FINS2624 Notes - Lecture 1 PDF

**[FINS2624 Notes]** **Lecture 1: Introduction to Bond Pricing and Term Structure of Interest Rates I** **LO1: Bond Characteristics** - A bond is a certification specifying a **debt obligation** for fixed sum between the Issuer (borrower) and bondholder (lender) - Obligates the issue...

**[FINS2624 Notes]** **Lecture 1: Introduction to Bond Pricing and Term Structure of Interest Rates I** **LO1: Bond Characteristics** - A bond is a certification specifying a **debt obligation** for fixed sum between the Issuer (borrower) and bondholder (lender) - Obligates the issuer to make specified payments (interest or principal payments) on designated dates - The coupon rate, maturity date, and par value of the bond are part of the **bond indenture**, which is the contract between the issuer and the bondholder. - *Default Risk* *Risk that the issuer will not repay their debt obligations* A bond has five key parameters in its **price** in its claim of future cash flows: 1. **Term (T):** The period of time to the maturity of the bond 2. **Face Value (FV) or par value:** The principal or loan amount of the bond, typically repaid in full as one large cash flow at maturity 3. **Coupon (C):** Series of smaller cash flows paid before maturity (interest payments) Note: Coupon Rate = C/FV 4. **Coupon Frequency:** No. of times per annum the coupon is paid (typically semi-annual) 5. **Yield to Maturity (YTM):** The actual (market) interest rate applied to discount the cash flows from the bond **LO2: Bond Pricing** Two approaches to pricing debt: **Fundamental and arbitrage pricing**. We use arbitrage pricing for bonds and derivatives **Fundamental Pricing** **Arbitrage Pricing** -------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------- Prices are set in supply demand equilibrium Replicate future cash flows of an asset with a portfolio of similar assets with known prices (the replicating portfolio or synthetic asset) The properties of an asset tell us what that price is likely to be **No Arbitrage Condition:** The market value of the asset we are trying to price should equal the market value of the replicating portfolio Final product of arbitrage pricing is the present value of the cash flows **Arbitrage** (*essentially free money)* is a trade that requires either: - Require **no net capital outlay** (zero net cash flow upfront) AND generate **risk free positive cash flow** in the **future** (ʼno cost no riskʼ - Wesley Deng 2024) - A positive and risk-free cash flow today with no net outlay (zero cash flow) in the future **Law of One Price:** Two assets with the same payoff (identical cashflows) should have the **same price in equilibrium**. - Two identical bonds should have the **same price** - If not undertake an **arbitrage trade** where you simultaneously buy the cheaper bond and sell the more expensive bond. The **replicating portfolio** (synthetic asset) should exactly match the cash flows the asset to be priced. *See the following example on a risk free one year zero coupon bond that pays \$100* \ [\$\$P = \\frac{100}{1 + 0.1} = 90.9\$\$]{.math.display}\ - The alternative investment that mimics the cash flow exactly risk free is a bank account earning 10% interest. - If the price was different (i.e. bond trades at \$80.90 opportunity for an arbitrage trade. Selling a bank deposit means borrowing the money - We buy the cheaper instrument and sell (or short) the more expensive instrument Free \$10 (arbitrage profit) - In actuality, this will be very short-lived - Increased demand for bond will raise its price - Borrowing will increase bank borrowing rate - True arbitrage opportunities are very rare and short lived as traders will very quickly capitalise on these opportunities re-establishing **equilibrium.** - There is no wrong price, the prices are **internally inconsistent** and violate the **law of one price (no arbitrage principle)** **Arbitrage Pricing: General Case**  The above concepts apply to the standard bond pricing formula where we apply the risk free rate from the market in **yield to maturity (YTM)** *r* with annuities: - First bit is annuity factor, second is PV factor - Discount all coupon payments and repayment at maturity - In Excel, =-PV (Make sure the PV is negative, most of the time answer will be in the negative) **LO3: Yield/Return Measures** - **Definition:** Bonds internal rate of return - Bond price is **inversely proportional** to interest rates *YTM*  - **Discount:** P \< FV → CR \< YTM (lower coupon payments, capital gain on face value) - **Par:** P = FV → CR = YTM - **Premium** = P \> FV → CR \> YTM (higher coupon payments, capital loss) If actual interest rates are constant and equal to the YTM, then the annualized actual return over any holding period will be the YTM. However, does not occur usually because: - Actual market interest rates **do not equal the YTM** - Actual market interest rates **do not stay constant** Thus, to measure actual return on a bond: - **Holding Period Return (HPR)** - **Realised Compound Yield (HPR Annualised)** **YTM** **HPR** ------------------------------------------------------------------------------------------------ ---------------------------------------------------------------------------------- Annualised average return if the bond is held to maturity Rate of return over particular investment period Assumes **one constant** future reinvestment rate Depends on coupon rate, bond price, maturity Allows for a **changing** reinvestment rate Bond price is **inversely proportional** to interest rates *YTM* and face value Depends on future interest rates and bond price at the end of the holding period All variables are **readily** available Can only be forecast and is **not readily** observable **Example Calculation of HPR** \ [\$\$HPR = \\frac{P\_{1} - P\_{0} + C}{P\_{0}}\$\$]{.math.display}\ If interest rates fall, coupons will be reinvested at lower rates and the return (HPR) will hence be lower. Thus, the annualised HPR = realised compound yield - Changes in rates affect HPR because coupons are re-invested at different rates to the YTM. - If *YTM* is unchanged, then *HPR* = *YTM*. - If *YTM* increases ⇒ *HPR* decreases \ [\$\$Annualised\\ HPR = {(1 + HPR)}\^{\\frac{1}{\\text{Years}}} - 1\$\$]{.math.display}\  **Steps in Summary** 1. Reinvest all interim cash flows to the end of holding period (with potentially varying interest rates) 2. Calculate aggregate cash flows at end of holding period 3. Calculate HPR 4. Annualise the return **Note:** - Realised compound yield \< YTM if interest rates **fall** - Realised compound yield \> YTM if interest rates **rise** - Realised compound yield = YTM if interest rates **remain constant** **LO4: Introduction to Term Structure of Interest Rates** - Term structure of interest rates essentially refers to how interest rates vary over different investment horizons (maturities) expressed on a **yield curve**. - A line that plots yields (interest rates) of bonds with equal credit rating but differing maturities - The slope helps predict the direction of interest rates/economic expansion or contraction that could result - A coupon bond can be viewed as a collection of maturing zero coupon bonds of the same term - **Bond stripping** is the process of spinning of each coupon and principal payment as its **own separate zero-coupon bond** **Price of a t-year zero-coupon bond** **Types of Yield Curve** **Pure Yield Curve** **On the Run Yield Curve (commonly published)** --------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------- The pure yield curve uses stripped or zero-coupon bonds The on-the-run yield curve uses recently issued coupon bonds selling at or near par The pure yield curve may differ significantly from the on-the-run yield curve On-the-run bonds have the greatest liquidity in the market In reality it may be difficult to derive the pure yield curve, as relevant zeros may not be available in the market The financial press typically publishes on-the-run yield curves **Spot Rate (Pure Yield)** - The spot rate is the interest rate today (i.e. at time 0) for a *t*-period zero coupon bond (i.e. starting today and ending at time *t*) - Conceptually, the spot rate (*y~t~*) can differ from the YTM (y) of a t-period bond because the **YTM applies to any type of bond** whereas the **spot rate** specifically refers to **zero-coupon bonds**   Generalised Method to calculate yield for **1-period zero** and **2-period coupon** 1. Find *y*~1~ from the price equation of a one period zero coupon bond. 2. Substitute into *y*~1~ for the 2-period coupon bond and find *y*~2~. Repeat this recursive process (domino effect) to find the spot rate for any *t* period coupon bond **Lecture 2: Term Structure of Interest Rates II and Duration** Readings - BKM 15 Term Structure of Interest Rates: 15.2 -- 15.4 - BKM 16 Managing Bond Portfolios: 16.1 -- 16.3 **LO1: Future Interest Rates** When given a series of spot rates, we can use them to derive interest rates predicted in the future. - For example: *y~1~* and *y~2~* are spot rates for 1-year and 2-year investments respectively. - Therefore, from *y~1~* and *y~2~* we can infer the interest rate for an investment starting from the end of year 1 and ending at the end of year 2 - Depicted through the notation '~1~*y*~2~', '~2~*y*~6~' etc We can replicate the cash flows of long-term bonds by re-investing the cash flows from short-term bonds. Note key notation: - ~0~*y*~t~ (or simply *y*~t~) is the spot rate at time 0 for a *t*-period bond (maturing at t) - ~1~*y*~t~ is the spot rate at time 1 for a (*t-1*)-period bond If everything is certain (~1~y~2~ is known for certain time at time 0), the **two investment returns should be equal.** **Spot Rates vs Short Rates** The Spot rate is the **geometric average** of its **component short rates**. In the equation below, to find y~4~, need to multiply all the rates together and then take 4^th^ root \ [\$\$\\sqrt\[4\]{(5\\% \\times 7.01\\% \\times 9.025\\% \\times 11.06\\%)} \\approx 8\\%\$\$]{.math.display}\  YTM on zero coupon bonds is called the spot rate - the rate that prevails today for a time period corresponding to the zero's maturity. The short rate for a given time interval refers to the interest rate for that **interval** available **at different points in time** - For example, the two year spot rate is an average of year 1's short rate and year 2's short rate **Forward Rates** - Under **uncertainty**, we can still infer the **expected** future interest rate - These expected interest rates are known as **Forward Rates** - Forward rates are agreed upon interest rates commencing at some point in the future for a defined future time period - Called forward interest rate rather than future short rate because it not need be the interest rate that actually will prevail at the future date  If most individuals are short-term investors, bonds must have prices that **make f~2~ greater than E(r~2~).** The forward rate will embody a **premium** compared with the expected future short-interest rate. This liquidity premium **compensates** short-term investors for the **uncertainty about the price** which they will sell their bonds in the future. - 1 year Forward rate in one year is given by the equation:  **LO2: Term Structure Hypotheses** **LO2.1: Expectations Hypothesis** - The market has an **expectation** that the (certain) cash flow from the 2-year zero will equal the expected cash flow from the 1-year rollover. - A common version states that the **forward rate equals the market consensus expectation** of the future short interest rate; that is, f~2~ = E(r~2~), and liquidity premiums are zero - **Note: Remember the powers on the brackets** However, the yield curve is typically **upward sloping** - If EH were to hold, this means the **market expects actual interest rates** in the **future** to be **higher than today almost all the time** - Perhaps there is some other explanation for the typically upward sloping yield curve **LO2.2: Liquidity Preference Hypothesis** - The **Liquidity preference hypothesis (LPH)** refers to the case where most investors in the market have **shorter horizons** and therefore **prefer short-term investments** to **avoid the liquidity risk** of holding **long-term bonds** - In the first example, for investors with shorter term horizons, Strategy 2 (the buy and sell strategy) was riskier. We call this **Liquidity Risk** - In the second example, for investors with longer term horizons, Strategy 1 (the reinvestment strategy) was riskier. We call this **Reinvestment Risk**  **Summary of Theories** **LO3: Interest Rate Risk (Must Memorise)** Interest Rate Risk refers to how bond prices change in response to interest rate movements. This **interest rate sensitivity** has a number of aspects: 1. **Bond prices** and **yields** are inversely related **P** ↑ **YTM** ↓ 2. Bonds are **more (upwardly) price sensitive** to **interest rate** falls than **(downwardly)** price sensitive to interest rate increases (**convexity**) 3. **Long-term bonds** are **more price sensitive** than **short-term bonds** to a **change in interest rates** i.e. Δ**P** ↑ **T** ↑ 4. As **maturity increases**, **price sensitivity increases at a decreasing rate** 5. **Low coupon bonds** are **more price sensitive** than high coupon bonds to a change in interest rates i.e. Δ**P** ↑ **C** ↓ 6. **Bonds with