Module 2: Portfolio Risk and Return: Part I (Very Important Module for Portfolio Management)
Example 1.1 Money-Weighted Rate of Return An investor purchases a share for $50 today. At the end of the year, she purchases another share for $60. At the end of Year 2, she sells both shares for $65 each. At the end of each year during the holding period, she also receives a dividend of $1 per share. What is her money-weighted rate of return on the investment? (Important)
(Go back and check the answer on textbook) [A bit different from examples in Corporates issuer] Keystroke: 50 [ENTER] Initial Cash OutlayCF0 = −50 [↓] 59 [+/−][ENTER] Period 1 cash flowC01 = −59 [↓] [↓] 132 [ENTER] Period 2 cash flowC02 = 132 [IRR] [CPT] Calculate IRR IRR = 13.86
Definitions/Component: Indifference Curve [Part 3/3] (Must review textbook here, pg 24-25!) (Job on the exam is to identify, not memorize)
+ "risk-averse investor" (general rule) would have a relatively steep indifference curve (significant extra return required to take on more risk). [A (very high), +, -> Risk Premium (higher)] + "less risk-averse investor" (general rule) would have a flatter indifference curve (lower extra return required to take on more risk). [A (medium), + -> Risk Premium (lower)] + "risk-seeking investor" (exception) would have an indifference curve with a negative slope. (Her utility increases with higher return and higher risk) (Want to max return and risk) [Negative slope] [A (is negative), -] + "risk-neutral investor" (exception) would have a perfectly horizontal indifference curve. Her utility does not vary with risk. [Slope is 0] [A ( is 0)] ---------------------------------------------------------- Note: The risk aversion coefficient (A, in the utility function) and the slope of the indifference curve are "positively related".
Calculation: Basic Variance, Covariance, Standard Deviation (Recap!!!) (Easy, scan through, identify)
+ Population Variance: σ^2 Σ of (Actual - Expected)^2 / n (expected: Population mean) + Sample Variance σ^2 Σ of (Actual - Expected)^2 / n-1 (expected: sample mean) + Standard Deviation σ σ or = Var^1/2 or [Σ of (Actual - Expected)^2 / n-1]^1/2 + Covariance: (only gives you direction, not degree) Σ of (AV -EV)(AV-EV) / (n-1) or Correlation Coefficient * σ1 * σ2 + Correlation Coefficient Covariance / (σ1 * σ2)
Definitions/Calculation: Application of Utility Theory to Portfolio Selection [Part 1/5] + E(R) portfolio (from Risky Asset and the Risk-Free Asset) (This is sort of a intro for the part 2) (Job on the exam is to identify, not memorize)
+ Risk Free Asset: risk = 0, σ = 0 + Expected Return for a Portfolio Containing a Risky Asset and the Risk-Free Asset: 1) Risky Asset and the Risk-Free Asset 's "covariances" or "correlation coefficient" is 0. Why is it important? To get the expected return, it's going to be the weighted average of the expected returns of the risk-free and the risky. That would be the expected return. -> ** E(R) portfolio = w of rf * E(R) of rf + w of risky *E(R) of risky = Wrf * E(R)rf + (1 - Wrf) *E(R)risky It's pretty simple, only way the test can make it more challenging: -> given: 5% = w of rf * 2% + w of risky *7% -> 5% = w of rf * 2% + (1 - w of rf) *7% -> 5% = 2%x + 7% - 7%x -> -2% = -5%x -> x = 2/5% -> x = 40%
Definition: Markowitz Efficient Frontier: [Basically, the upper half of MVF] (Theory here, no math, Theory a bit tough, quickly scan over, understand the general concept and you're good to go)
+ the upper half (above "Global Minimum Variance Frontier") of the MVF is Markowitz Efficient Frontier. It contains all the possible portfolios that rational, risk averse investors will consider investing in. Note regarding term: When efficient frontier appear, just think of the MVF. + Note: Slope of the curve decreases as we move to the right. [that's why it's a curve] The additional return attained as investors take on more risk to the right declines. Additional risk premium is declining as we go further and further to the right. + [From CFA questions explanation: Basically, The efficient frontier is the set of all attainable risky assets with the highest expected return for a given level of risk or the lowest amount of risk for a given level of return. ]
Definitions/Calculation: Utility Theory [Part 1/2] (Important, MUST MEMORIZE)
- ... Utility (Value) is a measure of the relative satisfaction that an investor derives from a particular portfolio. [For example, a risk-averse investor obtains a higher utility from a definite outcome relative to an uncertain outcome with the same expected value.] [How happy is the client going to be?] -> U = E(R) − 1/2* A* σ^2 Tip: 1/2* A = "Risk Aversion Factor", if it's "+", He's risk averse. U = Utility of an investment E(R) = Expected return σ^2 = Variance of returns A = Additional return required by the investor to accept an additional unit of risk Tip: "A" is a measure of risk aversion. It is higher for investors who are more risk averse, as they require larger compensation for accepting more risk.
