03-Measuring Returns and Risk _ 2014

8/10/2019 03-Measuring Returns and Risk _ 2014

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Measuring Returns and Risk

Marriott School of Management

Fin 410

Fall 2014

Rob Schonlau

Last updated Sept 8, 2014


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Your friends temporarily entrusted you with

2 million to invest. They asked you to

invest it in the best possible manner

From the last 2 lectures, you should be aware of a few differenttypes of assets you could choose to invest in and the financialmarkets where those assets are traded.

Now that you know some of your options, the next logicaldiscussion would be about how you should go about choosingbetween the various assets. But before we can do that we need tofirst review/learn some tools and measures.

Before we cover the financial theory behind optimal portfolioformation we need to first review some measures of financialperformance and risk. Lectures 3 and 4 emphasize themathematical measures we use for risk and return.

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Lecture 3 outline How do you summarize an assets financial performance?

Review of terminology and how to calculate various returnmeasures.

Look at the historical returns to various asset classes. Introducerisk and return concepts (1st pass).

Lecture 4 provides a review of some of the statistical concepts that

are useful for thinking about an assets performance and risk.

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How do you best summarize the financial

performance of an investment? Payoff, profit,

or returns?

Assume you bought a financial asset for P0and then after holdingthe asset for a time you sell it for P1. Both prices are in dollars.

Three ways to describe the financial performance:Payoff: P1

Profit: P1- P0

Return:

gross returns: P1/P0

net returns : (P1- P0)/P0 or (P1/P0)-1

Holding Period Return: (P1P0+ dividends)/P0

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Example: Assume you buy and sell 1 share

of Cisco and share of Apache stock

Cisco stock: You buy 1 share now for $100 and sell it in 3 months for$110.

Apache stock: You buy 1 share now for $200 and sell it in 3 months for$215.

What is the correct measuring stick for performance?Payoff: $110 $215Profit: $10 $15Return:

Gross: 110/100 (Cisco) and 215/200 (Apache)Net: (110-100)/100 and (215-200)/200

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Gross Returns

Gross returns measurepayoff as a percentage of the initialinvestment. Gross returns are simply payoff/price.

The gross return from buying Cisco is calculated as 110/100=110%meaning you end up with 110% of what you initially invested.

Gross returns above 100% are good. Gross returns below 100%indicate losses.

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Net Returns

Net returns measureprofitas a percentage of the initialinvestment. Net returns are calculated simply as(payoff/original price) -1.

For example the net return from buying Cisco would becalculated as 110/100 - 1 = 10%. This means your investmentgrew by 10%.

Net returns above 0 are good. Net returns below 0 signal aloss. Net returns are the growth rate of your investment.

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Two ways to annualize returns:

APRs vs EARs

Returns are quoted in a variety of ways. Annual percentagerates (APRs) ignore compounding and effective annual rates(EARs) include compounding.

EARs represent actual growth rates while APRs do not. When areturn is an actual growth rate, we call it an effective return.

Sometimes EARs are called annual percentage yields (APYs).

For example, an account that says it pays a 10% APR will notactually grow your deposit by 10% over a year if there iscompounding during that year.

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Rates can be compounded (and measured)

at different time intervals

It is also common to work with rates that are compounded, ormeasured, semi-annually, monthly, or even more often.

When you are dealing with returns over non-annual intervals it

is especially important to distinguish between effective and non-effective rates.


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Review: Time value of money formulas with

different compounding intervals.

These formulas are used when you have a single cash flow to move

through time as opposed to a stream of cash flows.

PV = present value

FV = future value

r = annual interest rate (APR)

n = number of compounding intervals per year

r/n = interest rate per compounding interval

y = number of years

= 1

=

1

= 1

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APR and Effective Rates

If n1 then

You are earning interest on interest during the year. r the effective annual rate (EAR). I.e. the effective growth

of your money over the year is not r because of compoundinterest.

r/n = the effective interest rate over the compoundinginterval.

