The stock of business adventures sells for $40

Compute the Facebook’s expected daily return and the standard deviation using its realized return from Jan 1, 2015 to June 30,2019. (Please watch the Class video for page 21 of CH5 for more instructions).

a) Collect historical price data.

You will begin by collecting historical stock price data for the company. Historical stock prices are available on the internet through at the http://finance.yahoo.com website. At this website you will put the ticker symbol for your company in the symbol box and press “go”. At this point you will see a page that has financial information about your company. On the left-hand side of the screen there will be a series of option which you can choose to get more information about your company. At this point you want to look under “quotes” and choose “historical prices”. The ticker of Facebook is FB.

At this point, you will be given some options about the time period and frequency of the data you are collecting. You want dailydata beginning January 1, 2015 and ending June 30, 2019. Once you have specified this time period, push the “get prices” button. After the historical price information appears, you will see an option “download to spreadsheet” at the bottom of the historical price table. Select this option to bring the information into an excel file.

b) Calculate daily returns.

The formula to compute daily returns is as followngs:

(P2-P1)/P1 =

c) Estimate the expected rate of return and Standard deviation.

The excel function for the expected rate of return is “=average ()”

The excel function for the standard deviation is “=stdev()”

Ch5 Portfolio Theory -Risk and Return

Liang (Kevin) Guo

Learning Objectives

Be able to calculate ex post and ex ante risk and return statistical measures, such as holding period return, average returns, expected returns, and standard deviation.

Understand the difference between time-weighted and dollar-weighted returns, geometric and arithmetic averages.

Be able to construct portfolios of different risk levels, given information about risk free rates and returns on risky assets.

Be able to explain the CML theory.

Table of Contents

5.1 Rates of Return

5.2 Risk and Risk Premiums

5.3 Inflation and Real Rates of Return

5.4 Asset Allocation Across Risky and Risk Free Portfolios

5.5 Passive Strategies and The Capital Market Line (CML)

5.1 Rates of Return

Considering one-single period investment: regardless of the length of the period.

Holding period return (HPR): measuring Ex-Post (Past) Returns over one-single period.

HPR = [PS - PB + CF] / PB

where

PS = Sale price (or P1)

PB = Buy price ($ you put up) (or P0)

CF = Cash flow during holding period ( Such as dividend, interest)

Example: You put up $50 at the beginning of the year for an investment. The value of the investment grows 4% and you earn a dividend of $3.50. What is your HPR?

Annualizing HPRs

Annualize a holding period return: translate it into percentage per year.

(1) Without compounding (Simple or APR): HPRann = HPR/n

(2) With compounding: EAR

HPRann = [(1+HPR)1/n ]-1

where n = number of years held

Annualizing HPRs for holding periods of greater than one year

Example: Suppose you buy one share of a stock today for $45 and you hold it for two years and sell it for $52. You also received $8 in dividends at the end of the two years. What is the annual rate of return with and without compounding?

HPR =

(1) Annualized w/out compounding

(2) The annualized HPR assuming annual compounding is (n =2 ):

(

Annualizing HPRs for holding periods of less than one year

Example: Suppose you buy one share of a stock today for $45 and you hold it for 3 months and sell it for $52. You also received $8 in dividends at the end of the two years. What is the annual rate of return with and without compounding?

HPR =

(1) Annualized w/out compounding

(2) The annualized HPR assuming annual compounding is (n =0.25 ):

Investment Returns over multiple periods

The holding period return (HPR) is a simple measure of investment return over a single period.

But how to measure the performance of a mutual fund over the last ten-year period?

Several measures to find the average investment return for a time series of returns .

(a) Arithmetic average return (simple Time-weighted average)

(b) Geometric average return (Geometric time-weighted average)

(c) Dollar-weighted return

(a) Arithmetic Average Return (AAR)

(a) Arithmetic average (simple Time-weighted average)

Arithmetic means are the sum of returns in each period divided by the number of periods.

Ignore compounding (Ignore reinvestment)

Used to forecast next-period return

n = number of time periods

(b) Geometric Average Return (GAR)

(b) Geometric average (Geometric time-weighted average return)

Consider reinvestment (compounding)

n = number of time periods

Example: Continuing previous example, what is geometric average return?

( c) Dollar-Weighted Return (DWR)

(c )Dollar-weighted return

It is the internal rate of return on an investment.

IRR method: (i.e. find the discount rate that makes the NPV of the net cash flows equal zero.)

This measure of return considers both security performance and changes in investment (accounting for cash flow).

If different amounts of money were managed in the portfolio for each period it may be useful to see the Dollar weighted returns.

The DWR gives you an average return based on the stock’s performance and dollar amount invested each period.