Calculation: Return Measure: Money-Weighted Return or Internal Rate of Return 4/7 [The important one out of all 5] Tip: 1) We're measuring the performance from the client perspectives. 2) This is skewed by client decision: Timing of deposit and withdrawal to the account (Recap in corporate issuer, go through IRR calculation in Corporate Issuer as well) (Important, MUST MEMORIZE)
- ... the money-weighted return(IRR) accounts for the amount of money invested in each period and provides information on the return earned on the actual amount invested Drawback: it does not allow for return comparisons between different individuals or different investment opportunities. For example, two investors in the same mutual fund could have different money-weighted returns if they invested varying amounts in different periods. Calculation: (On the calculator), basically just find IRR with the same way in corporates issuer, but pay attention to the inflow./outflow, especially the dividends. (Check example 1.1) Note: Whenever we see client perspectives, we use money weighted rate of return.
Definitions: Capital Allocation Line [Part 4/5] & E(R) portfolio (we draw a line from this equation to CAL) (Important, be familiar with this) (You must check notebook here, also these 1 & 2 great examples are what we will expect on the exam!!!)
- Applying the portfolio expected return, we draw out a line. This line is called "Capital Allocation Line" Using the following formula from part 3: ** E(R) portfolio = RFR + [E(Ri) − RFR)/ σi] *σ port or **E(R) portfolio = RFR + Sharpe ratio *σ port (Note: Sharpe ratio or slope is constant) + At Point RFR, the portfolio consists only of the risk-free asset. Its expected return equals RFR and variance equals zero. + At Point A, the portfolio consists only of the risky asset. Its expected return equals E(Ri) (risky asset expected return) and variance equals σi^2 (Risky asset variance) + Risk-return combinations beyond Point A can be obtained by borrowing at the risk-free rate (1 − wi < 0) and investing in the risky asset (wi > 1).
Definition: Two-Fund Separation Theorem
- Two-Fund Separation Theorem: states that regardless of risk and return preferences, all investors hold some combination of the risk-free asset and an optimal portfolio of risky assets. Therefore, investment problem can be broken down into two steps: 1) Investing Decision: Identify his optimal risky portfolio 2) Financing decision: determines where exactly on the optimal CAL she wants her portfolio to lie. Her risk preferences (as delineated by her indifference curves) determine whether her desired portfolio requires borrowing or lending at the risk-free rate. [First, what portfolio? we will try out best to get on CAL and Portfolio M. The point is that Portfolio M is not possible to attain. It's only theory, but we will try out best to get that] [Second, Looking back to CAL and EF graphs, you see how we always mention which side she wants to be on? the borrower or lender? This is what is this part talking about.]