(r/n)*n = APR = r

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Summary: Rates of Return

Conventions for Annualizing Rates of Return

APR = Per-period rate Periods per year 1 + EAR = (1 + Rate per period)

1 + EAR = (1 + Rate per period)n

= (1 + )

n

APR = [(1 + EAR)1/n1]nAPR

n


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Example: Effective returns and future value

Your investment will earn an effective return of 5% per year. Yourinitial investment is $100.

After the first year, what is the value of investment?

After the third year, what is the value of investment?

What is the effective 3-year return? 115.76/100-1 = 15.76%

10505.1100

76.11505.1100 3

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Example: Converting different time intervals for

effective rates. If the 3-year effective rate (ER3) is

15.76%, what is the 1-year effective rate (ER1)?

Suggested 3-step approach to solving the problem:

Step 1: Break the problem into its individual parts using notation:

- 2 growth rates (ER1and ER3)

- 2 different units of time (1 vs 3 years)Step 2: Assume you will make two $1 investments: one that willgrow at the ER1rate each year and the other that will grow at the ER3rate over the 3 year period.

Step 3: Choose an overall investment period that is divisible by both

units of time. In this case 3 years is the lowest number divisible byboth 3 and 1. Set the future value of the two investments over thechosen investment period equal so that you can solve for theunknown rate.

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Example continued: If the 3-year effective rate (ER3)

is 15.76%, what is the 1-year effective rate (ER1)?

Step 1: Break the problem into its individual pieces:

ER1= unknown and ER3=15.76% 2 different units of time (1 vs 3 years)

Step 2: Assume you will make two $1 investments over a 3 yearperiod: one that will grow at the ER1rate each year and the otherthat will grow at the ER3rate over the 3-year period. In this examplethe future values would be:

FV = 1(1+ER1)3 and FV = 1(1+ER

3)1= 1(1+.1576)1

Step 3: Set the future value of the two investments equal so thatyou can back out the unknown rate.

1(1+ER1)3= 1(1+.1576)1

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Example: EAR

Suppose you pay 1% interest on your credit card

balance each month. What is the EAR?

Step 1: Break the problem into its individual pieces:

ER1mo= 1% and ER12mo = unknown 2 different units of time (1month vs 12 months)

Step 2: Assume you will make two $1 investments over a 12 monthperiod: one that will grow at the ER1morate each month and theother that will grow at the ER12morate over the year. Note that ER12mo= EAR. In this example the future values would be:

FV = 1(1+ER1mo

)12= 1(1+.01)12 and FV = 1(1+ER12mo

)1

Step 3: Set the future value of the two investments equal so that youcan back out the unknown rate.

1(1+.01)12= 1(1+ER12mo)1

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Example: APR

Assume a bank charges 5% semi-annually.

What is the APR?

5% is the effective 6 month rate. Because interest is appliedevery six months this means that n=2. APRs dont accountfor compound interest.

Effective 6 month rate = r/n and .05= r/2 which means that r= 10%.

What is the effective annual return (EAR) on the loan?

(1+ r/n )n*y -1 = (1.05)2*1 -1 = 10.25%

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Concept check

(Q1) What is the difference between APR and EAR?

(Q2) If you have an initial investment of $1 and an effectiveannual rate of growth of 10%, what is your investment worth at theend of 1 year?

(Q3) Assume you have an initial investment of $100 and anaccount that provides an APR of 10% with quarterly compounding.What is your investment worth at the end of 1 year? What is theEAR?

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Concept check cont.

(Q4) The 3-year effective rate is 20%. What is the 2-year effectiverate?

(Q5) Assume a bank charges 5% semi-annually. What is theAPR? What is the effective annual return on the loan?

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Averaging returns over time: arithmetic vs

geometric means

In summarizing an assets financial performance over multiple

time intervals it is common to take the average of the returns.

There are two common approaches to averaging that youshould be familiar with: arithmetic and geometric.

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Arithmetic average

Assume you have 10 years of annual return data (r1, r2, , r10) for anasset and you want to summarize the annual historical returns forthis asset. The arithmetic average is calculated as the simpleaverage of the 10 yearly returns.

Arithmetic average =+++

0

The arithmetic average ignores compounding.

