CFP- Level-1-IP and Asset Management (Global) Chapter 1-6

Standard Deviation for a Two Asset Portfolio • ������������= ������12 ������12 + ������22 ������22 + 2������1������2������1������2������������,������ Any correlation less than 1 then portfolio standard deviation will decrease Friday, 15 July 2022 Investment Planning & Asset Management

Principles of Investment Risk Standard Deviation of a two Asset Portfolio Asset Expected Standard Portfolio A Return Deviation Weight 8% 5 70% Z 14% 15 30% The correlation coefficient is -0.50 between the two assets A & Z, ������������ = ������������2���������2��� + ������������2������������2 + 2������������������������������������������������������ ������������ ������������ = (.70)2 (5)2 + (.30)2 (15)2 + 2 .70 .30 (−������. ������)(5)(15) ������������ = .49 25 + .09 225 − 23.625 ������������ = 12.25 + 20.25 − 15.75 ������������ = 16.75 = 4.09268% Calculate the standard deviation of a two asset portfolio Investment Planning and Asset Management

Covariance Covariance identifies the likelihood of two variables—investment returns, in our context—moving together (in the same direction at the same time), over time. If we think about how diversification can reduce overall portfolio risk, covariance is the primary measurement of which assets when combined should give the best risk-reduction results, without necessarily having a negative impact on performance. Friday, 15 July 2022 Investment Planning & Asset Management

Principles of Investment Risk Covariance Positive covariance means the assets being evaluated have moved “in sync.” Negative covariance shows that the assets have tended to move in opposite directions. Generally, the higher the number, the more the assets move either together (positive number) or apart (negative number). ������ ������������ − ������)(������������ − ������ ������ − 1 ������������������������������ = ������=1 Investment Planning and Asset Management

Covariance Where: I = each period n = number of periods ������������ = return of x for period I (Stock 1) ������ = average value of x ������������ = return of y for period I (Stock 2) ������ = average value of y Friday, 15 July 2022 Investment Planning & Asset Management

Correlation Coefficient For portfolio construction, it is important to know covariance as well as the degree of correlation. Like the idea that standard deviation is a more useful number than variance, correlation is a bit easier to internalize than covariance. Covariance can give us a number like 36. We know that there is a positive relationship, but we don’t know how strong it is. Friday, 15 July 2022 Investment Planning & Asset Management

Principles of Investment Risk Correlation Coefficient The correlation coefficient between two variables identifies their degree of correlation (i.e., relationship), using a scale of +1 (perfectly positive) and -1 (perfectly negative). With correlation falling between –1 to +1 we understand the strength of the relationship much quicker.

Principles of Investment Risk Correlation Coefficient ������������������ ������, ������ ������������������ ������, ������ ������������������������������ = ������������������������ ������������ ������������������ = ������������������������ If the standard deviation of stock X is 20%, the standard deviation of bond Y is 8%, and the covariance between the two is -0.00480, what is the correlation coefficient? ������������������ ������, ������ ������������������ = ������������������������ −0.00480 ������������������ = 0.20 ������ 0.08 = −0.30 Investment Planning & Asset Management

Risk on portfolio of 2 securities • Mr. Client portfolio consists of stock X and stock Y. The weight of both is 50% each. Over a year it is expected that they will perform as per the given table. Friday, 15 July 2022 Investment Planning & Asset Management

Computation of Expected return and Risk on portfolio • Let us calculate the: • A) Expected Return of X, Y • (Answer: X = 4%, Y= 10%) • B) Expected Return from Portfolio • (Answer: 7%) • C) Correlation Co-efficient • (Answer: Cor (x,y) .907) • D) Standard Deviation of X, Y and Portfolio • (Answer: SD of X= 13.379%, SD of Y=14.832%, SD of Portfolio 13.77%) • E) Co-variance of Securities • (Answer: Coy (x,y) =180) V Friday, 15 July 2022 Investment Planning & Asset Management

Formula for SD on 2 securities • ������������= ������12 ������12 + ������22 ������22 + 2������1������2������1������2������������,������ Friday, 15 July 2022 Investment Planning & Asset Management

Calculation of Coefficient of correlation (Rxy) Step Properties 1 Go to SETUP, Come down to STAT and make it ON. STAT has to be “ON” 2 In questions with probability, STAT has to be “ON” Not “OFF” 3 Press ―”STAT” mode 4 A+BX: EXE 5 XY Frequency 1 40 40 0.10 2 10 20 0.20 30 10 0.40 4 -5 0 0.20 5 -10 -20 0.10 6 Press ―AC Shift Stat 7 Press ―”7.Req” 8 Press ―”3.r (small r)” 9 Press ―”EXE”, the answer ―0.907 will be displayed Friday, 15 July 2022 Investment Planning & Asset Management