lower YTM** are **more price sensitive** than bonds with higher YTM to a change in interest rates ie Δ**P** ↑ **YTM** ↓  **LO4: Duration** **LO4.1: Definition and Calculation** **Formal measure of the i/r risk** - The "effective" maturity of a bond is known as its **Duration** - Duration is a measure of the **weighted average time** that cash flows (CF) on a bond are received - In other words, it measures "effective" maturity by taking into account when the payments on a bond are actually made - As all cash flow on a zero-coupon bond is back-ended, it is always the case that **duration = maturity** for zero coupon bonds - **For coupon bonds**, it is always the case that **duration \< maturity** - Duration is a key concept for at least 3 reasons: - 1\. A measure of the **effective maturity** (or "payback") of a bond - 2\. A measure of the **interest rate sensitivity** of a portfolio - 3\. An essential tool in **immunizing portfolios** from **interest rate risk** Duration is a measure of interest rate risk - it is the **sensitivity or change in the market value** of a bond to a change in interest rates given by:  **Steps for calculating** 1. Calculate CF's on the bond 2. Discount each CF by the YTM to derive PV of each CF (the sum of the CF PV's is the Bond Price) 3. Divide each period's CF PV by Bond Price to derive weights 4. Multiply each weight by the time period (ie 1, 2, 3...n) 5. Sum the total **LO4.2: Determinants** - **Rule 1:** The duration of a **zero-coupon** bond **equals its time to maturity** - **Rule 2:** All else being equal, a bond's **duration is *higher*** when the **coupon rate** is ***lower* D** ↑ **C** ↓ - The higher the coupon, the **higher the weights** put on those cash flows received before maturity relative to the principal repayment at maturity - This "front-ends" the return and so shortens the payback on the bond (**shortening its effective maturity**) - **Rule 3:** All else being equal, a bond's duration generally ***increases*** with its **time to maturity D** ↑ **T** ↑ (note that it *increases* at a *decreasing* rate) - **Rule 4:** All else being equal, the duration of a coupon bond is ***higher*** when the bond's **yield to maturity** is *lower* **D** ↑ **YTM** ↓ - If **YTM decreases**, the **PV of all CFs increase**s, but the value of CFs in later periods increases relatively more (**stretching its effective maturity**) - If **YTM increases**, the **PV of all CFs decreases**, but the value of CFs in later periods decreases relatively more (**shortening its effective maturity**) - **Rule 5:** The duration of a **perpetuity** is equal to: (1 + *y*) / *y* **LO4.3: Duration as a measure of i/r risk**  **LO4.4: Convexity** The relationship between bond prices and yields is not linear. However, the duration-price relationship below implies it is linear. Duration (at a specific yield point) is essentially the **slope of the price-yield curve only at that point** - Therefore, the **duration-price relationship** (duration rule) above is a good approximation for only **small changes in bond yields** - Bonds with **greater convexity** have **more curvature** in the price-yield relationship and the **duration rule** is a **less accurate approximation** of interest rate sensitivity  **Portfolio Duration** We can think of a coupon bond as a portfolio of zero-coupon bonds. Then the duration of this coupon bond is just the weighted average of durations of the component zeros - Similarly, a portfolio of bonds can be viewed as an aggregate bond - In this case, the CF of this aggregate bond is the sum of CF's of its individual component bonds. **Asset-Liability Matching** Say we have a known future liability to meet: - One way to make sure we have sufficient funds to cover the future liability is to buy a zero-coupon bond that pays exactly the amount of the liability when it is due - Most of the time this zero-coupon bond will be difficult to find in the market Instead we may have to buy coupon bonds (whose maturity may match up to when the liability is due). Now we are exposed to interest rate risk: - **Higher** **interest rates** are beneficial if we have taken on **reinvestment risk**, since you can reinvest at better rate - **Lower** **interest rates** are beneficial if we have taken on **liquidity risk** (or *price risk* as labelled in BKM), since we can sell bonds at a higher price Inverse relationship between duration and coupon rate Duration is a function of time to maturity D = f(TTM) **Tutorial 3** - YTM can be the weighted average of y1, y2, y3, y4 - **Liquidity premium is biased** - The longer the bond, the bigger the liquidity risk premium, since the bond is being hold for longer time  **Duration** - Inverse relationship between bond price and interest rate - Bond price is exposed to the risk of interest rate changes - Interested in measuring **sensitivity of the market value** of a bond to a change in interest rates  **Lecture 3 -- Markowitz Portfolio Theory** Readings - BKM 5 Risk, Return and the Historical Record: 5.4 -- 5.5 - BKM 6 Capital Allocation to Risky Assets: 6.1, 6.5 - BKM 7 Optimal Risky Portfolios: 7.1 -- 7.2, 7.4, Appendix B **LO1: Measuring expected return and risk** **Key Assumptions** - For simplicity we are focused only on one-period investment - Keeps things simple compared to multi-period dynamic optimisation - Called "static" analysis - Fundamentals are the same for multiple periods - Two assumptions - 1\. One Period - 2\. Normally Distributed The return on any investment is uncertain. However, we can significantly simplify our analysis under uncertainty if we assume returns are **normally** **distributed:** - Importantly this means future scenarios for an investment can be assessed solely based on what it will return on average (mean - µ), how different all possible outcomes could be (volatility/standard deviation - σ) and how its payoffs move with other assets (correlation - ρ) - A normal distribution assumes symmetrical returns around the mean **Expected Return** - The 'reward' from an investment is a return - Returns = Price~t~ -- Price~t-1~ - But usually focus on **percentage return** \ [\$\$\\%\\ Return = \\frac{\\text{Price}\_{t} - \\text{Price}\_{t - 1}}{\\text{Price}\_{t - 1}}\$\$]{.math.display}\ - On a forward-looking basis under uncertainty, we form return expectations by multiplying the return under each possible state *r(s)* with the associated probability of that state *p(s)*. Expected return *E*(*r*) **Risk** - We seek to maximise return because return maximises wealth - However, risk is faced in a world of uncertainty - In finance, risk usually refers to the possibility that realised outcomes differ (better or worse) than what