|
PureBytes Links
Trading Reference Links
|
Most treasury rates are quoted on the so-called "discount" basis. The
discount rate is not the same as the true rate. In his 1978 business-school
textbook "Investments," William Sharpe explains (page 41) the difference
between the discount rate and the true rate thus:
"Despite the truth-in-lending law, other methods are still used to summarize
interest rates. One time-honored procedure is the 'bank discount' method.
If someone borrows $100 from a bank, to be repaid a year hence, the bank
will discount the interest of, say $8, and lend the borrower $92. According
to the bank discount calculation, this is an interest rate of %. Not so.
The borrower only receives $92, for which he or she must pay $8 interest.
The true interest rate (APR) must be based on the money the borrower
actually gets to use. In this case the true rate is 8.7%, since 8/92 =
0.087.
It is a simple matter to convert the a "bank discount" interest rate to a
true (or APR) interest rate. If the discount rate is Rd, then the true rate
R = Rd / (1-Rd), where interest rates are expressed as decimals. The
previous example would be 0.08/(1-0.08) = 0.087."
The example provided by Mr. Lance Fisher calculates as: 0.0182/(1-0.0182) =
0.01854.
The difference is trivial when the numbers are small, as is the case
currently, in the example provided by Lance, and could be looked at as
"splitting hairs." However, when the interest rates are larger, as they
were in the late 70s or early 80s, then the difference between the discount
rate and the true rate can be significant, as in the example provided by Dr.
Sharpe in his text. Also, the annualization procedure of multiplying a
quarterly rate by four (or a monthly rate by 12, etc.) is only an
approximation which is OK for small interest rates (as now) but would lead
to substantial inaccuracy if the interest rates are significant. The
mathematically correct calculation is:
Ra = (1+Rq)^4 -1, where Ra = annual rate, and Rq = quarterly rate, with
quarterly compounding.
For Rq = 10% = 0.10, we have Ra = 1.1^4 -1 = 0.4641 = 46.41%. Simply
multiplying would yield 0.1*4 = 0.4 = 40%. To me, the difference between
46% and 40% is significant.
Leon Torban.
-----Original Message-----
From: Alex Matulich [mailto:alex@xxxxxxxxxxxxxx]
Sent: Friday, March 29, 2002 10:30 AM
To: omega-list@xxxxxxxxxx
Subject: Re: Unbelievable / Sharpe ratio
Lance wrote:
>Sorry for the length of this post, but based on the variability of the
>"risk-free rate" information that I found, I can't help but be skeptical
>of any claim of a particular Sharpe Ratio for any given system if the
>assumed "risk-free rate" is not provided along with the performance
>result.
It's not as big of a deal as you might think. The Sharpe ratio's
inclusion of the risk free rate amounts to nothing more than a fudge
factor to make the calculation more conservative. The higher you
make the risk-free rate, the more the calculation will penalize
low-return portfolios because the excess return compared to standard
deviation will be a smaller number. The number can even go negative
if you make the risk free rate high enough.
When I do stuff like this, I always assume a risk free rate of 5% per
year. Some years it's lower (like now) some years it may be higher, but
overall it's a good number for calculations.
>- Should the Discount Rate or the Investment Rate be used to calculate
>Sharpe Ratio? I would assume Investment Rate, but will everyone else
>assume the same?
Splitting hairs. This isn't an exact science. Standard deviations
assume the underlying data is normally distributed, whereas often
it's not. If you're starting out with possibly invalid assumptions,
small differences in risk free rate don't matter one whit. What you
want is some consistent basis for comparison. Using a constant rate
works fine.
>- Pardon my ignorance, but am I correct in assuming that the above
>posted 91 Day T-Bill rate is annualized (I can't believe the risk free
>rate is 1.854% every 3 months)?
To get an idea of the T-bill annualized rate, look at what a 3-month
T-bill costs, subtract that cost from $1 and multiply by 4. You
can use the T-Bill futures quotes for this if you want, it's close
enough.
--
,|___ Alex Matulich -- alex@xxxxxxxxxxxxxx
// +__> Director of Research and Development
// \
// __) Unicorn Research Corporation -- http://unicorn.us.com
|