Principles of Investment Risk Coefficient of Determination (R2) Earlier, we identified two major divisions of risk: systematic (market-related) and nonsystematic (non- market-related). Systematic risk is undiversifiable, while nonsystematic risk can be mitigated through diversification. It is valuable to be able to determine how much of a portfolio’s (or asset’s) risk is diversifiable and how much is undiversifiable. Investment Planning & Asset Management

Principles of Investment Risk Coefficient of Determination (R2) R-squared measures the degree to which the fund's performance can be attributed to the performance of the selected benchmark index. R-squared is reported as a number between 0 and 100. A hypothetical mutual fund with an R-squared of 0 has no correlation to its benchmark at all. A mutual fund with an R-squared of 100 matches the performance of its benchmark precisely. The calculation to measure this is called the coefficient of determination, R2, or R-squared. As the name implies, something is squared. That “something” is the correlation coefficient. Investment Planning & Asset Management

Selection of weight of 2 securities with perfect negative (-1) coefficient of correlation • When Rxy is -1 (Perfect negative) • Weights of each security can be selected in the way that return is good but risk (SD) becomes 0 • We can create a risk less portfolio by selecting weights • Wx= SDy / (SDx+Sdy) • Coefficient of correlation= -1 (Perfect Negative), Compute the expected return of a portfolio constructed to drive the Standard Deviation of portfolio return to zero. Stock A E(r) =15% SD = 24% Stock B E(r) =17% SD= 28% Friday, 15 July 2022 Investment Planning & Asset Management

Solution • The weights that drive the standard deviation of portfolio to zero, when the returns are perfectly negatively correlated are: • Weight of A= SD of B/SD of A+SD of B • =28/(24+28) • Weight of A=0.538 • Weight of B= 1-0.538= 0.462 • The Expected return= 0.538*15+0.462*17 • = 8.07+ 7.85=15.92% • Risk on Portfolio= 0 Friday, 15 July 2022 Investment Planning & Asset Management

Practice questions • The following information is available Expected Return Stock A Stock B Standard Deviation 18% 12% 15% 9% • Coefficient of correlation= 0.70 1) What is the Covariance between Stock A and B 2) What is the expected return and Risk of a portfolio in which A and B have weights of 60% and 40% respectively Friday, 15 July 2022 Investment Planning & Asset Management

Solution 1) Covariance (A,B) = 15 (SD A)* 9 (SD B)*0.70 (RAB) • = 94.5 2) Expected Return= WA*Return A+ WB*Return B = 0.60*18+0.40*12 = 15.60% Risk of Portfolio = √(WA2*Var A+WB2*Var B+ 2 WA*WB*Covariance of A & B) = (0.60)^2*(15)^+ (0.40)^2*(9)^2+2*0.6*0.4*94.5 = √81+12.96+45.36 = √139.32 = 11.80 Friday, 15 July 2022 Investment Planning & Asset Management

Principles of Investment Risk Beta Beta can be used to measure the systematic risk. On the surface, beta is easy to understand. If an asset or portfolio has a beta of 1.00 against its benchmark(s), and the market goes up 10.00%, the asset can be expected to also rise 10.00%. Likewise, if the market drops 10.00%, the asset would be expected to drop 10.00%. The asset or portfolio matches perfectly to the benchmark(s). Investment Planning & Asset Management

Principles of Investment Risk Beta However, when the correlation is not perfect, we could expect the asset to have more or less movement than the benchmark as it moves. As an example, a beta of 1.50 means the asset will be 50% more volatile than the market, while a beta of 0.70 means the asset will be just 70% as volatile as the market. When the market moves up 10%, a beta of 1.5 will cause the asset to rise 15% (0.10 x 1.50), and when it moves down 10%, a beta of 0.70 will cause the asset to fall 7.0% (0.10 x 0.70). As a rule, clients who identify themselves as being conservative will want to focus on assets that have a beta below 1.0, or at least very close to 1.0 on the upside. Investment Planning & Asset Management