is expected - We seek not to avoid risk, but to incorporate it appropriately into decision making - Think of return as the "reward" and risk as the "cost" of that reward - Risk is not how much you 'lose' or 'gain', will always be a positive number, since it is a range of variability **Measuring Risk** Risk is a **measure of volatility** of our returns. Volatility is simply the amount of (squared) deviation from our expectations. It is given by the **variance equation**:  Historical data is used to estimate future return and risk - **The higher the variance, the higher the risk** **LO2: Risk and Risk Aversion** **Mean-Variance Criterion** - If asset returns are **normally distributed**: Mean and variance (or standard deviation) of returns ("the first two moments") are sufficient to describe asset return and risk - This is known as "Mean-Variance" analysis - Expected return (*y*-axis) and standard deviation (*x*-axis) Mean-variance criterion states that **Portfolio A dominates Portfolio B** if: \ [*E*(*r*~*A*~) ≥ *E*(*r*~*B*)~ and *σ*~*A*~ ≤ *σ*~*B*~]{.math.display}\ **Dominance** - 2 dominates 1; has a higher return - E(R~2~) \> E(R~1~), [*σ*~2~ = *σ*~1~]{.math.inline} - 2 dominates 3; has a lower risk - E(R~2~) = E(R~3~), [*σ*~2~ \ E(R~3~), [*σ*~3~ = *σ*~4~]{.math.inline} **Risk Aversion** - Portfolio theory rests on the assumption investors are **Risk Averse**: - That is, they follow the mean-variance criterion -- for the same level of E(r) investors will **choose** the asset with the **lowest risk (𝜎)** - **Risk neutral** investors **judge assets solely by their E(r)** and are **indifferent to risk** while **risk seekers** prefer **higher levels of risk** - **Portfolio attractiveness** **increases** with **E(r)** and **decreases** with **risk (𝜎)** - What happens when return increases with risk? - Individual investors have different degrees of risk aversion - It will depend on each investor's individual risk-reward trade-off - Hence the need to understand investor preference and utility **LO3: Preference and Utility** - **Utility** is a measure of **satisfaction or welfare/happiness of an investor** - When an investor has a **preference** for Asset A over Asset B, we say that Asset A provides the investor with **greater utility** - A utility function assigns a **value to each outcome** so **that preferred outcomes get higher values** (i.e., higher utilities) - We model utility as depending only on **wealth** for convenience - Essentially we assume that the more money an individual has, the better he/she is able to achieve preferred outcomes - Therefore, in finance: - More is better - Surer is better -- risk aversion **Utility Function** - Wealth depends on **investment returns** and is therefore **uncertain** - Investment returns are uncertain unless we **invest in risk-free assets** - As wealth is dependent on risk and return, we can also define utility based on the risk/return properties of the assets we will invest our wealth in - We assume a quadratic utility function for investments (contains return and risk components: - For a risk free portfolio, *U* = *r*, as *r* is a known constant and σ^2^ = 0 - Hence, *U* for a risky investment can be interpreted as its **certainty equivalent return** - Risk aversion doesn't mean we won't take risk, it means we put a **higher price (return)** **on taking risk.** 'Highest return' for 'minimal risk' high utility for investor  **Indifference Curves** - We can illustrate our preferences by indifference curves - Curves plotted in the risk-return or "mean-variance" (*E(r)-* σ) space that **connect points giving equal utility** - Note that two indifference curves with different utility levels never intersect - Choosing any point which lies on the graph will have **same utility**, despite differing levels of expected return and risk (e.g. U=25% line) - An indifference curve is a **graphical representation of the utility function** for different levels of risk and return which offer the same utility value - Addresses the question of the risk/reward trade-off (like PPF from eco) - Called an Indifference "Curve" because the utility function is a quadratic equation: - **Risk averse investors** (*A* \> 0) will be **convex/upward sloping** - **Risk seekers** (*A* \< 0) will be **concave/downward sloping** - **Risk neutral investors** (*A* = 0) will be **linear and flat**  **LO4: Forming Portfolios** - What is our objective when we make an investment: - To achieve the optimal investment outcome that maximises our utility - Utility from an investment depends on its expected return and risk - Construct the **optimal portfolio** which has minimum risk, maximum return, therefore providing the highest utility (among all possible portfolios) - Portfolio Return: Weighted average returns - Portfolio Variance: Tricky **Portfolio Return** - The return of a portfolio of assets, *r~P~* is a weighted average of the returns of the assets that make up the portfolio: - Similarly, the expected return of a portfolio is a weighted average of the expected returns of its component assets:  **Portfolio Risk** - Portfolio risk is not as straight forward, as it involves accounting for how the assets in the portfolio move with each other -- which is known as their **covariance** - The covariance of two variables refers to their **tendency** to be **higher or lower** than their **respective mean values** at the same time: - It is the **sum of deviations** from the mean of 2 assets in different states: **Covariance and Correlation** \ [Cov(*X*,*Y*) = *ρσ*~*X*~*σ*~*Y*~]{.math.display}\  **For 2 assets** For n assets  But can only be tested up to n=3 **Perfect Correlation**  **Diversification** - Combining two or more imperfectly correlated assets in one portfolio is called **diversification** - The risk reduction is called a **diversification benefit.