-S_{1}^{u}\right)\)
• \(p_{1}^{d}=\max \left(0, X-S_{1}^{d}\right)\) Video: https://youtu.be/bXEC-78y AU

Learning Module 1: Alternative Investment Features, Methods, and Structures

GP Compensation Structure

Ignoring management fee; no catch-up clause

\[ r_{G P}=\max \left[0, p\left(r-r_{h}\right)\right] \]

Ignoring management fee; with catch-up clause

\[ r_{G P}=\max \left[0, r_{c u}+p\left(r-r_{h}-r_{c u}\right)\right] \]

where:

• \(r_{G P}=\mathrm{GP}^{\prime}\) s rate of return
• \(p=\) Performance fee as a percentage of total return
• \(r=\) Single-period rate of return
• \(r_{h}=\) Hard hurdle rate
• \(r_{c u}=\) Catch-up clause

Learning Module 2: Alternative Investment Performance and Returns

Multiple on Invested Capital

\[ M O I C=\frac{\text { Realized value of investment }+ \text { Unrealized value of investment }}{\text { Total amount of invested capital }} \]

Leveraged Portfolio Return

\[ r_{L}=r+\frac{V_{b}}{V_{c}}\left(r-r_{b}\right) \]

where:

• \(r=\) Periodic rate of return on invested funds
• \(r_{b}=\) Periodic cost of borrowing
• \(V_{b}=\) Amount of borrowed funds
• \(V_{c}=\) Amount of cash (investor’s own capital)

Investor’s Return Net of Fees

\[ \begin{gathered} r_{i}=\frac{P_{1}-P_{0}-R_{G P}}{P_{0}} \\ R_{G P}=\left(P_{1} \times r_{m}\right)+\max \left[0,\left(P_{1}-P_{0}\right) \times p\right] \end{gathered} \]

where:

• \(P_{0}=\) Beginning-of-period asset value
• \(P_{1}=\) End-of-period asset value
• \(p=\mathrm{GP}\) performance fee
• \(R_{G P}=\mathrm{GP}^{\prime}\) s return in current terms
• \(r_{m}=\) GP’s management fees as a percentage of assets under management

Calculating Hedge Fund Fees and Returns

Management Fee Based on Beginning Market Value

\[ \underset{\text { Fee }}{\text { Management }}=\underset{\text { Fee }}{\%}=\stackrel{\text { Management }}{\text { Beginning Market }} \]

Management Fee Based on Ending Market Value

\[ \underset{\text { Fee }}{\text { Management }}=\underset{\text { Fee }}{\% \text { Management }} \times \stackrel{\text { Ending Market }}{\text { Value }} \]

Incentive Fee Calculated Independent of Management Fee

\[ \begin{gathered} \text { Incentive } \\ \text { Fee } \end{gathered}=\begin{gathered} \text { \%Incentive } \\ \text { Fee } \end{gathered} \times \text { Gain } \]

Incentive Fee Calculated Net of Management Fee

\[ \begin{gathered} \text { Incentive } \\ \text { Fee } \end{gathered}=\begin{gathered} \% \text { Incentive } \\ \text { Fee } \end{gathered} \times(\text { Gain }- \text { Management Fee }) \]

Incentive Fee with Hard Hurdle (Independent of Management Fee)

\[ \begin{gathered} \text { Incentive } \\ \text { Fee } \end{gathered}=\begin{gathered} \% \text { Incentive } \\ \text { Fee } \end{gathered} \times(\text { Gain }- \text { Hurdle }) \]

Incentive Fee with Hard Hurdle (Net of Management Fee)

\[ \begin{gathered} \begin{array}{c} \text { Incentive } \\ \text { Fee } \end{array}=\begin{array}{c} \% \text { Incentive } \\ \text { Fee } \end{array} \times(\text { Gain }- \text { Management Fee }- \text { Hurdle }) \\ \text { Hurdle }=\text { Hurdle Rate } \text { × } \text { Beginning market value } \end{gathered} \]

Note: 1) No incentive is paid if hedge fund incurs loss for the year. 2) Gain may be subject to high watermark.

¹ ## Learning Module 3: Investments in Private Capital: Equity and Debt

No formula.

Learning Module 4: Real Estate and Infrastructure

Loan-to-Value (LTV) Ratio

\[ L T V=\frac{\text { Mortgage liability }}{\text { Portfolio value }} \]

Required reduction in mortgage liability = Mortgage liability - Required mortgage liability

Learning Module 5: Natural Resources

No formula.

Learning Module 6: Hedge Funds

No formula.

Learning Module 7: Introduction to Digital Assets

No formula.

