Key Lessons from Financial Market Analysis (FMAx) Module 0: Pre-Course Basics – Financial Mathematics

1. Time Value of Money

• Lesson: The principle of time value of money states that a dollar today is more valuable than a dollar received in the future due to people's time preference, inflation, and uncertainty about future events.
• Example: A dollar today can be invested to earn interest, meaning it will grow to more than a dollar in the future.

2. Interest Rates and Their Determinants

• Lesson: Interest rates reflect people's time preferences, inflation expectations, and the uncertainty of future events. They are the cost of borrowing money over time.
• Example: Higher interest rates compensate for greater uncertainty or inflation expectations in the future.

3. Simple and Compound Interest

• Lesson: Simple interest is calculated on the initial principal only, while compound interest is calculated on both the initial principal and the accumulated interest.
• Example: A deposit of $100 at an 8% simple interest rate for 92 days will earn $102.02, while with compound interest, the total would be higher due to interest-on-interest.

4. Nominal vs. Effective Interest Rates

• Lesson: The nominal interest rate is quoted based on annual compounding, while the effective interest rate accounts for compounding within the year. The effective rate is usually higher than the nominal rate when interest compounds more than once a year.
• Example: A 12% nominal interest rate compounded monthly results in an effective annual rate of 12.68%.

5. Net Present Value (NPV)

• Lesson: NPV measures the value of future cash flows discounted to the present. Investments are assessed based on the NPV, where a higher NPV indicates a more profitable investment at a given interest rate.
• Example: If cash flows are $500 yearly for 15 years at a 5% interest rate, the NPV determines the worth of those future payments in today's terms.

6. Internal Rate of Return (IRR)

• Lesson: The IRR is the discount rate that makes the NPV of an investment’s cash flows equal to zero. It represents the average interest rate earned on an investment over its life.
• Example: The IRR is used to compare different investment opportunities and determine the rate of return that sets the present value of cash inflows equal to the initial investment.

  Key Lessons from Financial Market Analysis (FMAx) Module 1: Pricing Money Market Instruments

1. Understanding the Money Market

• Lesson: The money market is where short-term financial instruments with maturities of less than one year are traded. It plays a key role in providing liquidity and funding for various participants, including governments, corporations, and financial institutions.
• Example: Treasury bills (T-bills) are government debt instruments traded in the money market with typical maturities ranging from 90 to 360 days.

2. Key Money Market Instruments

• Lesson: Common money market instruments include Treasury bills (T-bills), certificates of deposit (CDs), and repurchase agreements (repos). Each has distinct characteristics, such as no coupons for T-bills and deposit insurance for CDs.
• Example: A certificate of deposit (CD) can be issued in any denomination with maturities from 3 months to 5 years, often backed by deposit insurance.

3. Pricing Fixed-Income Securities

• Lesson: The price of a fixed-income security, such as bonds, is calculated by discounting its future cash flows (interest payments and principal repayment) back to the present value using the yield or required return.
• Example: A zero-coupon bond is priced by discounting its face value (maturity amount) by the market yield over its term.

4. No Arbitrage Condition

• Lesson: The principle of no arbitrage ensures that identical assets with the same cash flows and risk profiles should have the same price. Any price differences create arbitrage opportunities, allowing traders to profit without risk until prices equalize.
• Example: If two bonds with the same cash flows have different prices, traders can buy the cheaper bond and sell the more expensive one, forcing the prices to converge.

5. Day Count Conventions

• Lesson: Different financial markets use various day count conventions to calculate accrued interest. The most common methods are 30/360, actual/actual, and actual/360.
• Example: In the U.S., Treasury bills typically use the 30/360 convention, while in European markets, actual/360 or actual/365 is more common.

6. Yield vs. Discount

• Lesson: Returns on fixed-income securities can be quoted in terms of yield or discount. The yield measures the return as a percentage of the purchase price, while the discount reflects the difference between the purchase price and face value.
• Example: For a $100 T-bill purchased for $80, the yield would be 25% while the discount is 20%.