Tips on Calculating Dollar Weighted Returns

Initial Investment is an outflow

Ending value is considered as an inflow

Additional investment is an outflow

Security sales are an inflow

Dollar-Weighted Return (Example)

Example: You initially buy one share of mutual fund AAA at $50, in one year collect a $2 dividend, and you buy another share at $53. In two years you sell the stock for $54, after collecting another $2 dividend per share. What is dollar-weithted return?

NPV = $0 = -$50 - $51/(1+IRR) + $112/(1+IRR)2

Solve for IRR:

IRR is average annual dollar weighted return.

Dollar-weighted return vs Time-weighted return

Example: You initially buy one share of mutual fund AAA at $50, in one year collect a $2 dividend, and you buy another share at $53. In two years you sell the stock for $54, after collecting another $2 dividend per share. What are the time-weighted average return (Arithmetic Average Return and Geometric Average Return )? What makes it different from dollar-weighted return?

Time-weighted return (TWR) assumes you buy ONE share of the stock at the beginning of each period and sell ONE share at the end of each period. TWRs are thus independent of the dollar amount invested in a given period.

DWR vs TWR (cont.)

TWR cash flows:

HPR for year 1 = HPR for year 2 =

To calculate TWRs:

(1) Arithmetic Average Return =

(2) Geometric Average Return =

Q: When should you use the DWR and when should you use the TWR?

- If you want to measure the performance of your investment in a fund, including the timing of your purchases and redemptions you should calculate the DWR instead of TWR.

Year 0-1 Year 1-2
0 1 1 2
-$50 $ 2 -$53 $ 2
+$53 +$54
Arithmetic Average Return (AAR) vs Geometric Average Return (GAR)

Q: When should you use the GAR and when should you use the AAR?

A1: When you are evaluating PAST RESULTS (ex-post):

Use the AAR (average without compounding) if you ARE NOT reinvesting any cash flows received before the end of the period.

Use the GAR (average with compounding) if you ARE reinvesting any cash flows received before the end of the period.

A2: When you are trying to estimate an expected return (ex-ante return):

Use the Arithmetic Average Return (AAR)

Class Discussion

(1) If you want to measure the performance of your investment in a fund, including the timing of your purchases and redemptions you should calculate the __________.

(2) If you desire to forecast performance for next period, the best forecast will be given by the ________.

(3) If you always reinvest your dividends and interest earned on the portfolio. Which method provides the best measure of the actual average historical performance of the investments you have chosen?

A. Dollar weighted return B. Geometric average return C. Arithmetic average return D. Index return

5.2 Risk and Risk Premiums

Risk = Uncertainty or potential variability in future cash flows

How likely/closely will the realized return be to the expected return?

To quantify risk, we can begin with the question: What holding period return are possible? And how likely are they?

To determine the variability, we calculate the standard deviation of the distribution of realized returns.

Indicates the dispersion around the expected (historical average) return

Two approaches to estimate the standard deviation.

Scenario Analysis

- Requires analysts’ estimates for probability & outcomes

(b) Using historical data

Assume the past data will extend into the future

Scenario Analysis

(a) Scenario Analysis: Describes a probability distribution of future returns

List all possible economic outcomes (scenarios)

Specify both the probability (likelihood) of each scenario and the HPR the security will realized in that scenario.

Example: The stock of Business Adventures sells for $40 a share. Its likely dividend payout and end-of-year price depend on the state of the economy by the end of the year as follows.

Economic states Probability Dividend Stock price HPR
Boom 1/3 2 50
Normal conomy 1/3 1 43
Recession 1/3 0.5 34
The list of possible HPRs

with associated prob is

called the probability

Distribution of HPRs.

Scenario Analysis (Formula)

Scenario analysis

Requires analysts’ estimates for probability & outcomes

Subjective Expected Return (mean) is the weighted average of all the possible returns, weighted by the probability that each return will occur.

Subjective Variance is the expected value of the squared deviation from the mean.

Standard deviation is square root of variance, describing the expected value of deviation from the mean.

E(R) = Expected Return

VAR(R) = Variance

Pi = probability of a state

Ri = return if a state occurs

Example (Scenario Analysis)

The stock of Business Adventures sells for $40 a share. Its likely dividend payout and end-of-year price depend on the state of the economy by the end of the year as follows. Calculate the expected HPR and standard deviation of the HPR.

Economic state Prob Dividend Stock price HPR Column B * Column E Deviation from Mean Return Column B * squared deviation
Boom 1/3 2 50
Normal Economy 1/3 1 43
Recession 1/3 0.5 34
Column sums Expected return = Variance =
Square root of variance = Standard deviation =
Using historical data

(b) Using historical data

Assume the past data will extend into the future

Estimating Expected HPR (E[r]) from ex-post data. Use the arithmetic average of past returns as a forecast of expected future returns

Expected return is arithmetic mean of historical realized return

Variance is the expected value of the squared deviation from the mean.

Standard deviation is square root of variance, describing the expected value of deviation from the mean.