Definitions/Calculation: Coming up with simple regression model after part 1 & 2 using risky and risk free asset [Part 3/5] (Important, MUST MEMORIZE)
- Two-asset portfolio comprising various weights in risk-free and risky, how do we get the expected return on that portfolio? (simple regression model for a portfolio) ** E(R) portfolio = RFR + [E(Ri) − RFR)/ σi] *σ port or **E(R) portfolio = RFR + Sharpe ratio *σ port [(E(Ri) − RFR)/ σi] = slope [This is also the formula for "Capital Allocation Line"] RFR = risk free rate of return from risk free asset E(Ri) = expected return of risky σi = standard deviation of risky σ port = from last part, we know it's only (w*σ risky)
Intro: Indifference Curve [Part 1/3] (Must review textbook here, pg 24-25!) (Job on the exam is to identify, not memorize)
- risk-return tradeoff that an investor is willing to bear can be illustrated by an "indifference curve". An investor realizes the same total utility or satisfaction from every point on a given indifference curve. - Using -> U = E(R) − 1/2* A* σ^2
Example 1.3 Time-Weighted Return An investor purchases a share for $50 today. At the end of the year, she purchases another share for $60. At the end of Year 2, she sells the shares for $65 each. At the end of each year in the holding period, she also receives $1 per share as dividend. What is her time-weighted rate of return?
1) HPY1 = [(60 + 1)/50] − 1 = 22% HPY2 = [130 + 2)/120] − 1 = 10% 2) (1 + time-weighted rate of return)2 = (1 + HPY1)(1 + HPY2) = (1.22)(1.10) Time-weighted rate of return = [(1.22) (1.10)]^ 0.5 − 1 = 15.84%.
Example 1.2 Computation of Returns An analyst gathered the following information regarding a mutual fund's returns over 5 years: (Check pictures for data) Calculate the holding period return for the 5-year period. Calculate the arithmetic mean annual return. Calculate the geometric mean annual return. How does it compare with the arithmetic mean annual return? Calculate the money-weighted annual return.
1) ** Holding period return = [(1.25) (1.1) (0.9) (1.05) (1.2)] - 1 = 55.93% 2) Arithmetic mean annual return = (0.25 + 0.1 - 0.1 + 0.05 + 0.2) / 5 = 10% 3) Geometric mean annual return = {[(1.25) (1.1) (0.9) (1.05) (1.2)]^1/5} - 1 = 9.29% 4) The following cash flows are used to compute the money-weighted rate of return. CF0 = −$40; CF1 = $15; CF2 = −$16.5; CF3 = −$20.5; CF4 = $43.5; CF5 = $36 40 [+/−] [ENTER] [↓] 15 [ENTER] [↓] [↓] 16.5 [+/−] [ENTER] [↓][↓] 20.5 [+/−] [ENTER] [↓][↓] 43.5 [ENTER] [↓] [↓] 36 [ENTER] [IRR] [CPT] IRR = 7.97% (In Q4, that's not how much they put in each year. That's how much in total at the beg including the withdrawal)
Calculation/Definition: Other Return Measures and Their Applications (Job on the exam is to identify, not memorize)
1) Gross and Net Returns: (recap from FRA) + Gross Return: calculated before deductions before any expense and taxes + Net Return: calculated after expense but before taxes (= Pre-Tax Nominal Returns) 2) Pre-Tax and After-Tax Nominal Returns: + Pre-Tax Nominal Returns: it's basically synonymous with "Net Return" + After-Tax Nominal Returns: simply [net return or pretax nonminimal return * (1 - tax rate)]. Simply after tax return. 3) Real Return = [(1 + After-Tax Nominal Returns) / ( 1 + Inflation)] - 1 + It is useful in comparing returns across "time period" + It is useful in comparing returns among "countries" + after-tax real return is what an investor receives as compensation for postponing consumption and assuming risk after paying taxes on investment returns. 4) Leveraged Return: = Total Asset / Total Equity The leveraged return is computed when an investor uses leverage (by either borrowing money or using derivative contracts) to invest in a security.