** This concept is at the heart of portfolio theory. Sophisticated version of "not putting all your eggs in one basket" - The lower the correlation the better (same portfolio return for lower portfolio risk) - We typically want to combine many assets with low (ideally negative) correlations to maximize the diversification benefits - **Low correlated assets** are **unlikely to return below their respective means** at the same time (cancels out some portfolio risk) - If **correlations are negative**, they even work as **insurance for each other** (a riskless hedge may be possible) - The **more assets** we have in our portfolio, the **risk will reduce** **Lecture 4 -- Optimal Portfolios** Readings: - BKM 6 Capital Allocation to Risky Assets: 6.2 -- 6.6 - BKM 7 Optimal Risky Portfolios: 7.3 -- 7.4 **LO1: Optimal portfolios -- no risk-free asset** **Definition:** Provides the best risk and return combination for the investor (minimum risk minimum variance portfolios) **Minimum Variance Frontier (MVF)** By combining the assets in different proportions we can construct portfolios with risk-return profiles which are on the red line: - The red line is known as the **minimum variance frontier (MVF)** -- the **lowest risk** portfolio at each level of return. The procedure to derive it is: 1. Set **target expected portfolio return**, e.g. 10% 2. **Optimise portfolio weights** to **minimise variance** at this level of return 3. Repeat at **different return levels** till we have plotted the frontier  - Any portfolio or asset combination **below the MVF** will be **dominated** by a portfolio **on the MVF** - Similarly, any portfolio below the turning point of the MVF will be **dominated** by one **above the turning point** - The **turning point** is the portfolio (asset combination) that has the **lowest possible risk level** - It is known as the **Global Minimum Variance Portfolio (GMVP)** **Efficient Frontier** - As each portfolio on the MVF above the GMVP **dominates those below**, we **discard the portfolios with Expected Returns below the GMVP** - The part of the MVF above the GMVP is called the **Efficient Frontier** - It is the section of the MVF above the GMVP (to plot it we would need to identify the GMVP first) - **Risk averse** investors should only choose portfolios on the **efficient frontier** - Where along the frontier should we invest? Pick the **point** which provides the **highest utility**, represented by the **highest attainable indifference curve** - The portfolio marked by the **star** gives the highest possible utility - Only applies when we only have risky assets to invest in - This portfolio lies on the **tangent point** between the **efficient frontier** and the **highest attainable indifference curve**  - **Adding a risk-free asset** can improve our allocation options and enhance our optimal portfolio - Although **all investors face the same efficient frontier**, they will each have **different utility functions** (due to different **risk aversion coefficients *A***) - They will therefore have **different indifference curves** - So, the optimal risky portfolio may **differ** between investors **Key Notes:** - Frontier is derived from optimum portfolio frontier (like a curve) - When **weighting for asset is negative**, it means you **borrow money** - E.g. w~A~ = -1, w~B~ = 2 Want to invest 200% into Asset B, but only 100% of money, borrow 100% of asset A, use this money to invest 200% into Asset B - To find GMVP: - **Note: σ~DE~ = Cov(D,E)** **LO2: A Complete Portfolio** **Introducing the Risk-Free Asset** - **Short-term Government bills** (T-bills) are often considered as the **risk-free asset**, as they have almost no default risk - Some market practitioners also use the Government bond rate as the risk-free asset for long-term investments - The return on the risk-free asset (the "risk free rate") is denoted ***r~f~*** (generally positive) **Complete portfolio *C*** where our capital is **fully allocated** into a **combination of risky and risk-free assets.** The return on our complete portfolio *C* is: \ [*r*~*C*~ = (1−*y*)*r*~*f*~ + *yr*~*P*~]{.math.display}\ - Denote the fraction invested in the risky asset - Denote the fraction invested in the risk-free asset (1 -- *y*) - *y* could be any non-negative constant (it can even be higher than 1) **Expected Return** - If the market is in **equilibrium** (demand = supply), then **expected return** = **required rate of** **return** determined by the market - If an investment is viewed as "unattractive", investors will **require a higher return to buy it**, which results in a **lower price** - **Ceteris Paribus,** a low price implies higher required return higher expected return - ↑ Required Return (higher risk) → ↓ Price → ↑ Expected Return The Expected Return on the Complete Portfolio *C* is given by: \ [*E*(*r*~*C*~) = *E*\[(1−*y*)*r*~*f*~ + *yr*~*P*~\]]{.math.display}\ \ [*E*(*r*~*C*~) = (1−*y*)*r*~*f*~ + *yE*(*r*~*P*~)]{.math.display}\ \ [*E*(*r*~*C*~) = *r*~*f*~ + *y*\[*E*(*r*~*P*~) − *r*~*f*~\]]{.math.display}\ - Note: Red = Risk Premium **Complete Portfolio Risk**  - ***y*** represents the **fraction invested in risky assets** - If we drew a straight line connecting the risk-free rate *r~f~* to the risky asset *P*, ***y*** would tell us **where along the line we sit** - If ***y* \> 1**, it means **borrowing at the risk free rate** (instead of investing in the risk-free asset) and **investing the proceeds into risky assets**. In this case, we are taking a **levered position** in the risky asset **LO3: Optimal Risky Portfolio (*P\**)** - Capital Allocation Line (CAL~P~) -- refers to complete portfolios - Both efficient risky portfolios, *P~1~* and *P~3~*, are dominated by Complete Portfolios of the risk-free asset and the (inefficient) risky portfolio, *P~I~*. - Hence, efficient risky portfolios are no longer efficient if we further consider complete portfolios that include the risk-free asset - The risk-free asset has **significantly increased our investment opportunities**  - Clearly it is **not optimal** to **combine the risk-free asset** with **an interior (inefficient) risky portfolio**, *P~I~* (i.e. A portfolio underneath the efficient frontier) - For any portfolio of the risk-free asset and an interior portfolio, we can always find a portfolio of the risk-free asset and an efficient portfolio that dominates (same return but lower risk, or same risk but higher return) **Principles to note:** - The **higher the slope of CAL** to an efficient risky portfolio (steeper the CAL), the **better the risky portfolio** is to form a complete portfolio - For any complete portfolio on a lower-slope CAL, you can always find a **dominating complete portfolio** on the **higher-slope CAL** - We are interested in drawing CAL to risky portfolio *P* where: 1. The complete portfolios along the CAL dominate all other portfolios (the steepest possible CAL) 2. The investment opportunity is attainable **Finding P\*** - The point of tangency between CAL (tangent line) and efficient frontier is called the **Optimal Risky Portfolio, *P\**** - **Note terminology: CAL~P\*~ = CAL** (same thing) - Only risky portfolios on the efficient frontier should be considered. The CAL links the risk-free asset to a portfolio on the efficient frontier - The steeper the CAL the better e.g. *CAL~2~* clearly dominates *CAL~1~* - The optimal risky portfolio *P\** is on the steepest possible CAL **and** the efficient frontier -- it is the point of tangency from *r~f~* to the efficient frontier **LO4: Separation