Example (Using historical data)

Compute the Facebook’s expected daily return and the standard deviation using its realized return from May 18, 2014 to June 30,2017.

How to interpret Standard Deviation?

Standard deviations are useful for ranking the investments from riskiest to least risky.

High standard deviation indicates the high risk

About two-thirds (68.26%) of all possible outcomes fall within one standard deviation above or below the average

About 95% of all possible outcomes fall within two standard deviations above and below the average.

Example: How to interpret the standard deviation of precious example? (Expected return =10.8% and S.D.=16.37%)

23

Normal Distribution

Risk is the possibility of getting returns different from expected.

 measures deviations above the mean as well as below the mean.

Returns > E[r] may not be considered as risk, but with symmetric distribution, it is ok to use  to measure risk.

Frequency distributions of annual HPRs

Which is the most risky and which is the least risky?

Risk Premium

Risk Premium is additional return we must expect to receive for assuming risk.

The risk premium is the difference between the expected return of a risky asset and the risk-free rate.

Excess Return or Risk Premium= E[r] – rf

The risk free rate is the rate of return that can be earned with certainty.

Risk premium depends on level of risk associated with the assets. As the level of risk associated with asset increases, we will demand additional expected returns.

Risk Aversion

Risk aversion is an investor’s reluctance to accept risk.

The degree to which investors are willing to invest risky asset depends on their risk aversion.

Risk premium on risky assets is to induce risk-averse investor to hold these assets.

Risk premium an investor demands of a risky portfolio also depends on their risk aversion.

Quantifying risk aversion:

E(rp) = Expected return on portfolio p

rf = the risk free rate

0.5 = Scale factor

A = risk aversion, between 2 to 4

The larger (lower) A is, the more risk averse (tolerant) the investors are, the larger (smaller) will be the investor’s added return required to bear risk.

Sharpe (reward-to-volatility) Ratio

Sharpe (reward-to-volatility) Ratio: Risk-return trade-off, measure risk-adjusted performance

The Sharpe ratio tells us whether a portfolio's returns are due to smart investment decisions or a result of excess risk.

Class discussion: Considering two portfolios. Portfolio A generated a return of 15% and a 25% standard deviation last year while Portfolio B generated a return of 18% and a 32% standard deviation last year. T-bills were paying 4% last year. Which portfolio do you prefer?

Higher Sharpe measure indicates a more efficient portfolio

5.3 Inflation and Real Rates of Return

The average inflation rate for the last 40 years was about 4%.

For the last 40 years, this relatively small inflation rate reduces the terminal value of $1 invested in T-bills from a nominal value of $10.08 to a real value of $1.63.

Nominal rate of interest determines how much more money you will have while real rate of interest represents the rate of increase in your actual purchasing power, after adjusting inflation.

Fisher effect (Approximation): nominal rate  real rate + inflation rate

We can express their precise relationship as follows (exact Fisher effect): :

( 1+ nominal interest rate) = (1+ real rate of interest) (1+ rate of inflation)

Class discussion (Interest rate)

Example: what is the nominal rate of interest if real interest rate is 10% and inflation rate is 4%?

Example: what is the real rate of interest if nominal interest rate is 10% and inflation rate is 4%?

Historical Real Returns & Sharpe Ratios

Real returns have been much higher for stocks than for bonds

Sharpe ratios measure the excess return to standard deviation.

The higher the Sharpe ratio the better.

Stocks have had much higher Sharpe ratios than bonds.

Real Returns% Sharpe Ratio
Series
World Stock 6.00 0.37
US Large Stock 6.13 0.37
Small Stock 8.17 0.36
World Bond 2.46 0.24
Long term Bond 2.22 0.24
Motivation (Portfolio Theory)

What is a Portfolio and Why is it useful?

A portfolio is simply a specific combination of securities, usually defined by portfolio weights that sum to 1:

Weights can be positive (long positions) or negative (short positions).

Example

Your investment account of $100,000 consists of three stocks: 200 shares of stock A, 1,000 shares of stock B, and 750 shares of stock C. Your portfolio is summarized by the following weights:

Asset Shares Price/Share Dollar Investment Portfolio Weight
A 200 $50 $10,000 10%
B 1000 $60 $60,000 60%
C 750 $40 $30,000 30%
Total $100,000 100%
Motivation (Portfolio Theory)

Why not Pick the Best Stock instead of forming a portfolio?

We don’t know which stock is best!

Portfolios provide diversification, reducing unnecessary risks.

Portfolio can enhance performance by focusing bets.

Portfolios can customize and manage risk/reward trade-offs.

How do we construct a “Good” portfolio?

What does “good” mean?

What characteristics do we care about for a given portfolio?

Risk and return trade-offs

Investors like higher expected returns

Investors dislike risk

Question: How can we choose portfolio weights to optimize the risk/reward characteristics of the overall portfolio?

