Yes - I see.
I had started with daily rates and expected to be
able to match the annual rate by doing the same calculation
two different ways but in fact the two ways are not
equivalent.
I had actually done this originally in a spreadsheet so
it was less clear.
On 12/11/06, Roger Hui <[EMAIL PROTECTED]> wrote:
Given:
stkret=: 365$3^%365
bndret=: 365$2^%365
You are expecting that the following two
expressions should give the same results:
*/(0.60*stkret)+0.40*bndret
2.55099
{: (0.60**/\stkret)+0.40**/\bndret
2.6
The latter is mathematically equivalent to:
(0.60**/stkret)+0.40**/bndret
2.6
So why to you think the following expressions
should be the same?
*/(0.60*stkret)+0.40*bndret
(0.60**/stkret)+0.40**/bndret
Some examples to show that they are not the same.
(Imprecision due to floating point numbers is
clearly not a factor.)
s=: 1 2 [ b=: 3 4
*/(0.6*s) + 0.4*b
5.04
(0.6**/s) + 0.4**/b
6
s=: 2 2 [ b=: 5 5
*/(0.6*s) + 0.4*b
10.24
(0.6**/s) + 0.4**/b
12.4
s=: 1r2 1r2 [ b=: 1r3 1r3
*/ (3r5*s) + 2r5*b
169r900
(3r5**/s) + 2r5**/b
7r36
----- Original Message -----
From: Devon McCormick <[EMAIL PROTECTED]>
Date: Sunday, December 10, 2006 7:12 pm
Subject: Re: [Jgeneral] exp(y). sin(y) and accuracy.
> I am not making this assumption at all.
>
> On 12/9/06, Chris Burke <[EMAIL PROTECTED]> wrote:
> >
> > The mistake is the assumption that:
> >
> > (x^r)+(y^r) is the same as (x+y)^r
> >
> > i.e.
> >
> > */365 $ +/0.6 0.4 * 3 2 ^ %365
> > 2.55099
> >
> > */365 $ (+/0.6 0.4 * 3 2) ^ %365
> > 2.6
> >
> >
> > Devon McCormick wrote:
> > > Here's a real-life example I encountered recently. I've
> simplified the
> > > annual return
> > > numbers to 200% for stocks and 100% for bonds (*3 2,
> respectively) to
> > make
> > > the
> > > math more easily checkable.
> > >
> > > I start with a series of daily returns for a stock and bond index:
> > >
> > > */stkret=. 365$3^%365 NB. Daily stock returns - triple
> over the
> > year
> > > 3
> > > */bndret=. 365$2^%365 NB. Daily bond returns - double
> over the year
> > > 2
> > >
> > > I want to use the daily returns of a 60/40 stock/bond
> benchmark based on
> > > these
> > > index returns to calculate an annual return:
> > >
> > > NB. 60/40 stock/bond benchmark - naive calculation:
> > > */bmkret1=. (0.60*stkret)+0.40*bndret
> > > 2.5509869
> > > NB. is demonstrably wrong:
> > > 0.6 0.4 +/ . * 3 2
> > > 2.6
> > >
> > > NB. Alternate version (using cumulative returns) gives correct
> answer:> > {:bmkretcum=. (0.60**/\stkret)+0.40**/\bndret
> > > 2.6
> > >
> > > So, I've introduced a noticeable loss of precision simply by
> multiplying> > each series by a number less than one before
> accumulating the returns.
> > > This error is easy to make but harder to catch when using real
> return> > numbers
> > > if we don't have some independent calculation of the final result.
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--
Devon McCormick
^me^ at acm.
org is my
preferred e-mail
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