Property** - Different portfolios on CAL correspond to different allocations (*y*) in the optimal risky portfolio (*P\**), and so carry **different amounts of risk**: σ~c~ = *y*σ~P\*~. - An investor will choose their optimal portfolio on CAL (and therefore their optimal risk allocation *y*), depending on their **risk preference** - Therefore, the **Separation Theorem** states **portfolio optimisation** can be derived in two separate steps: 1. Find the **optimal risky portfolio**, *P\** (**common to all investors**) 2. Determine the **share of our wealth** we **invest** in the optimal risky portfolio *P\** (ie the "optimal risky share" *y\**) based on our **individual risk tolerance** (**specific to the individual investor**) **Step 1: Choosing *P\** - maximizing the Sharpe Ratio** The slope of the CAL *S~P~* tells us the incremental return/reward we get - *E*(*r*) - for taking on incremental risk 𝜎. We can calculate this slope from our two known points on the line:  The **slope of the line** (CAL) is known as the **Sharpe Ratio** **Step 2: Choosing the optimal risky share *y***Different investors have different risk aversion coefficients, utility functions and indifferent curves  Combining return and risk on complete portfolio formula with utility formula, we get:  **Borrowing Constraint** - The optimal risky allocation *y\** can be **\>1** for **(less risk averse) investors.** This means investors take a **levered position** in the optimal risky portfolio, i.e. borrowing to invest - In reality, although we can invest at the risk-free rate (by buying government bonds) **we generally can't borrow at the risk-free rate** - Let's assume that we are able to borrow at some higher interest rate than the risk-free rate ***r~b~* \> *r~f~*** - Then it means that for those (less risk averse) investors who would like to borrow to invest, the section of CAL above *P\** is not obtainable, as those levered positions are based on *r~f~* Therefore, a new optimal risky portfolio (*Pb\**) and associated CAL should be derived for the borrowing rate *r~b~.* Only the section of CAL above *P~b~\** is relevant, as this is the CAL for borrowing to take a levered position in the optimal risky asset - Assume no borrowing constraints  **Separation Theorem Implications** - In market equilibrium all investors hold the same optimal risky portfolio *P\** - If, in equilibrium, all investors are holding the same risky portfolio, this must be the **market portfolio *M*** which comprises all assets - The weight of each asset in *M* is the asset's total market value divided by the total value of *M* - Each investor holds a small fraction of this portfolio - Since ***P**\** is the market portfolio *M*, *M* has the **highest possible Sharpe ratio** - The rational way to **increase risk** (and return) is to **increase leverage** and **invest more in *M*** (rather than deviating from *M* and buying risky assets in different weightings to their weightings in *M*) **The Market Portfolio *M*** - Since every investor holds *M* for their risky asset allocation, the attractiveness of a stock is determined by how it contributes to the return and risk of this portfolio - The contribution an individual asset makes to portfolio **return** **is simply proportional** to its weight - However, its contribution to portfolio **risk** **depends on its covariance** with the other stocks in the portfolio - So the risk of an individual asset is no longer measured just by its own variance or volatility, but rather its covariance with all other assets in the market portfolio *M* -- in equilibrium this is now our key measure of risk - Key insight into **Capital Asset Pricing Model (CAPM)** **Lecture 5 -- Capital Asset Pricing Model (CAPM)** Readings: - BKM 9 Capital Asset Pricing Model: 9.1 **LO1: Derivation of the CAPM** **Market Portfolio** - Under Markowitz model, every rational investor invests along the CAL regardless of risk aversion level - In market equilibrium, all rational investors hold the **same optimal risky portfolio P\*** - If all investors hold the **same P\*,** this must be the **market portfolio *M*** and this must comprise all assets **Capital Market Line (CML)** - Since every investor invests in *M*, the common CAL associated with this is called the CML. **CML is equivalent to the CAL**. It is an aggregation of **EVERY** investor's CAL which are all the same - M is equivalent to P\*. It is the aggregation of every investor's optimal risky portfolio P\* which are all the same - **Every investor invests along the CML**. Movements along the CML represent different allocations between the market portfolio M and the risk-free asset **CAPM Assumptions** CAPM is derived under a similar equilibrium model framework as Markowitz: 1. Investors are **rational mean-variance optimizers** as per Markowitz 2. Investors are **price takers** - no investor is large enough relative to the market to influence equilibrium prices 3. Investors have a **common planning horizon of a single period** 4. Investor have **homogeneous expectations** on the statistical properties of all assets (same expected returns and covariances) 5. There is a **risk free asset** available to all with **no borrowing constraints** 6. All assets are **publicly traded** (which is not the case in reality, for example, human capital is typically viewed as untradeable) 7. **Perfect capital markets** - there are no financial frictions such as short selling constraints, transaction costs, taxes etc Under these assumptions, all investors **derive the same efficient frontier, CAL, and *P\****. Since everyone holds *P\**, *P\** is the market portfolio *M* of ALL risky assets **How does Markowitz lead to CAPM?