  Key Lessons from Financial Market Analysis (FMAx) Module 2: Bond Pricing

1. Bond Definition and Characteristics

• Lesson: Bonds are a type of fixed-income security where the bondholder lends money to the issuer for a specific period at a fixed or variable interest rate. Bonds promise regular coupon payments and return the principal at maturity.
• Example: A bondholder lends $1,000 to a corporation at an annual 5% interest rate. The bondholder receives periodic interest payments and the principal at maturity.

2. Present Value and Net Present Value (NPV)

• Lesson: Present value (PV) is the current value of a series of future cash flows discounted at a given interest rate. NPV is the difference between the present value of future cash flows and the initial investment.
• Example: An investor assesses a bond by calculating the PV of its coupon payments and principal to determine if the bond price is fair based on the yield.

3. Bond Pricing Formula

• Lesson: The price of a bond is the present value of its future cash flows, including coupon payments and the principal repayment, discounted by the yield-to-maturity (YTM).
• Example: A bond paying $50 semi-annually with a $1,000 face value is priced by discounting these payments using the YTM.

4. Yield-to-Maturity (YTM)

• Lesson: YTM represents the bond’s internal rate of return (IRR) and is the interest rate that makes the present value of a bond’s cash flows equal to its current market price.
• Example: A bond with a 5% coupon and a market price of $950 has a YTM higher than the coupon rate because the investor earns a higher return from buying it at a discount.

5. Premium, Par, and Discount Bonds

• Lesson: A bond trades at a premium if its coupon rate is higher than the YTM, at par if the coupon equals the YTM, and at a discount if the coupon rate is lower than the YTM.
• Example: A bond with a 9% coupon trading at $1,200 is a premium bond since its coupon rate exceeds the market YTM.

6. Bond Investment Risks

• Lesson: Bonds are subject to various risks, including interest rate risk (the risk that bond prices fall as interest rates rise), reinvestment risk, and default risk (the risk that the issuer will not meet its payment obligations).
• Example: An investor holding a bond during an interest rate hike may experience capital losses as bond prices decline.

  Key Lessons from Financial Market Analysis (FMAx) Module 3: Bond Price Sensitivity

1. Bond Price Sensitivity to YTM Changes

• Lesson: The price of a bond is inversely related to changes in the yield-to-maturity (YTM). As YTM increases, the bond price decreases, and as YTM decreases, the bond price increases. This relationship is convex, meaning bond prices react more significantly to declines in YTM than to increases.
• Example: For a one-period bond delivering $1, the bond price is calculated as ( P = \frac{1}{1 + y} ). If YTM rises, the bond price falls.

2. Duration as a Measure of Price Sensitivity

• Lesson: (Macaulay) Duration measures the average time an investor must wait to receive the bond’s cash flows. It indicates the bond price's sensitivity to YTM changes—the longer the duration, the more sensitive the bond price.
• Example: A zero-coupon bond’s duration is equal to its maturity, while a coupon bond has a duration less than its maturity.

3. Properties of Duration

• Lesson: Several factors affect a bond's duration:
  • Lower coupon rate → Higher duration
  • Longer maturity → Higher duration
  • Lower YTM → Higher duration
• Example: A zero-coupon bond has a duration equal to its maturity, while a coupon bond with a lower coupon rate has a higher duration than a bond with a higher coupon rate.

4. Convexity and Price Sensitivity

• Lesson: Convexity accounts for the curvature in the price-yield relationship. It adjusts the bond price sensitivity for large YTM changes, improving the accuracy of price forecasts. Bonds with higher convexity have a greater price increase when YTM falls and a smaller price decrease when YTM rises.
• Example: A bond with higher convexity experiences a greater price increase when interest rates fall compared to a bond with lower convexity.

5. Immunization and Barbell Strategy

• Lesson: Immunization matches the durations of assets and liabilities to protect a portfolio from interest rate changes. The Barbell strategy increases convexity by constructing a portfolio with bonds of varying durations, providing better price protection during YTM fluctuations.
• Example: A Barbell portfolio constructed from 2-year and 10-year U.S. Treasuries has higher convexity than a 5-year bond with the same duration.