VOLUME 9: PORTFOLIO MANAGEMENT

Learning Module 1: Portfolio Risk and Return: Part I

Expected Return on Asset

\[ 1+E(R)=\left(1+r_{r F}\right) \times[1+E(\pi)] \times[1+E(R P)] \]

where:

• \(r_{r F}=\) Real risk-free rate
• \(E(\pi)=\) Expected inflation
• \(E(R P)=\) Expected risk premium for the asset

Utility on Investment

\[ U=E(R)-\frac{1}{2} A \sigma^{2} \]

where:

• \(U=\) Utility of investment
• \(E(R)=\) Expected return of investment
• \(A=\) Risk aversion coefficient
• \(\sigma^{2}=\) Variance of investment (Note: Substitute \(\sigma\) in decimals)

Capital Allocation Line (CAL)

For a portfolio of risky assets (Weight: \(w_{i}\) ) and risk-free asset:

\[ E\left(R_{p}\right)=R_{f}+\left[\frac{E\left(R_{i}\right)-R_{f}}{\sigma_{i}}\right] \sigma_{p} \]

where:

• \(R_{f}=\) Rate of return on risk-free asset
• \(E\left(R_{i}\right)=\) Expected return of risky asset
• \(E\left(R_{p}\right)=\) Expected return of portfolio
• \(\sigma_{i}=\) Standard deviation of risky asset’s returns
• \(\sigma_{p}=\) Standard deviation of portfolio’s returns \(=w_{i} \times \sigma_{i}\)
• \(\frac{E\left(R_{i}\right)-R_{f}}{\sigma_{i}}=\) Market price of risk

Two-asset portfolio

Portfolio expected return, \(E\left(R_{p}\right)=w_{1} R_{1}+w_{2} R_{2}\)

Portfolio variance, \(\sigma_{p}^{2}=w_{1}^{2} \sigma_{1}^{2}+w_{2}^{2} \sigma_{2}^{2}+2 w_{1} w_{2} \operatorname{Cov}\left(R_{1}, R_{2}\right)\)

Portfolio standard deviation, \(\sigma_{p}=\sqrt{w_{1}^{2} \sigma_{1}^{2}+w_{2}^{2} \sigma_{2}^{2}+2 w_{1} w_{2} \operatorname{Cov}\left(R_{1}, R_{2}\right)}\)

Note: 1) \(\operatorname{Cov}\left(R_{1}, R_{2}\right)=\rho_{12} \sigma_{1} \sigma_{2}\) 2) \(n\) securities requires \(n\) variances and \(\frac{n(n-1)}{2}\) covariances

Video: https://youtu.be/IUwulZ9ONCO

Foreign Asset

Return of a foreign asset in domestic currency

\[ R_{D}=\left(1+R_{l c}\right) \times\left(1+R_{F X}\right)-1 \]

Standard deviation of return of a foreign asset in domestic currency

\[ \sigma_{D}=\sqrt{\sigma_{l c}^{2}+\sigma_{F X}^{2}+2 \times \rho \times \sigma_{l c} \times \sigma_{F X}} \]

where:

• \(R_{l c}=\) Return of foreign asset (in local currency)
• \(R_{F X}=\) Change in exchange rate (FX rate quoted as domestic currency/foreign currency)
• \(\sigma_{l c}=\) Standard deviation of foreign asset’s returns
• \(\sigma_{F X}=\) Standard deviation of the exchange rate (DC/FC)
• \(\rho=\) Correlation coefficient between returns on foreign asset and exchange rate

Portfolio of Many Risky Assets

\[ \sigma_{p}^{2}=\frac{\bar{\sigma}^{2}}{N}+\frac{N-1}{N} \overline{\operatorname{Cov}}=\frac{\bar{\sigma}^{2}}{N}+\frac{N-1}{N} \rho \bar{\sigma}^{2} \]

where:

• \(N=\) Number of assets in portfolio
• \(\bar{\sigma}^{2}=\) Average variance
• \(\overline{\operatorname{Cov}}=\) Average covariance

Learning Module 2: Portfolio Risk and Return: Part II

Capital Market Line (CML)

\(E\left(R_{p}\right)=w_{f} R_{f}+\left(1-w_{f}\right) E\left(R_{m}\right)=R_{f}+\left[\frac{E\left(R_{m}\right)-R_{f}}{\sigma_{m}}\right] \sigma_{p}\) \(\sigma_{p}=\left(1-w_{f}\right) \sigma_{m}\)