Definitions/Component: Utility Theory [Part 2/2] (Job on the exam is to identify, not memorize)
1) Key Point: Utility cannot be compared across individuals because it is a personal concept. 2) Utility Theory Assumption: + Investors are generally risk averse, but prefer more return to less return. + Investors are able to rank different portfolios based on their preferences and these preferences are internally consistent: Ex: A =B, B=C, then A = C 3) Utility Theory Conclusion: + Utility is unbounded on both sides—it can be highly negative (risk seeking) or highly positive (risk averse). + Higher return results in higher utility. (true for all investor) + Higher risk results in lower utility. (True if you're risk averse) + The higher the value of "A," the higher the negative effect of risk on utility. 4) Important notes regarding "Risk Aversion Coefficient, "A": + "A" is positive for a risk-averse investor. Additional risk reduces total utility. [General rule] + "A" is negative for a risk-seeking investor. Additional risk enhances total utility. [Exception] + "A" is 0 for a risk-neutral investor. Additional risk has no impact on total utility. [Exception]
Example 3.1 (Check textbook pg. 37-38) (The graph is on the other side, it's part of the question to answer) 1) Which of the above points is not achievable? 2) Which of the portfolios will not be chosen by a rational, risk-averse investor? 3) Which of these portfolios is most suitable for a risk-neutral investor? 4) Why is gold held by many rational investors as part of a larger portfolio, when it is shown in the graph to lie on the inefficient part of the feasible set? (quickly scan over, common sense, understand the general concept and you're good to go)
1) Portfolio A lies outside the feasible set and therefore is not achievable. 2) Portfolios C and F will not be chosen by a rational, risk-averse investor. This is because Portfolio D provides higher return (25%) than both of them for the same level of risk (30%). [C and F are "inferior} Portfolios C and F are the only investable points that do not lie on the capital allocation line. 3) Portfolio D is most suitable for (less risk averse) a risk-neutral or risk seeking investor who does not care about risk and wants the highest possible return. Portfolio G is the single point of tangency. This looks like what? The efficient frontier. Portfolio B is for more risk-averse investor Portfolio E is RFR portfolio 4) Although gold lies on the inefficient part of the feasible set, it is still held by many rational investors as part of a larger portfolio. This is because gold has low or negative correlation with many risky assets, which helps to reduce the overall risk of the portfolio
Example 1.5 Computation of Special Returns Continuing from Example 1.2, suppose that the mutual fund spends a fixed amount of $600,000 every year on expenses that are unrelated to the manager's performance. Given that an investor faces a tax rate of 25% and that the inflation rate is 3%, answer the following questions: What is the annual gross return for the fund in Year 1? What is the after-tax net return for the investor in Year 2? Assume that all gains are realized at the end of the year and that taxes are paid immediately at that time. What is the expected after-tax real return for the investor in Year 5? What is the net return earned by investors in the fund over the 5-year period?
1) The fixed expenses of $600,000 would cause the gross return to be higher than net return by 1.5% (600,000 / 40,000,000). Therefore, gross return would equal 26.5%(1.5% + 25%). 2) After-tax return (Year 2) = 10% × (1 − 0.25) = 7.5% 3) After-tax return (Year 5) = 20% × (1 − 0.25) = 15% 4) After-tax real return (Year5) = (1+0.15)(1+0.03)−1=1.1165−1=0.1165=11.65% 5) The HPY for the fund over the 5-year period is computed after considering all direct and indirect expenses. The net return is 55.93%.