  Key Lessons from Financial Market Analysis (FMAx) Module 4: Term Structure of Interest Rates

1. Term Structure of Interest Rates (TSIR)

• Lesson: The term structure of interest rates (TSIR) represents the relationship between the yield-to-maturity of zero-coupon bonds and their respective maturities. It is also known as the "spot curve."
• Example: The spot curve shows how interest rates for bonds of different maturities (e.g., 1-year vs. 10-year) vary over time.

2. Shapes of the Yield Curve

• Lesson: Yield curves can take various shapes—upward-sloping, downward-sloping, or flat—each providing insights into future interest rates, inflation expectations, and economic activity.
• Example: An upward-sloping yield curve indicates that long-term interest rates are higher than short-term rates, often signaling economic expansion and rising inflation expectations.

3. Forward Interest Rates

• Lesson: Forward interest rates reflect the market’s expectations of future spot rates. They are tightly linked to current spot rates through the no-arbitrage condition.
• Example: If the 2-year spot rate is 2% and the 3-year spot rate is 3%, the forward rate between years 2 and 3 will be calculated as 5.03%, representing the market’s expected return for the additional year.

4. Pure Expectations Hypothesis

• Lesson: This hypothesis suggests that the shape of the yield curve reflects market expectations of future interest rates, assuming investors are risk-neutral and only care about expected returns.
• Example: Under this hypothesis, an upward-sloping yield curve indicates that the market expects interest rates to rise in the future.

5. Term Premium

• Lesson: The term premium is the additional return investors require to hold longer-term bonds, compensating them for the price risk and convexity of long-term bonds.
• Example: Investors may require a term premium to hold a 10-year bond over a 1-year bond because longer-term bonds are more sensitive to interest rate changes.

6. Bootstrapping for Spot Rates

• Lesson: The bootstrapping method is used to construct the spot rate curve from coupon bonds. This method relies on the assumption that no arbitrage exists in the market.
• Example: By using the prices of coupon bonds with different maturities, bootstrapping allows you to derive spot rates for each maturity, even if direct quotes for those rates are unavailable.

  Key Lessons from Financial Market Analysis (FMAx) Module 5: Equity Pricing

1. Equity Financing vs. Bond Financing

• Lesson: Equity financing allows a company to raise capital by selling ownership stakes (shares) in the business, whereas bond financing involves borrowing with the promise to repay with interest. Equity holders are residual claimants, benefiting from profits or bearing losses, while bondholders have a fixed claim.
• Example: In equity financing, shareholders of a successful company earn dividends and capital gains if the company's stock price rises, unlike bondholders, who receive fixed interest payments regardless of company performance.

2. Intrinsic Value of Equity (Discounted Dividend Model)

• Lesson: The intrinsic value of a share of equity is the present value of all future dividends, discounted back using the market capitalization rate (k). This valuation reflects the firm’s expected earnings and dividend payout policy.
• Example: If a firm is expected to pay increasing dividends over time, the value of its shares will be higher when discounted using the DDM.

3. Market Capitalization Rate (k) and Risk

• Lesson: The market capitalization rate (k) reflects the investor's required rate of return, factoring in the risk associated with the investment. Riskier firms require a higher rate of return to attract investors, which affects the valuation of their equity.
• Example: A company with high volatility in earnings may have a higher k, leading to a lower valuation of its shares compared to a stable firm.

4. Price-to-Earnings (P/E) Ratio

• Lesson: The P/E ratio measures a firm's market price relative to its earnings. Firms with higher growth prospects, profitability, or lower risk tend to have higher P/E ratios. It helps investors assess whether a firm's stock is over- or under-valued.
• Example: A company with a high return on investment (ROI) and low risk is likely to have a higher P/E ratio because investors expect more significant earnings growth.

5. Beta and Market Volatility

• Lesson: Beta measures a firm's sensitivity to overall market movements. A firm with a beta greater than 1 is more volatile than the market, while a beta less than 1 indicates less volatility. Beta helps assess the risk premium required for holding a firm's equity.
• Example: A tech company with a beta of 1.5 will experience more significant price swings in response to market changes compared to a utility firm with a beta of 0.8.

