Expected Return unit of measurement?

[fofx]

Hello,

I'm going through some notes on measuring risk for an investment. I have two values for each scenario in the investment: Return and Probability. I know the product of these two values is the expected return for that scenario...and then adding them altogether to gives the Expected Return for the investment.

My question is. Is expected return a percentage or an absolute value (i.e. $)? I suppose the answer is it depends on whether return is a % or $? My question is, if the metric isn't provided (i.e. return is just '4' or '35'), is this typically a % or a value in dollars?

Thanks.

 

[scose-no-doubt]

Suppose it depends on whether you're looking for expected return or expected value. EV will be monetary.

 

[fofx]

Suppose it depends on whether you're looking for expected return or expected value. EV will be monetary.

I'm calculating Expected Return. However the column titled "return" for a specific scenario doesn't state whether it's a percentage or currency amount.

 

[Joey25]

Hello,

I'm going through some notes on measuring risk for an investment. I have two values for each scenario in the investment: Return and Probability. I know the product of these two values is the expected return for that scenario...and then adding them altogether to gives the Expected Return for the investment.

My question is. Is expected return a percentage or an absolute value (i.e. $)? I suppose the answer is it depends on whether return is a % or $? My question is, if the metric isn't provided (i.e. return is just '4' or '35'), is this typically a % or a value in dollars?

Thanks.

It depends on the investment. If it's a long-only portfolio, then you would probably calculate:

log(Price at end) - log(Price at beginning) all weighted by the probabilities as you said.

A big problem occurs when it's a combination of long and short, with only a small exposure to the market. The percentage in this case would be huge - as if the person has hardly invested anything. Therefore in these circumstances, either the dollar amount is used or some sort of fudge like the percentage return on the sum of the absolute values of the long and shorts.

 

[scose-no-doubt]

Expected return will be % term.
Expected value will be $ term.

 

[fofx]

I'm probably not being clear. There is:

1. Return
2. Probability
3. Expected Return.

Summing the (Return * Probability) for each scenario gives the Expected Return.

Is 'Return' a percentage or amount?

I know Expected Return is a percentage, my question is about 'Return'.

 

[Joey25]

The most common representation is log(Price today) - log(Price yesterday), where log is the 'natural logarithm'. This gives the 'continuously compounded return', which is the most common representation.

The second most common is percentage return, and the least likely is dollar amount - but it does depend on the context.

Edit: the continuously compounded and percentage are both quoted as percentages, so without further information you won't be able to distinguish. However, you can rule out the dollar amount since you say you know that Expected Return is a percentage.

 

[scose-no-doubt]

For the third time... if you are being asked for expected return the return figures will be % if you are being asked for expected value they will be in $

Expected value = P*$R
Expected return = p*%R

What are you being asked for? return? value?

 

[fofx]

I am calculating Expected Return in order to calculate Variance and Standard Deviation.

scose-no-doubt - the first time you wrote

"Suppose it depends on whether you're looking for expected return or expected value. EV will be monetary."

I know EV is monetary and ER is percentage, however what you were implying (without actually stating) was that if ER is being calculated then Return is a percentage. However implying something isn't particularly helpful.

Again, with your second post, you write:

"Expected return will be % term.
Expected value will be $ term."

You seem to have missed the part about the impact this has on the metric of "return".

On your third post, you actually stated what you implied in your first post. It may be clear to you what you meant, but your first 2 posts were far from clear.

..I should add that I appreciate your help of course. Thank you.

 

[scose-no-doubt]

$

 

[Joey25]

I am calculating Expected Return in order to calculate Variance and Standard Deviation.

The Standard Deviation is in the same units as Expected Return, e.g. % or dollar

The Variance is in the units (%^2) or (dollar^2).

Volatility is the annualised standard deviation of percentage returns (where again percentage is mostly common used using naturally logs).

I would guess they want this:

1) Daily % returns using the natural log formula I gave you (weighted by probabilities)

2) The daily standard deviation of those percentages, and then sqrt(250)*daily standard deviation to give you 'volatility'.

 

[Joey25]

Is the problem like this?

State of World (Good) : Return 100, Probability 0.8
State of World (Bad) : Return 50, Probability 0.2

If so,

Solution: Expected return = (100*0.8) + (50*0.2) = 90

Variance = 0.8*( (100-90)^2 ) + 0.2*( (50-90)^2 ) = 400
Standard Deviation = sqrt(variance) = 20

The dimension of the return doesn't affect the calculation.