The arithmetic average provides the best prediction for the nextsingle period return assuming future returns will be drawn from thesame distribution as historical returns.

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Geometric average

Assume you have n periods of returns: (r1, r2, , rn)

The geometric average (rg) is defined as the nth root of the productresulting from multiplying a series of returns together as follows:

= 1 1 2 1 / 1

Note that the product within the brackets is the cumulative returnover the n periods that the investor experienced and includes the

effects of compound interest.

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Geometric average intuition, cont.

The geometric average represents the per-period return that if ithad occurred for each of the n years would give the same n-yearcumulative return as the actual observed sequence of historicalreturns did.

For example, a 10-year observed cumulative return would becalculated as follows: 1 1 2 1 0 1

To solve for the geometric average return think of a single return(rg) occurring repeatedly in each of the n periods that would

produce the same n-year cumulative return as observed. So1 1 2 1 1 = 1 1 1 1

1 1 2 1 1 = 1

1

Rearrange terms: = 1 1 2 1 / 1

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Dollar-weighted returns

The textbook presents 3 potential measures of performance:arithmetic average of returns, geometric average of returns,and dollar-weighted returns.

When we want to account for varying dollar amounts undermanagement we calculate the IRR for the cash flows in and

out of the portfolio.

See pages 112/113 for an example.

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Lecture 3 outline

How do you summarize an assets financial performance?

Review terminology and how to calculate various returnmeasures.

Look at the historical returns to various asset classes. Introduce

risk and return concepts (1st pass).

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Which of these asset classes are most

risky?

Large firm common stock

Long-term government bonds

Treasury bills

Small firm common stock Long-term corporate bonds

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What is risk?

Riskis related to the probability of obtaining outcomes that are fardifferent than the expected value. In some sense risk measuresthe likely variability of the possible results.

In this class we will use measures of dispersion (standarddeviation, variance) as proxies of risk. We will refine our measuresof risk in subsequent lectures.

Risk and investment decision:In the presence of risk one does not know beforehand what thereturn on any asset will be. However, you can form expectationsabout the possible return outcomes associated with each assetand can assign probabilities to each outcome.


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The mean-variance framework

The variance of returns for any investment measures the disparitybetween actual and expected returns.

Expected Return

Low Variance Investment

High Variance Investment


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General preferences and the mean-variance

framework

All else equal, people prefer higher expected returns (higheraverages). Thus given a choice between two investments of thesame risk they will choose the one with higher expected return.

All else equal, people prefer lower risk (lower variance)investments. Thus given a choice between two investments withthe same expected return they will choose the one with lower risk.

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What kinds of returns are commonly seen in

the market?

Using annual returns

1926-2009Large Stocks

(World)

LargeStocks(US)

SmallStocks(US)

LT Bonds(US)

Geometric Ave

9.43

9.57

11.6

5.37

Arithmetic Ave 11.23 11.63 17.43 5.69

Standard Deviation 19.27 20.56 37.18 8.45

1968-2009Large Stocks

(World)

Large

Stocks(US)

Small

Stocks(US)

LT Bonds(US)

Geometric Ave 9.9 9.32 10 7.96

Arithmetic Ave 11.77 10.89 13.47 8.44

Standard Deviation 19.36 17.95 27.41 10.34

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http://www.smartmoney.com/map-of-the-market/http://www.smartmoney.com/map-of-the-market/


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The distribution of annual returns for U.S.

Large Stocks (S&P 500), Small Stocks,

Corporate Bonds, and Treasury Bills, 1926-

2004.

-60% -50% -40% -30% -20% -10% 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% >100%

Annual Return

Frequency

(#

ofye

ars)

3-mo Treasury Bills

AAA Corporate Bonds

S&P 500

Small Stocks


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Rates of Return on Stocks, Bonds, and Bills


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Concept check, cont.

(Q6) You observe the following annual returns on an asset: 10%,5%, -5%, -3%, 15%, 10%. Write the formula for the arithmetic andgeometric averages.

(Q7) What two assumptions do we make about the averageinvestors preferences with regard to asset investment choice?

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03-Measuring Returns and Risk _ 2014