Definitions/Component/Quick Recap: Optimal Investor Portfolio (This information sort of just review what we went through in Lesson 2 and 3!) (quickly scan over, common sense)
1) line CAL- M represents the best portfolios available to an investor. The portfolios along this line contain the risk-free asset and the optimal portfolio, Portfolio M, with varying weights. ** 2) In the graph, investor's indifference curve, which is tangent to the CAL-M at Point C. Therefore, the optimal investor portfolio for this particular investor is Portfolio C on the CAL-M. [Basically, again, any IDC that's tangent to CAL is the optimal portfolio] 3) A more risk-averse investor, the optimal investor portfolio would lie closer to the y-axis (a higher proportion invested in the risk-free asset} A less risk-averse investor's optimal portfolio would lie closer to Portfolio M, and further away from the y-axis. 4) The only decision that the investor makes is how to divide her funds between the risk-free asset and the Portfolio M.
Summarization: Constant Correlation with Changing Weights (Take a look at the graph as an example) (Check Textbook -g 32-33 for other graphs!!) (Important) (quickly scan over, common sense)
Basing on the graph: Assuming correlation is "0", 1) When we see the same "σ port" (or risk) in different portfolios, we choose the highest E(R) or (I doubt it will come to this) 2) What's the best to do on the exam when we see different weight? Assuming all portfolios are within client level risk aversion, we choose the highest "risk premium" per unit of risk assumed, known as "Sharpe ratio" Lets say RFR = 2%, we find "Sharpe Ratio, [E(R) - RFR] / σ port": F = (0.005 - 0.002)/0.07 = .428 G = (0.00625 - 0.02)/0.05 = .85 H = (0.075 - 0.02) / 0.05 = 1.1 I = (0.0875 - 0.02) / 0.07 = 0.964 J = (0.1 - 0.02) / 0.09 = 0.889 + H has the highest Sharpe ratio, we will pick this first out of all these portfolios. Then, we rank the portfolios with Sharpe ratios.
Calculation: Return Measure: Geometric Return 3/7 (Recap in Quant) (Job on the exam is to identify, not memorize) (Important)
Geometric Return/Mean: R = {[ (1+R1) × (1+R2) ×...× (1+Rn)]^1/n} − 1 Why geometric mean? the return accounts for compounding of returns.
Definitions: Historical Return and Risk (Easy, scan through, identify) (This is Trivial Pursuit)
Historical Mean Return and Expected Return - What's happened historically, your historical average and what's expected in the future, there's no guarantee past performance will generate same in the future. (common sense)
Calculation: Return Measure: Holding Period Return 1/7 Part 2 Multiple Period How do we define the period? +Single period (day, week, month , year, be careful with their defined period) +Multiple period (Very similar to what we learned in Quant) (Job on the exam is to identify, not memorize) (Important)
Holding Period Return on investment in a Multiple period: Step 1: We calculate each sub period first: = (PEnd - PBeg + Periodic Income or D ) / PBeg or = (Δ Value + Periodic Income) / PBeg Step 2: then, we find cumulative return: => R = [ 1+ R1] * [ 1+R2] * [1 + R3] * ..... [1 +Rn] - 1
Calculation: Return Measure: Holding Period Return 1/7 Part 1 Single Period How do we define the period? +Single period (day, week, month , year, be careful with their defined period) +Multiple period (Very similar to what we learned in Quant) (Job on the exam is to identify, not memorize) (Important)
Holding Period Return on investment in a single period: = (PEnd - PBeg + Periodic Income or D ) / PBeg or = (Δ Value + Periodic Income) / PBeg Periodic income = Dividend of that period Note: We have assumed that the dividend is paid at the end of the period -> [No Reinvestment Income] --------------------------------------------------------- Exception: dividend is paid at the beginning of the period = Δ Value + Periodic Income + Reinvestment Inc) / PBeg (this wasn't covered in the textbook, it's extra from instructor)
Definitions: Other investment characteristics (Easy, scan through, identify) (This is Trivial Pursuit)