Principles of Investment Risk Beta The formula to calculate a beta is to divide the covariance of the asset and the market by the variance of the market. β = ������������������������������ ������ ������������ − ������ ������������ − ������ ������������������������������������������������: ������2= (������������ − ������)2 ������2 ������ ������ − 1 (������ − 1) ������������������������������ = ������=1 Recall that we can restate covariance as the correlation coefficient multiplied by the standard deviations. We used this in the formula for the standard deviation of a two- asset portfolio. ������������������������������ = ������������������ ������������ ������������ ������������������ ������, ������ ������������������ = ������������������������ Also, we could rewrite ������2 ������ = ������������ ������������ So, we can adjust the beta formula: =β ������������������ ������������ ������������ ������������ ������������ Investment Planning & Asset Management

Principles of Investment Risk Beta The Steps to Calculate a Beta: 1) Calculate Average Return for the Index; 2) Calculate Average Return for the Stock; 3) Calculate the Covariance; 4) Calculate the Variance of the Market; 5) Beta = Covariance / Variance Mkt. Investment Planning & Asset Management

Practice question on Beta • Covariance between security X and Market is 16 and Variance of Market security is 44.50. Calculate Beta. • β = ������������������������������ ������2 ������ • = 16/44.50 • =0.359 Friday, 15 July 2022 Investment Planning & Asset Management

How to compute Systematic Risk • As you all know Beta helps in computing Systematic Risk of Security • Market Beta is always 1 • Systematic Risk of Security= (Beta of security*SD of Market)^2 • The return of Index over five consecutive months was found to be 1.8%, 7.1%, ,-0.5%, 0.1%, 0.1% • The return on a security over the same period was found to be 6.1%, 4.7%, -3.3%, 6.8% and -2%. Compute Systematic Risk on security Friday, 15 July 2022

Solution • SD of Security(i) = 4.7479 • SD of Market(m) = 3.1276 • Rim= 0.4263 • Cov= 4.7479*3.1276*0.4263=6.3303 • Beta of Security= Cov (Market and Security)/(SD of market) 2 • = 6.3303/9.7818 • =0.6471 • Systematic Risk= (Beta of Security*SD of Market) 2 • = (0.6471*3.1276) 2 • = 4.096 Friday, 15 July 2022 Investment Planning & Asset Management

• THANKS A LOT • Chapter 3 is over here • All the very best Friday, 15 July 2022 Investment Planning & Asset Management

Chapter 4: Investment Performance Management Friday, 15 July 2022 Investment Planning & Asset Management

Evaluating Performance • Once the investment is done and portfolio has been constructed, how does the investment advisor know how well the investment choices are performing over time? • One easy method is to compare current year-end value to the beginning value. To some extent, that’s all the information many clients want. If a scheme has given a return of 8%, the client feels good until they compare it to the scheme’s benchmark return of 11% for the same period. • Similarly, a stock that is down 5% looks bad until it is compared with a 10% sector or market drop. • Additionally, this approach does not address how the return is viewed on a risk-adjusted basis, and gives no context as to whether clients are on track to achieve their financial goals. • With this in mind, the investment advisor needs to be able to evaluate investment and portfolio performance more definitively. • To begin this process, we will explore these types of investment return: • Weighted average, • Dollar (currency) weighted or internal rate of return (IRR), • Time-weighted and holding period. Friday, 15 July 2022 Investment Planning & Asset Management

Weighted Average Return A weighted average return is the average of the returns of each holding in the portfolio, adjusted by the percentage weighting of each holding in the portfolio. The weights are proportional to the value of each holding within the portfolio, to take into account what portion of the portfolio each individual return represents in calculating the contribution of that holding to the return on the portfolio. Friday, 15 July 2022 Investment Planning & Asset Management

Weighted Average Return Holding Initial value Annual Weighting Annual return x return Weighting Fund A Rs.40,000 Fund B Rs.20,000 10% 0.4 10 x 0.4 = 4.0% Fund C Rs.30,000 6% 0.2 6 x 0.2 = 1.2% Fund D Rs.10,000 8% 0.3 8 x 0.3 = 2.4% Total Rs.100,000 14% 0.1 14 x 0.1 = 1.4% 1.000 9.00% Friday, 15 July 2022 Investment Planning & Asset Management

Time Weighted Return • The geometric, or time-weighted, return is most appropriately used for evaluating the performance of investment managers. • An investor’s transactions in a portfolio and the portfolio’s returns over a four-year period are below: 1 23 4 Investment return for -10% 25% -15% 5% the year Friday, 15 July 2022 Investment Planning & Asset Management

• Geometric average return = • ������ 1 + ������1 x (1 + ������2) x … 1 + ������������ − 1 Friday, 15 July 2022 Investment Planning & Asset Management