** 1. Everyone will hold market portfolio M and M will comprise all assets 2. Therefore, every individual asset should be assed based on its contribution to market portfolio return and market portfolio risk 3. The contribution an individual asset makes to market **return** **is simply proportional** to its weight. However, its contribution to market **risk depends on its covariance** with the other stocks in the market 4. So individual asset risk is no longer measured just by its own variance or volatility, but rather its covariance with all other assets in the market portfolio *M* -- **in CAPM equilibrium this is now our key measure of risk** **Derivation of the CAPM equation**  **Interpretation of the CAPM** - So an asset *i*'s expected return above the risk-free rate (its "risk premium") equals its **Risk x the Price of Risk** - Its Risk is measured by its relative contribution to the variance of *M*, its beta: 𝛽*~i~* = *Cov*\[*r~i~*, *r~M~*\] / 𝜎^2^~M~ which is based on how it co-varies with *M* - Price of Risk is *M*'s expected return above the risk-free rate: *E*(*r~M~*) -- 𝑟*~f~* (often referred to as the **market risk premium**) - In equilibrium, **more risky assets** are compensated with **higher expected returns** to make them **equally favourable** to less risky ones - Note the following convention: ***r*** refers to absolute (raw) return; ***R*** refers to the risk premium i.e. the **excess returns** above the risk-free rate. With this notation the CAPM can be stated simply as: - *E*(*R~i~*) = 𝛽~𝑖~*E*(*R~M~*) - That is, asset *i*'s risk premium is linearly related to the market risk premium according to its 𝛽 **Beta 𝛽** - Rather than running covariances between each individual asset in the portfolio, as we do under the Markowitz model, we estimate each asset's return as a **function of its covariance** with the market return under CAPM - Using this **one common factor** significantly reduces the calculations we need to make - Importantly, the β relates each asset's return to the market return - β can be \1. - β*~i~* \> 1 means that *Cov*(*r~i~ , r~M~*) \> 𝜎^2^𝑀 - the asset contributes **more risk** than the average asset (or is **more risky than the market**) - β*~i~* \< 1 means that *Cov*(*r~i~ , r~M~*) \< 𝜎^2^𝑀 - the asset contributes **less risk** than the average asset (or is **less risky than the market**)  **LO2: Regression Formulation of the CAPM** **Decomposing Total Risk**  **Systematic and Unsystematic risk** **Systematic Risk / Market Risk / Non-diversifiable Risk** - Risk attributable to **market-wide risk sources** (macro factors) which **remains even after extensive diversification** - Recall that β is a measure of an asset's covariance with the market. Systematic risk is the part of an asset's risk that is in common with the market, and thus **unable to be diversified** - Examples include: - Market-wide events such as an oil embargo, interest rate rise, recession or an earthquake - These events affect all firms, so we say that these risks are systematic - Since all firms are affected, their returns would move similarly - **Therefore we cannot diversify the risk away** **Unsystematic Risk / Firm-Specific Risk / Diversifiable Risk / Idiosyncratic Risk** - The part of an asset's risk that is **specific to the asset itself** - Since this risk comes from sources that do not affect the market, it can be diversified away in the market portfolio - Examples include: - Suppose that a retail store experiences poor sales due to unfashionable stock, or an accidental fire burns a plant down. This would be bad for the respective firm's stock - But it is unlikely that other firms would be affected - Since other firms are unaffected, we could diversify such risks away by combining many stocks in a portfolio As we **add more assets** to our portfolio through **diversification**, we **eliminate unsystematic (firm-specific) risk** and we **converge** toward the **systematic (market) risk level** which cannot be diversified away. **CAPM Prices Systematic Risk** - The CAPM is an equilibrium model. In equilibrium, the CAPM predicts that all rational investors have **diversified and eliminated unsystematic risk** - The CAPM therefore **only prices systematic risk** - If all investors hold *M* and *M* includes all assets, by definition all investors have eliminated unsystematic risk - Investors are therefore **not compensated** for **taking on unsystematic risk** (because it should be diversified away in equilibrium) - Systematic risk is measured by the 𝛽 - The implication is that we **should always hold well-diversified portfolios** - The CAPM recommends a **passive strategy** -- ie hold the market portfolio ("meet the market, don't try to beat the market") **LO3: Security Market Line (SML)** The **Security Market Line (SML)** is captured in the **E(r)-β space** because systematic risk (measured by β) is the relevant risk to measure when we have well diversified portfolios under a CAPM equilibrium. This relationship is **linear** as indicated by the CAPM equation.  - **Movements up and down** the SML represent the **reward of higher return E(r)** at the **cost of higher systematic risk β** - Indicates a **positive relationship** between systematic risk and expected excess return above r~f~ (i.e. the risk premium) - An asset with β=0.5 contributes **half the risk** of the average asset and gets **half the reward** in terms of expected excess return above rf - An asset with β=2 contributes **double the risk** of the average asset and gets **double the reward** in terms of expected excess return above rf **SML vs CAL** - Individual assets typically plot **under the CAL**, since for efficient portfolios located on the CAL, all unsystematic risk is diversified away, meaning **total risk = systematic risk** - An asset's risk to the **left of the CAL** is **systematic** (and earns expected returns), but its risk to the **right of the CAL** is **unsystematic** (and earns no extra expected return) **SML v CML** - **\[Key\]** Both the CML and SML **start from r~f~**, but the **CML accounts for total risk** (σ) while the SML only accounts for **systematic risk β** - **\[Key\]** CML only includes efficient/optimal assets. SML: Includes all assets in the market - CML's *x*-axis is σ but **SML's** *x*-axis is **β** - The **β** of the market portfolio *M* is always 1 but the market σ*~M~* may change. The SML **does not show** the level of σ*~M~* as it scales σ*~M~* to 1 - ***E*(*r~M~*) is the same** for both the CML and SML. It is determined by the CML and used by the SML - A **change in σ*~M~*** has **no effect on the SML** (except where it changes *E*(*r~M~*)) but it **would change** the slope of the **CML**  **LO4: Applications, assumptions and critiques of the CAPM** **Assumptions and Critiques** - CAPM relies on several strong assumptions in particular that **all investors are rational mean-variance optimizers**, and **all assets are tradeable**. However, these **assumptions easily fail in reality**, and so does CAPM. - The CAPM predicts expected return which may differ from actual return. But the basic principles of portfolio theory and CAPM should hold: - Investors should hold **diversified portfolios** to reduce risk - Relevant risks are **systematic ones** that are **not diversifiable** - An **assets' expected returns** should be determined by its **systematic risk** **Applications** - The CAPM has implications for portfolio construction - **Avoid unsystematic risk** by holding a **well-diversified portfolio** - **Monitor** the amount of **systematic risk** of investments - The CAPM allows us to **evaluate investment performance** - Relate returns to a **return benchmark** with **similar