Investment Assumptions: + Returns follow a normal distribution (which is fully described by its mean and variance). + Markets are informationally and operationally efficient. Distributional Characteristics + In reality, however, returns are not always normally distributed. 1) Skewness: (recap from quants) concentrated on the left, it is referred to as right skewed or positively skewed. concentrated to the right, it is referred to as left skewed or negatively skewed. 2) Kurtosis: refers to fat tails or higher-than-normal probabilities for extreme returns (recap from quants)
Test Bank:
M2L1: (Haven't done it yet} M2L2: (Haven't done it yet} M2L3: (Haven't done it yet}
Calculation: Return Measure: Portfolio Return 7/7 (a simply weighted return) (recap from quants and this is very common)
Portfolio Return: Rp = w1R1 + w2R2 ... wnRn
Definitions/Component/Summarization: Indifferences Curves + Capital Allocation Line [Part 5/5] (Job on the exam is to identify, not memorize, be very familiar tho!) (Important) (You must check textbook here)
Question: which of these numerous portfolios that lie along the CAL will actually be chosen by the investor? (Look at the graph) Answer: Combining indifference curves and the CAL, indifference curves represent the investor's utility function, while the CAL represents the risk-return combinations of the set of portfolio + Portfolios (IDC1) that lie below the CAL may be invested in, but then the investor would not be maximizing the potential return given the level of risk she is willing to take. [Simpler word: IDC intersects 2 points with CAL is "inferior". On the exam, it's a signal that's "inferior" and we don't want it] + Portfolios (IDC3) that lie above the CAL are most desirable, but cannot be attained with the given assets. [simpler word: ideal portfolio but not possible] + ** The indifference curve that has a "single point of tangency" (Point M, IDC2) with the capital allocation line-- that's going to be the one that has the optimal portfolio. [Simpler word: IDC that's "tangible" to CAL, that's the best portfolio so to speak]
Definitions/Calculation: Standard Deviation of a Portfolio Containing a Risky Asset and a Risk-Free Asset [Part 2/5] (Job on the exam is to identify, not memorize, but be very familiar) (Important part!)
Quick recap: Portfolio Variance, σ^2: = w(1)^2*σ^2(1) + w(2)^2*σ^2(2) + 2*w1*w2*(Cov or Corre Coeff *σ1*σ2) Now, a portfolio with risky asset and risk free asset: Applying the portfolio variance, when we have a "risk free" asset its σ or risk is 0. It means, the entire right hand side of the equation is 0. ** σ^2 = = w1^2*σ^2(1) + w2^2*σ^2(2) + 2*w1*w2*(Cov or Corre Coeff *σ1*σ2) ** It basically equal = σ^2 = w(1)^2*σ^2(1), risk free part is 0.
Definitions: Risk tolerance: (Easy, scan through, identify)
Risk tolerance: refers to the level of risk that an investor is willing to accept to achieve her investment goals. + lower the risk tolerance, the lower the level of risk acceptable + lower the risk tolerance, the higher the risk aversion.
Calculation: Return Measure: Arithmetic or Mean Return 2/7 [Literally simple average] (Recap in Quant) (Job on the exam is to identify, not memorize)
Simple Average: R = [R1 + R2 +R3 ... ] / N Why arithmetic return? easy and it gives statistical properties. remember from quants? we use this for standard deviation and the dispersion around it. Note: (recap from quants) It's skewed by the outliers Recap: Coefficient of Variation: σ / expected value Recap: Arithmetic Mean ≥ Geometric Mean ≥ Harmonic mean
Calculation: ** Variance of a Portfolio of Assets (which we can find standard variation from variance) (check example on pg 21) (Instructor way) (Important, MUST MEMORIZE)
Step 1) Portfolio Variance = σ^2 = w^2*σ^2 + w^2*σ^2 + 2*w1*w2 * (σ1*σ2*corre coeff) or σ^2 = w2*σ2 + w^2*σ2 + 2*w1*w2* Cov1,2 Step 2) Portfolio Standard Deviation = σ = [ w2*σ2 + w^2*σ2 + 2*w1*w2* Cov1,2] ^ 1/2 or [Var] ^ 1/2 ---------------------------------------------------- Note: Depends on your Cov is + / -. If it's negative or 0, we get diversification. [ looking at the formula above, corre coeff is very important. Its negative or positive determines the portfolio.] From textbook: The second part (2w1w2Cov1,2) shows us that portfolio standard deviation is also dependent on how the two assets move in relation to each other (covariance or correlation)