  Key Lessons from Financial Market Analysis (FMAx) Module 6: Asset Allocation and Diversification

1. Portfolio Allocation and Importance of Diversification

• Lesson: Portfolio allocation is critical for investors as it allows for diversification, reducing risk without sacrificing returns. Diversification involves combining assets with imperfect correlation to minimize overall portfolio risk.
• Example: By investing in both bonds and equities, where returns are not perfectly correlated, an investor reduces risk compared to holding only one asset type.

2. Markowitz Portfolio Theory

• Lesson: Harry Markowitz's portfolio theory suggests that the optimal portfolio balances risk (variance) and expected return. Investors should aim to maximize returns for a given level of risk, or minimize risk for a given level of return.
• Example: An investor chooses the combination of bonds and equities that provides the highest return for a given risk level using historical return and variance data.

3. Risk and Correlation

• Lesson: The risk of a portfolio depends on the variances of the individual assets and the correlation between them. Diversification gains are greater when assets are less correlated. If assets are perfectly correlated, no risk reduction occurs.
• Example: In a portfolio of two assets with a correlation coefficient of 0.2, the portfolio's risk will be lower than the weighted sum of the individual risks, offering a diversification benefit.

4. Optimal Portfolio with Risk-Free Asset

• Lesson: When a risk-free asset is available, the goal is to maximize the excess return over the risk-free rate, adjusted for risk (Sharpe ratio). The optimal portfolio combines risky assets and the risk-free asset to achieve this.
• Example: By allocating part of the portfolio to a risk-free bond and part to equities, an investor can adjust their risk tolerance while aiming for higher returns.

5. International Diversification

• Lesson: Investing across borders adds diversification benefits due to lower correlations between domestic and international assets. However, currency fluctuations introduce additional risks.
• Example: A Japanese investor diversifying into U.S. stocks might face both asset risk and exchange rate risk, which could either enhance or diminish overall returns depending on currency movements.

  Key Lessons from Financial Market Analysis (FMAx) Module 7: Introduction to Risk Management

1. Value at Risk (VaR)

• Lesson: VaR is a risk management tool that estimates the potential maximum loss over a specific time period with a certain probability. It is used to quantify the risk of a portfolio and is typically expressed in monetary terms.
• Example: A 95% VaR of $1 million over a one-day horizon means that there is a 95% chance the portfolio will not lose more than $1 million in a single day.

2. Historical Simulation for VaR

• Lesson: Historical simulation uses historical data to calculate VaR by generating returns and simulating potential losses based on past performance. It requires no assumptions about the distribution of asset returns.
• Example: By analyzing historical returns, an investor can estimate how their portfolio might perform under different market conditions and calculate the VaR without assuming normal distribution.

3. Delta-Normal VaR Approach

• Lesson: The Delta-Normal method assumes asset returns are normally distributed and calculates VaR based on standard deviations (volatility) from the mean. It is straightforward but may be inaccurate for assets with asymmetric or non-linear returns.
• Example: Using this method, a portfolio’s VaR is calculated by multiplying the standard deviation by a chosen confidence level (e.g., 95% or 99%).

4. Monte Carlo Simulation for VaR

• Lesson: Monte Carlo simulation generates random variables to model asset returns and assess potential portfolio performance under various hypothetical scenarios. It provides flexibility in dealing with non-linear payoffs and different distributions.
• Example: In Monte Carlo simulation, the future price of a stock might be simulated thousands of times to compute the expected loss in extreme scenarios, allowing for more complex risk analysis.

5. Expected Shortfall (ES)

• Lesson: Expected shortfall (also called conditional VaR) measures the average loss that occurs beyond the VaR threshold. It addresses one limitation of VaR by capturing tail risk and estimating the potential losses during extreme market events.
• Example: If the 95% VaR is $1 million, the expected shortfall might estimate the average loss on the worst 5% of days, providing more information about the size of potential extreme losses.