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how are returns computed?
Old 02-25-2005, 10:48 PM   #1
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how are returns computed?

When people quote 5 (or 10, 15,...) year
returns, how are these returns being actually computed?

The way I understand it, these numbers are
average returns. For example, let's follow 2
funds over a 5 year period.

Fund 1:
year1: 50%
2: -60%
3: 30%
4: 30%
5: -10%
average 5 year return: ((50+30+30)-(60+10))/5 = 8%

Fund 2:
years 1, 2,3,4,5: 8%
average 5 year return; 8%

The question then is: though both funds have
8% average 5 year returns, have these funds really
performed the same way? Let's suppose you invested
$10K on both funds. Some simple math shows:

fund 1 after 5 years: 9.12K
fund 2 after 5 years: 14.69K

Fund 1 lost $880 while fund 2 gained a hefty $4690
from the same $10K initially invested.

Are the numbers quoted by the financial industry
computed as average returns (as above)?
Is the financial industry using something that
often appears better than really is?
Also, isn't this the way the highly quoted
and widely accepted historical stock performance is computed?

Joe
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Re: how are returns computed?
Old 02-26-2005, 03:46 AM   #2
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Re: how are returns computed?

Returns should be calculated as follows:

(final value/initial value)^(1/# of periods)-1

As an example:
initial value: $1000
final value: $1500
Periods: 5 years

annualised return: (1500/1000)^(1/5)-1 = 0.0845 (or 8.45%)

Spanky


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Re: how are returns computed?
Old 02-26-2005, 05:17 AM   #3
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Re: how are returns computed?

OK, I'm a dummy...

What does the ^ symbol represent?

Thanks,
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Re: how are returns computed?
Old 02-26-2005, 06:43 AM   #4
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Re: how are returns computed?

Spanky has told you how returns should be calculated. Often, people use the average of the annual returns. That makes the numbers look better.

Spanky's formula is for the annualized return. If you invest an initial balance and leave it untouched for a year, your balance after one year is:
balance after one year = (initial balance)*(1+r) where r is the rate of return. The asterisk * means "multiplied by" or "times"

If you leave the money untouched for another year and if it has the same rate of return r, the balance after two years = (the balance after one year)*(1+r) = (initial balance)*(1+r)*(1+r) = (initial balance)*(1+r)^2 where ^ means "to the power".

[I just snuck in the answer to your question.]

If you leave the money untouched for a third year and if it has the indentical return r, the balance after three years = (the balance after two years)*(1+r) = ... = (initial balance)*(1+r)^3.

You may have noticed a pattern.

The balance after N years = (initial balance)*(1+r)^N.

Rearrange this and you have Spanky's formula.

The annualized return of a portfolio over N years is the value of r that would have produced the same final balance as you actually ended up with. It replaces a sequence of returns with a single number.

Here is an good approximation:

The annualized return r = the average return - 0.5*(the standard deviation or volatility of your portfollio)^2.

Your annualized return r equals the average only when there is no volatility. Any volatility reduces r. Advertisers like to use average returns because the numbers are bigger. Researchers often report average returns because averages are easier to calculate.

Have fun.

John R.
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Re: how are returns computed?
Old 02-26-2005, 07:26 AM   #5
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Re: how are returns computed?

Quote:
OK, I'm a dummy...

What does the ^ symbol represent?

Thanks,
^ means "raised to" or "to the power of..."

Example

3^3=3X3X3=27

Hope this helps.

JLP
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Re: how are returns computed?
Old 02-26-2005, 07:29 AM   #6
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Re: how are returns computed?

Quote:
Spanky has told you how returns should be calculated. Often, people use the average of the annual returns. That makes the numbers look better.

Spanky's formula is for the annualized return. If you invest an initial balance and leave it untouched for a year, your balance after one year is:
balance after one year = (initial balance)*(1+r) where r is the rate of return. The asterisk * means "multiplied by" or "times"

If you leave the money untouched for another year and if it has the same rate of return r, the balance after two years = (the balance after one year)*(1+r) = (initial balance)*(1+r)*(1+r) = (initial balance)*(1+r)^2 where ^ *means "to the power".

[I just snuck in the answer to your question.]

If you leave the money untouched for a third year and if it has the indentical return r, the balance after three years = *(the balance after two years)*(1+r) = ... = (initial balance)*(1+r)^3.

You may have noticed a pattern.

The balance after N years = (initial balance)*(1+r)^N.

Rearrange this and you have Spanky's formula.

The annualized return of a portfolio over N years is the value of r that would have produced the same final balance as you actually ended up with. It replaces a sequence of returns with a single number.

Here is an good approximation:

The annualized return r = the average return - 0.5*(the standard deviation or volatility of your portfollio)^2.

Your annualized return r equals the average only when there is no *volatility. Any volatility reduces r. Advertisers like to use average returns because the numbers are bigger. Researchers often report average returns because averages are easier to calculate.


Have fun.

John R.
I think the mutual fund companies do use Average Annual Total Returns, which is also the geometric average.

JLP
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Re: how are returns computed?
Old 02-26-2005, 02:24 PM   #7
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Re: how are returns computed?

Quote:
...
Here is an good approximation:

The annualized return r = the average return - 0.5*(the standard deviation or volatility of your portfollio)^2.
Excellent - I like this approximation.

Quote:
Your annualized return r equals the average only when there is no volatility. Any volatility reduces r. Advertisers like to use average returns because the numbers are bigger. Researchers often report average returns because averages are easier to calculate.
That's exactly my point! I'd say in most cases,
those number are quite deceiving. One should always
look at the volatity behind the returns, specially
if they are average returns.

That's a great post, btw, John.

Joe
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