Example 2.1 Computing Utility *** Investment Expected Return; E(r) Standard Deviation(s) A 8% 19% B 10% 24% C 17% 28% D 24% 32% Which investment will a risk-averse investor with a risk aversion coefficient of 5 choose? Which investment will a risk-averse investor with a risk aversion coefficient of 3 choose? Which investment will a risk-neutral investor choose? (we want to max return regardless of risk) Which investment will a risk-loving investor choose? (we want to max with highest risk) (Important)(We probably won't calculate like this, but we definitely want to be very familiar with this. Add on: Similar questions went on the CFA practice, so maybe there question like this afterall)
The following table shows the utility for risk-averse investors with A = 5 and A = 3. Utility at A = 5 Utility at A = 3 A −0.010 0.0259 B −0.044 0.0136 C −0.026 0.0524 D −0.016 0.0864 1) risk-averse investor with a risk aversion coefficient of 5 would choose Investment A. 2) A risk-averse investor with a risk aversion coefficient of 3 would choose Investment D. 3) A risk-neutral investor's risk aversion coefficient is 0. She wants the highest return possible. Therefore, she would choose Investment D. 4) A risk-loving investor likes both higher risk and higher return. Therefore, she would choose Investment D as well.
Calculation: Return Measure: Time-Weighted Rate of Return 5/7 Part 1 [The important one out of all 5] (Important, MUST MEMORIZE)
Time-Weighted Rate of Return: Step 1: We find each sub period return(We find the Holding period return) Step 2: We use Geometric return and we will find time-weighted rate of return. (add 1 to each return, multiply, raise to 1/n, and - 1) [Note: Time-Weighted Rate of Return is literally "Geometric Return" ... like how money weight is IRR] Add-on after CFA practice: If they ask time-weighted rate of return of the portfolio and give you 4 quarters periods, you don't need to raise it to the power, or meaning skipping geometric steps. You simply [(1+r1)(1+r2).....]
Definition: Return Measure: Time-Weighted Rate of Return 5/7 Part 2 [The important one out of all 5] Add-on: The most appropriate return metric for evaluating manager performance is the time-weighted return. (Job on the exam is to identify, not memorize)
Time-Weighted Rate of Return: measures the compounded rate of growth of an investment over a stated measurement period. In contrast to money-weighted return, the time-weighted return: 1) Is not affected by cash withdrawals or contributions to the portfolio. 2) Averages the holding period returns over time. -------------------------------------------------------- Takeaway: 1) time-weighted rate of return is preferred because it is not affected by the timing and amount of cash inflows and outflows. 2) Decisions regarding contributions and withdrawals from a portfolio are usually made by clients. Since these decisions are not typically in investment managers' hands, it would be inappropriate to evaluate their performance based on money-weighted returns. If a manager does have discretion over withdrawals and contributions of funds in a portfolio, money-weighted return might be a more appropriate measure of portfolio performance. 3) If funds are deposited into the investment portfolio prior to a period of superior performance, money-weighted return will be higher than time-weighted return. [Client made a good decision] If funds are deposited into the investment portfolio just before a period of relatively poor performance, money-weighted return will be lower than time-weighted return. [client made a bad discion]
Definitions/Component: Indifference Curve [Part 2/3] (Must review textbook here, pg 24-25!) (Job on the exam is to identify, not memorize)
Two points relating to indifference curves for risk-averse investors are worth noting: + They are upward sloping. This means that an investor will be indifferent between two investments with different expected returns only if the investment with the lower expected return entails a lower level of risk as well. + They are curved, and their slope becomes steeper as more risk is taken. The increase in return required for every unit of additional risk increases at an increasing rate because of the diminishing marginal utility of wealth. [-> which means you're going to demand higher risk premium] --------------------------------------------- + KeyPoint: Risk premium isn't constant. As risk (up), Risk premium (up) + Slope = % change in Dependent Variable/ % change in Independent Variable + (Look at the graph in the textbook!)