Investment Performance Management Time Weighted Return An investor’s transactions in a portfolio and the portfolio’s returns over a four-year period are below: Investment return for the year 1 2 3 4 -10% 25% -15% 5% Arithmetic average return: = ������ 1 + ������1 x (1 + ������2) x … 1 + ������������ − 1 (-10%+25%-15%+5%) / 4 = 5/4 = 1.25% Geometric average return: 4 1 − .10 x 1 + 0.25 x 1 − 0.15 x (1 + 0.05) 4 0.90 x 1.25 x 0.85 x (1.05) 4 1.00406 − 1 ������ 100 = 0.10141% You can do with compounding function also

Solution with the help of CMPD • Rs.100 invested becomes Rs. 90 after one year with a return of -10% • Rs.90 invested becomes Rs.112.50 after one year with a return of 25% • Rs.112.50 invested becomes Rs.95.625 after one year with a return of -15% • Rs.95.625 invested becomes Rs.100.40625 after one year with a return of 5% • PV=-100, Nper= 4, FV=100.40625 • Solve i=0.1014% Friday, 15 July 2022 Investment Planning & Asset Management

Dollar/Rupee -Weighted Return This return is also known as a money-weighted return or internal rate of return (IRR). Unlike the time-weighted approach to measuring investment returns, the money-weighted return also allows for the size and timing of cash flows into and out of an investment. This makes dollar-weighted returns or internal rate of return (IRR) the most appropriate measure of return to use with individual portfolios. Friday, 15 July 2022 Investment Planning & Asset Management

Investment Performance Management Dollar-Weighted Return New investment at beginning of the year 1 Year 3 4 CFO = -5,000 Investment return for the year 5,000 2 2,000 0 Withdrawal by investor at end of year -10% -15% 20% CF1 = -3,000 (new investment at 3,000 (1,000) 0 beginning of Year 2). 0 25% (1,500) CF2 = -500 (withdrawal of 1,500 at end Year 2, -2,000 new investment Year 12 3 4 beginning Year 3). Starting balance 0 4,500.00 7,875.00 7,393.75 CF3 = 1,000 (withdrawal of 1,000 at end of Year 3). New investment at the 5,000.00 3,000.00 2,000.00 0 beginning of year CF4 = 8,872.50 (balance at end of 5,000.00 7,500.00 9,875.00 7,393.75 Year 4). Net balance at the beginning -10% 25% -15% 20% of year IRR = 4.45531. Investment return for the year Investment gain (loss) (500.00) 1,875.00 (1,481.25) 1,478.75 0 (1,500) (1,000) 0 Withdrawal by investor at end 7,875.00 7,393.75 of the year 4,500.00 8,872.50 Balance at the end of year Friday, 15 July 2022 Investment Planning & Asset Management

Continuous Compounding • Continuous compounding is the mathematical limit that compound interest can reach if it's calculated and reinvested into an account's balance over a theoretically infinite number of periods. • While this is not possible in practice, the concept of continuously compounded interest is important in finance. • It is an extreme case of compounding, as most interest is compounded on a monthly, quarterly, or semiannual basis. Friday, 15 July 2022

Continuous Compounding Formula and Calculation of Continuous Compounding • Instead of calculating interest on a finite number of periods, such as yearly or monthly, continuous compounding calculates interest assuming constant compounding over an infinite number of periods. The formula for compound interest over finite periods of time takes into account four variables: • PV = the present value of the investment • i = the stated interest rate • n = the number of compounding periods • t = the time in years Friday, 15 July 2022

Continuous Compounding • FV = PV x e (i x t), • where e is the mathematical constant approximated as 2.7183. Friday, 15 July 2022

Example of How to Use Continuous Compounding • As an example, assume Rs.10,000 investment earns 15% interest over the next year. The following examples show the ending value of the investment when the interest is compounded annually, semiannually, quarterly, monthly, daily, and continuously. • Annual Compounding: FV = 10,000 x (1 + (15% / 1)) (1 x 1) = 11,500 • Semi-Annual Compounding: FV = 10,000 x (1 + (15% / 2)) (2 x 1) = 11,556.25 • Quarterly Compounding: FV = 10,000 x (1 + (15% / 4)) (4 x 1) = 11,586.50 • Monthly Compounding: FV = 10,000 x (1 + (15% / 12)) (12 x 1) = 11,607.55 • Daily Compounding: FV = 10,000 x (1 + (15% / 365)) (365 x 1) = 11,617.98 • Continuous Compounding: FV = 10,000 x 2.7183 (15% x 1) = 11,618.34 Friday, 15 July 2022