Mean Variance Analysis

Objective：

Assume investors focus only on the expected return and variance (or standard deviation) of their portfolios: higher expected return is good, higher variance is bad

Develop a method for constructing optimal portfolios

Portfolio Returns and Risk

The expected return on a portfolio is the weighted average of the expected returns of the individual assets in the portfolio.

The portfolio’s risk is measured by its return variance (Variance is more complicated:

36

Portfolio Return Variance

Portfolio variance is the weighted sum of all the variances and covariances:

There arenvariances, and n2 −n covariances􀂃Covariances dominate portfolio variance􀂃Positive covariances increase portfolio variance; negative covariances decrease portfolio variance (diversification)

5.4 Asset Allocation Across Risky and Risk Free Portfolios

Possible to split investment funds between safe and risky assets

Risk free asset rf : proxy; T-bills or money market fund

Risky asset portfolio rp: risky portfolio

Example. Your total wealth is $10,000. You put $2,500 in risk free T-Bills and $7,500 in a stock portfolio invested as follows:

Stock A you put 2,500

Stock B you put 3,000

Stock C you put 2,000

$7,500

Allocating Capital Between Risky & Risk-Free Assets

Weights in risky portfolio rp :

WA =

WB =

WC =

The complete portfolio includes the riskless investment and rp.

Wrf = 25% Wrp = 75%

Weights in the complete portfolio

WA =

WB =

WC =

Allocating Capital Between Risky & Risk-Free Assets

How much should be invested in the risky asset and risk free asset respectively?

Examine risk & return tradeoff

Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets

Depending on your level of risk you must merely choose between your weights of the risk free and the risky portfolio

Combined Portfolio Expected Return and Risk

Example: The information about T-bill and risky portfolio runs as follows:

Expected Return rate for T-bill, rf= 5%

Standard deviation for T-bill, rf= 0%

Expected Return rate for risky portfolio, rp= 14%

Standard deviation for risky portfolio, rp = 22%

Suppose you invest y of your total wealth in the risky portfolio

What is the expected return and standard deviation for the complete or combined portfolio?

Varying y results in E[rc] and c that are linear combinations of E[rp] and rf and rp and rf respectively.

E(rc) =y E(rp) + (1 - y) rf

c = yrp

E(r)

E(rp) = 14%

rf = 5%

22%

0

P

F

Possible Combinations of asset allocation choices

s

E(rp) = 11.75%

16.5%

y =.75

y = 1

y = 0

5-42

Risk-return Trade Offs

42

Using Leverage with Capital Allocation Line

Borrow at the Risk-Free Rate and invest in stock

Using 50% Leverage, which means y = 1.5

E(rc) =

c =

E(rC) =18.5%

33%

y = 1.5

E(r)

E(rp) = 14%

rf = 5%

= 22%

0

P

F

rp

) Slope = 9/22

E(rp) -

rf = 9%

CAL

(Capital

Allocation

Line)

s

Complete portfolio offers a return per unit of risk of 9/22.

5-44

Capital Allocation Line (CAL) and its Slope

Capital Allocation Line plots all risk-return combinations available by varying asset allocation between a risk free asset and a risky portfolio

44

Risk Aversion and Allocation

How much should be invested in the risky portfolio and risk free asset respectively? – depending on risk aversion and trade-off between risk and return.

Greater levels of risk aversion lead investors to choose larger proportions of the risk-free assets (risk free rate)

Lower levels of risk aversion (more risk tolerance) lead investors to choose larger proportions of the portfolio of risky assets

Willingness to accept high levels of risk for high levels of returns would result in leveraged combination

If the reward-to-volatility ratio increases, investors might well decide to take a greater position in the risky portfolio.

5.5 Passive Strategies and The Capital Market Line (CML)

How can investor choose the assets included in the risky portfolio?

Using either passive or active strategies

Passive strategy

The investor makes no attempt to actively find undervalued strategies nor actively switch their asset allocations.

Investment policy that avoids security analysis: securities are fairly priced

Two advantages compared to active strategy

Avoids the costs involved in the undertaking security analysis (active trading strategies may not guarantee higher returns )

free ride benefit: the activity of knowledge investors force prices to reflect currently available information

Capital Market Line (CML)

A simple passive strategy (Indexing strategy): Investing in a broad stock index (like S&P 500 index) and a risk free investment.

Indexing has become an extremely popular strategy for passive investors.

Capital Market Line: the Capital allocation line (CAL) provided by combinations of one month T-bills and a broad index of common stocks (or an index that mimics overall market performance).

What does Portfolio theory suggest?

Investors should only invest two passive portfolios

Short-term T-bills

Fund of common stocks that mimics a broad market index

Vary combinations according to investor’s risk aversion.

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Net -$50 -$51 $112

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