Intro/Definitions/Components: - Unless told otherwise, investors are "risk aversion". We assume they're risk-averse, and they will ask for higher "premium" if you want to assume more risk. Using Textbook Example: (Check the other side of flashcard) + Risk Aversion Investor + Risk Seeking Investor + Risk Neutral Investor (Job on the exam is to identify, not memorize)
Using Textbook Example: Option 1: He is guaranteed $25 in one year. Option 2: There is a 50% chance that he will get $50 in one year, and a 50% chance that he will get nothing. expected value in both these cases is $25, [option 2 = 50%*50 +50%*0 = 25] but there are three possibilities regarding the investor's preferences: 1) [General Rule] The investor may play it safe and go with Option 1. This behavior is indicative of "risk aversion". Risk-averse investors aim to maximize returns for a given level of risk and minimize risk for a given level or return. This is "Risk Aversion Investor" [Another word: Max their return and min their risk]. 2) [Exception] The investor may choose to gamble and go with Option 2. Such risk-seeking investors get extra utility or satisfaction from the uncertainty associated with their investments. This is "Risk Seeking Investor"[Another word: they Seek highest expected return and highest risk] 3) [Exception] investor may be indifferent between the two options, who seek higher returns irrespective of the level of risk inherent in an investment. This is "Risk Neutral Investor" [Another word: they Seek highest expected return irrespective of risk]
Calculation: Return Measure: Annualized Return 6/7 (recap from prior topics, we see this in fixed income) (Important, MUST MEMORIZE)
r annual = (1 + r period)^n − 1 Ex: Investment 1 offers a 5.5% return in 120 days Investment 2 offers a 6.2% return in 16 weeks Investment 3 offers a 7.3% return in 4 months = R = (1 + 0.055)^365/120 − 1 = 17.69% = R = (1 + 0.062)^52/16 − 1 = 21.59% = R = (1 + 0.073)^12/4 − 1 = 23.54% Let's say period > 1 year: 480 Days= R = (1.055)^365/480 - 1 = 4.15% 75 weeks = R = (1.062)^ 52/75 -1 = 4.26% 14 months = R= (1.073)^12/14 -1 = 6.23% =
Calculation/Definition/Quick Recap: Portfolio Risk, σ port (We've seen this in prior lessons already!!) (Check pg 30 & 31 for implication of diversification) (quickly scan over, common sense)
σ port = Portfolio Variance, σ^2: = w1^2*σ^2(1) + w2^2*σ^2(2) + 2*w1*w2*(Cov or Corre Coeff *σ1*σ2) You square root that, and it will be your σ + First part of the formula for the two-asset portfolio standard deviation (w^2*σ^2+w^2*σ^2) tells us that portfolio standard deviation is a positive function of the standard deviation and weights of the individual assets held in the portfolio. + Second Part (2*w1*w2*Cov1,2) shows us that portfolio standard deviation is also dependent on how the two assets move in relation to each other [basically shows us how are we diversified] -------------------------------------------------------- Combining what we knew to portfolio standard deviation: + maximum value for portfolio standard deviation will be obtained when the correlation coefficient equals +1. [Max risk] + Portfolio standard deviation will be minimized when the correlation coefficient equals −1. [Max diversification] + If the correlation coefficient equals zero, the second part of the formula will equal zero and portfolio standard deviation will lie somewhere in between. ----------------------------------------------------------- Conclusion: The risk (standard deviation) of a portfolio of risky assets depends on the asset weights and standard deviations, and most importantly on the correlation of asset returns. The "higher the correlation between the individual assets, the higher the portfolio's standard deviation".