Risk adjusted return Sharpe Ratio Three of the most widely used performance measures are the 1. Sharpe ratio, 2. the Treynor ratio and 3. the Jensen index (often referred to as “Alpha”). Friday, 15 July 2022 Investment Planning & Asset Management

Risk adjusted returns The Sharpe and Treynor ratios are used to compare the risk-adjusted return of an investment to the risk-adjusted return of a similar investment or market index. The formulas for calculating Sharpe and Treynor differ only in the type of risk represented in each. Both are relative measures of performance, meaning the information produced by either formula is useful only in comparison to another investment or index. The Jensen index, or Alpha, is used to compare actual returns to expected returns. Friday, 15 July 2022 Investment Planning & Asset Management

Sharpe Ratio If an investment’s return is 12% with a standard deviation of 14, with an available risk-free rate of 3%, the Sharpe Index (ratio) is .6428 ������������ = ������������ − ������������ ������������ = 12-3/14= .6428 When comparing investments based on this ratio, the higher the number the better the return/risk relationship for the investor. Friday, 15 July 2022 Investment Planning & Asset Management

Investment Performance Management Sharpe Ratio Return Fund 1 Fund 2 ������������ = ������������ − ������������ Standard Deviation 10% 6% ������������ Risk-Free Rate 15 12 3% 3% Sharpe Ratio Fund 1: (0.10 – 0.03)/0.15 = 0.4667 Sharpe Ratio Fund 2: (0.06 – 0.03)/0.12 = 0.25 All other factors being equal, Fund 1 would be the better choice based on the Sharpe ratio, because it produced more return per unit of risk that was taken. When comparing investments based on this ratio, the higher the number the better the return/risk relationship for the investor

Risk adjusted return Treynor Ratio The formula for =Tre������y������n���−���o������r������r������atio is: ������������������������������������������ ������������������������������ Where: rp = return of the portfolio rf = risk-free rate of return βp = beta of the portfolio Friday, 15 July 2022 Investment Planning & Asset Management

Risk adjusted return Treynor Ratio If an investment’s return is 10% with a beta of 0.85 with an available risk-free rate of 3%, the Treynor is 8.2352 ������������������������������������������ ������������������������������ = ������������ − ������������ ������������ ������������������������������������������ ������������������������������ = 10−3 = 8.23529 0.85 When comparing investments based on this ratio, the higher the number the better the return/risk relationship for the investor. The Treynor ratio can be used to compare different portfolios, funds or securities (provided they are of a similar nature to allow for reasonable comparison). Friday, 15 July 2022 Investment Planning & Asset Management

Investment Performance Management Treynor Ratio Return Fund 1 Fund 2 Standard Deviation 10% 6% Beta 15 12 0.85 0.3 Risk-Free Rate 3% 3% Treynor Ratio Fund 1: (10 – 3)/0.85 = 8.23529 Treynor Ratio Fund 2: (6 – 3)/0.3 = 10.0000 All other factors being equal, Fund 2 would be the better choice based on the Treynor ratio, because it produced more return per unit of risk that was taken. When comparing investments based on this ratio, the higher the number the better the return/risk relationship for the investor

Risk adjusted return Jensen Index Unlike the Sharpe and Treynor ratios, which are relative measures of performance, alpha is an absolute measure of performance, meaning that it can be used by itself. The simplest description of Jensen’s alpha (or index) is that it measures the difference between an investment’s actual returns and those that could have been earned by its benchmark, with a comparable beta. The Jensen index uses the capital asset pricing model (CAPM), covered later, as the basis of its measurement. Friday, 15 July 2022 Investment Planning & Asset Management

Investment Performance Management Jensen Index Jensen’s alpha = Rp – [Rf + βp(Rm - Rf)] ������������ = ������������ − ������������ + ������������ ������������ − ������������ Where: rp = actual return of the portfolio rf = risk-free rate of return βrm = return of the market (benchmark) = beta of the portfolio

Investment Performance Management Jensen Index An actively managed mutual fund with a beta of 1.1 achieves an annual return of 12%, compared to a 10% return by its benchmark index. If the risk-free rate of return is currently 3%, what is the fund’s alpha? Jensen’s alpha = Rp – [Rf + βp(Rm - Rf)] Α = 0.12 – [0.03 + (0.10 – 0.03)1.1] α = 0.12 – [0.03 + (0.07 x 1.1)] The fund returned 1.3% α= 0.12 – [0.03 + 0.077] more than was required, α = 0.12 – 0.107 given the level of risk α = 0.013 taken (positive alpha).