systematic risk** - Examine whether the unexpected component of realised return is **due to unsystematic risk** (in which case good or bad returns may be totally due to luck) - The CAPM can be used for **capital budgeting purposes** - Estimate the β for non-traded assets or projects from similar traded assets or firms, and then calculate required returns - Use this required return (discount rate) to discount estimated future cash flows on assets/projects and determine if it produces positive NPV CAL with the Max Sharpe Ratio = CML y\* is the risk loading on market portfolio. y\* = β M = All risky asset When you talk about covariance, sometimes it is just about systematic risk **Lecture 7 -- SIM and Factor Models** **LO1: Single Index Model (SIM)** **Stiglitz Paradox** - If everyone is a **passive CAPM investor**, there is **no incentive to do analysis**. If there is no analysis, there is **no channel for information** about the real economy to enter the capital market - Analysis is costly. Reasonable to believe prices deviate enough from their fundamental values for an analyst to achieve enough excess returns to reward them for their work **Single Index Model (SIM)** - Under the CAPM, ***M* should include all risky assets**, including for example an individuals' human capital, and all risky assets are tradeable - In reality, no observable market portfolio M containing all risky assets and not all risky assets are tradeable - Thus, try to pick a proxy for M when using the CAPM in practice - To emphasize that we are not dealing with the true market portfolio *M*, we sometimes refer to the resulting **empirical** model as the **Single Index Model, or the SIM** **SIM Regression** - To evaluate an asset's returns based on CAPM, we typically regress its excess returns against the market excess returns: - Similar to the CAPM, the above equation is stated in terms of excess returns (the premium to the risk-free rate) - Under CAPM, we expect that: - 𝛼~i~ = 0 - *E*(*ε~i~*) = 0 - *Cov*(*ε~i~* , *r~M~*) = 0 - *Cov*(*ε~i~* , *ε~j~*) = 0 - Difference from CAPM is the 𝛼 (Jensen's) and error terms *ε* (unsystematic risk) **Jensen's Alpha** - Traditional CAPM model 𝛼 = 0, if not then asset *i* is mispriced according to CAPM - We refer to the 𝛼 as Jensen's alpha or simply as the "alpha" - Often a **non-zero 𝛼** indicates our model **does not describe the full relationship between the variables** (e.g. the model may be incomplete) - In order to analyse mispricing, let's assume that the model is correct and a non-zero 𝛼 indeed indicates mispricing of that asset - Note: ***r*** refers to absolute (raw) return; ***R*** refers to the excess return, then the mispricing, *α~i~* can be given by:  **LO2: Active Investing** **Mispriced Assets** - If an asset is **correctly priced** by the CAPM, it should **plot on the SML** - In the graph below, assets A and C are mispriced, i.e. have a non-zero *α* - If we identify mispriced assets, we deviate from the market weights to exploit the mispricing, the *α* - This means selecting an optimal risky portfolio, *P\**, that isn't equal to *M* **Exploiting a Mispriced Asset** - Constructing an optimal risky portfolio P\* (comprising M and the mispriced asset) which will have a higher Sharpe ratio than M alone - ***M* has the highest Sharpe Ratio** when **all assets are correctly priced** - This is **no longer the case** when there are **mispriced assets (disequilibrium)** - We aim to construct a portfolio with a **higher Sharpe Ratio than the market**  - Denoting A as the mispriced asset, the return is given by: - The reward of active investing is the additional alpha generated - The risk/cost is the additional (unpriced) unsystematic risk introduced **Discussion** - We can think of this as if we were "buying" the benefit of mispricing at the cost of taking on unsystematic risk - We weigh these effects against each other (reward-to-risk) - The price at which we can "buy mispricing" is *α~A~*/σ^2^*~ϵA~* - This is the same principle of "buying" excess return by taking on risk when constructing *P\** (optimising the **reward-to-risk ratio**) - This ratio determines how much we **tilt our portfolio towards the mispriced asset** **Reward-to-Risk Ratio** We then derive our weighting in A according to its reward-to-risk ratio relative to the market M's reward-to-risk ratio: Lastly, we **adjust for the diversification benefit** of combining asset A with the market, based on Asset A's beta β*~A~*:  **Active Weights** - When *α* \> 0 asset is **underpriced,** we **buy** it (positive active weight) - When *α* \< 0 asset is **overpriced**, we **sell** it (negative active weight) - The active weight depends essentially on *w^0^~A~* - The higher is *w^0^~A~*, the more **extra return per unit** **of (unsystematic) risk** we get from our **active asset** (relative to the return per unit of risk we get from *M*) -- so we should **increase the weight in the active asset**  **Active Investing Example** 1. Calculate the *α* (the benefit) 2. Calculate the unsystematic risk (the cost) 3. Convert to a reward-to-risk ratio 4. Get the optimal weight of the active portfolio by relating the active portfolio's reward-to-risk to the market's reward-to-risk 5. Calculate the amount invested in this reconstructed risky portfolio and the amount invested in the risk-free asset **Portfolio *α* and β**  **Exploiting multiple mispriced assets** - First, we combine them to form an Active Portfolio (*AP*) - Then we treat the *AP* as a single mispriced asset, and follow the same procedure discussed previously to form *P\** comprising *M* and *AP* - Forming the (optimal) active portfolio AP of mispriced assets: - Each mispriced asset in the active portfolio is given a weight proportional to its reward-to-risk ratio:  - So the weight of the *i*th mispriced asset in the active portfolio would be: **Information Ratio**  **LO3: Factor Models** **Multi-Factor Model** - A general factor model expresses the excess return *Ri* on an asset as follows: - where, *F~j~* is the *j* th systematic factor, and *βj,i* is the loading of asset *i* on the *j*th factor - And the asset's expected excess return should be equal to  - where, is the risk premium of the *j* th factor **Fama-French Carhart 4 Factor Model** - Fama and French developed the three-factor model, Carhart expanded with the fourth factor - WML - A factor model commonly used in practice is the Fama-French-Carhart four-factor model that includes four portfolio returns as factors: - Market portfolio - Small-minus-big (size) portfolio **(SMB)** - High-minus-low (book-to-market ratio) portfolio (**HML)** - Book to market ratio: Book value of asset/Market value of asset (usually \