 

[fofx]

Is the problem like this?

State of World (Good) : Return 100, Probability 0.8
State of World (Bad) : Return 50, Probability 0.2

If so,

Solution: Expected return = (100*0.8) + (50*0.2) = 90

Variance = 0.8*( (100-90)^2 ) + 0.2*( (50-90)^2 ) = 400
Standard Deviation = sqrt(variance) = 20

The dimension of the return doesn't affect the calculation.

This is exactly the problem! And although you're right the dimension doesn't affect the calculation, it does mean I end up with 300 or 0.3 depending on whether I treated the return as a percentage or monetary value.

When I go onto calculate co-variance and then coefficient correlation, I end up with numbers > 1, and I know correlation coefficient should be between -1 and 1.

 

[Joey25]

This is exactly the problem! And although you're right the dimension doesn't affect the calculation, it does mean I end up with 300 or 0.3 depending on whether I treated the return as a percentage or monetary value.

When I go onto calculate co-variance and then coefficient correlation, I end up with numbers > 1, and I know correlation coefficient should be between -1 and 1.

The covariance can have any number and that's why people prefer the correlation coefficient. The formula for that for 2 assets is:

corr = covariance(A, B) / stdDev(A)*stdDev(B)

Is that what you are using?

I should add that the problem I wrote has 0 covariance because the events are mutually exclusive

 

[fofx]

When you write " covariance(A, B)" do you mean the product of the two variance values?

 

[Joey25]

The way you calculate covariance is (weighted by probability of event):

(Expected value of X - individual value of x) * (Expected value of Y - individual value of y)

For instance:

X1 = 1, X2 = 3, so E(X) = 2
Y1 = 4, Y2 = 2, so E(Y) =3

the 1 and 2 represent different states of the world, with probability 0.5 each.

Then, covar = (0.5*-1*1) + (0.5*1*-1) = -1


The second way of working it out is:

covar(X, Y) = E(XY) - E(X)E(Y)
= 5 - (2*3)
= -1

 

[fofx]

The way you calculate covariance is (weighted by probability of event):

(Expected value of X - individual value of x) * (Expected value of Y - individual value of y)

For instance:

X1 = 1, X2 = 3, so E(X) = 2
Y1 = 4, Y2 = 2, so E(Y) =3

the 1 and 2 represent different states of the world, with probability 0.5 each.

Then, covar = (0.5*-1*1) + (0.5*1*-1) = -1


The second way of working it out is:

covar(X, Y) = E(XY) - E(X)E(Y)
= 5 - (2*3)
= -1

So you calculate your expected return by: probability * return, and then sum them

For X
0.5*1 = 0.5
0.5*3 = 1.5

0.5+1.5 = 2. Therefore expected return for X is 2

For Y
0.5*4 = 2
0.5*2 = 1

2+1 = 3. Therefore expected return for Y is 3

The you find the product of the difference from expected return for each state:
State 1: (1-2) * (4-3) = -1
State 2: (3-2) * (2-3) = -1

Adding these two together (-1 + -1) = -2. Then average them gives -1

Is this correct?

 

[Joey25]

Yes - spot on!

Also, it's useful to know that the covariance of something with itself is the variance: cov(X,X) = var(X).

Once you're happy with the two asset case, for more it's just a case of forming a covariance matrix, where all the diagonal terms are variances and the others are the covariances.

 

[fofx]

Yes - spot on!

Also, it's useful to know that the covariance of something with itself is the variance: cov(X,X) = var(X).

Once you're happy with the two asset case, for more it's just a case of forming a covariance matrix, where all the diagonal terms are variances and the others are the covariances.

I must be doing something wrong with the data I have because I'm not getting a number between -1 and 1 for correlation.

 

[scose-no-doubt]

I am calculating Expected Return in order to calculate Variance and Standard Deviation.

scose-no-doubt - the first time you wrote

"Suppose it depends on whether you're looking for expected return or expected value. EV will be monetary."

I know EV is monetary and ER is percentage, however what you were implying (without actually stating) was that if ER is being calculated then Return is a percentage. However implying something isn't particularly helpful.

Again, with your second post, you write:

"Expected return will be % term.
Expected value will be $ term."

You seem to have missed the part about the impact this has on the metric of "return".

On your third post, you actually stated what you implied in your first post. It may be clear to you what you meant, but your first 2 posts were far from clear.

..I should add that I appreciate your help of course. Thank you.

sorry