Return-Generating Models

\[ E\left(R_{i}\right)-R_{f}=\beta_{i 1}\left[E\left(R_{m}\right)-R_{f}\right]+\sum_{j=2}^{k} \beta_{i j} E\left(F_{j}\right) \]

where:

• \(E\left(R_{i}\right)-R_{f}=\) Expected excess return on asset \(i\)
• \(k=\) Number of factors \(\beta_{i j}=\) Factor weights (also called factor loadings) \(E\left(R_{m}\right)=\) Expected return on market

The Single-Index Model

\[ E\left(R_{i}\right)-R_{f}=\left(\frac{\sigma_{i}}{\sigma_{m}}\right)\left[E\left(R_{m}\right)-R_{f}\right] \]

where:

• \(\frac{\sigma_{i}}{\sigma_{m}}=\) Factor loading (or factor weight)

Capital Asset Pricing Model

\[ E\left(R_{i}\right)=R_{f}+\beta_{i}\left[E\left(R_{m}\right)-R_{f}\right] \]

The Market Model

\[ R_{i}=\alpha_{i}+\beta_{i} R_{m}+e_{i} \]

Beta of security \(\boldsymbol{i}\)

\(\beta_{i}=\frac{\operatorname{Cov}\left(R_{i}, R_{m}\right)}{\sigma_{m}^{2}}=\frac{\rho_{i, m} \sigma_{i}}{\sigma_{m}}\) Portfolio beta, \(\beta_{p}=\sum_{i=1}^{n} w_{i} \beta_{i}\) Total variance \(=\) Systematic variance + Nonsystematic variance

\[ \sigma_{i}^{2}=\beta_{i}^{2} \sigma_{m}^{2}+\sigma_{e}^{2} \]

Total risk, \(\sigma_{i}=\sqrt{\beta_{i}^{2} \sigma_{m}^{2}+\sigma_{e}^{2}}\)

Arbitrage Pricing Theory (APT) Model

\[ E\left(R_{P}\right)=R_{F}+\lambda_{1} \beta_{P, 1}+\cdots+\lambda_{K} \beta_{P, K} \]

where:

• \(E\left(R_{P}\right)=\) Expected return on portfolio
• \(R_{F}=\) Risk-free rate
• \(\lambda_{j}=\) Risk premium for factor \(j\)
• \(\beta_{P, 1}=\) Sensitivity of the portfolio to factor \(j\)
• \(K=\) Number of risk factors

Fama-French Model

\[ E\left(R_{i t}\right)=\alpha_{i}+\beta_{i, M K T} M K T_{t}+\beta_{i, S M B} S M B_{t}+\beta_{i, H M L} H M L_{t} \]

Carhart Model

\[ E\left(R_{i t}\right)=\alpha_{i}+\beta_{i, M K T} M K T_{t}+\beta_{i, S M B} S M B_{t}+\beta_{i, H M L} H M L_{t}+\beta_{i, U M D} U M D_{t} \]

where:

• \(E\left(R_{i}\right)=\) Return on an asset in excess of the one-month T-bill return
• \(M K T=\) Excess return on the market portfolio
• \(S M B=\) Difference in returns between small-capitalization stocks and large-capitalization stocks (Size)
• \(H M L=\) Difference in returns between high-book-to-market stocks and low-book-to-market stocks (Value versus growth)
• \(U M D=\) Difference in returns of the prior year’s winners versus losers (Momentum)

Portfolio Performance Appraisal Measures

Sharpe ratio \(=\frac{R_{p}-R_{f}}{\sigma_{p}}\)

Treynor ratio \(=\frac{R_{p}-R_{f}}{\beta_{p}}\) \(M^{2}=\left(R_{p}-R_{f}\right) \frac{\sigma_{m}}{\sigma_{p}}+R_{f}=\) Sharpe ratio \(\times \sigma_{m}+R_{f}\) \(M^{2}\) alpha \(=M^{2}-R_{m}\)

Jensen’s Alpha, \(\quad \alpha_{p}=R_{p}-\left[R_{f}+\beta_{p}\left(R_{m}-R_{f}\right)\right]\)

Security Characteristic Line (SCL)

\[ R_{i}-R_{f}=\alpha_{i}+\beta_{i}\left(R_{m}-R_{f}\right) \]

Information ratio \(=\frac{\alpha_{i}}{\sigma_{e i}}\)

Learning Module 3: Portfolio Management: An Overview

No formula.

Learning Module 4: Basics of Portfolio Planning and Construction

No formula.

Learning Module 5: The Behavioral Biases of Individuals

No formula.

Learning Module 6: Introduction to Risk Management

No formula.

Footnotes

1. Video: https://youtu.be/ODKmCgsAAdc