Stock Examples - Background |
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| Every
stock in the market has a certain return. We assume that this return has a
normal |
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| distribution.
This means that we can completely describe this distribution by two terms.
The |
| first
term is the mean (expected return) and the second is the variance of
returns. We can also |
| compute
a covariance of returns between any pair of stocks -- those that move
together will |
| have
positive covariances, those that move in opposite directions will have
negative |
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| covariances.
If we know the expected return and variance of several stocks, we can put |
| together
a portfolio of these stocks that has a desired variance (risk) with a certain
expected |
| return.
The Solver is used to pick the portfolio with the smallest variance for a
certain |
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| expected
return. (Or it could be used to find
the highest expected return for a certain |
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| variance.) |
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| How
do we calculate the expected return and variance of a portfolio? Harry
Markowitz |
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| developed
a method that computes the portfolio variance as the sum of the individual
stock |
| variances
and covariances between pairs of stocks in the portfolio, weighted by the
relative |
| proportion
of each stock in the portfolio. From a mathematical point of view this is the
right |
| thing
to do. However, since all covariance terms between all stocks must be known,
this |
| requires
many calculations. A portfolio with 100 different stocks would require more
than |
| 5000
covariance terms, for instance. |
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| William
Sharpe devised another method of determining the expected return and variance
of |
| a portfolio. This
method assumes that the return of each stock is composed of two parts: |
| One
part (denoted by beta) is dependent on the overall market's performance, and
the other |
| (denoted
by alpha) is independent of the market. So we can write the expected return
as |
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Return = Beta * Market + Alpha + Residual |
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| In
this equation, alpha and beta are constants that are different for each
stock; the residual |
| term
is a random variable with an expected value of zero. When this formula is
used, a |
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| portfolio
with 100 stocks would only require 302 terms to fully describe its
distribution (100 |
| alphas, 100 betas, 100 residual variances, plus the market
return and the market variance). |
| Compared
to the more than 5000 terms of the Markowitz method, this is a big
improvement. |
| However,
although the Sharpe method requires less work, it is not as accurate as the |
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| Markowitz
method. |
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| Bond Examples
- Background |
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| Bonds
are generally considered less risky than stocks. If a bond is of high credit
quality, its |
| price
changes will depend almost entirely on interest rate changes. If interest
rates go up, the |
| price
of a bond goes down, while if interest rates go down, the price of a bond
goes up. |
| However,
most bonds pay interest in an annual (or semi-annual) coupon, and the
interest can |
| be
reinvested. If interest rates go up, the price of the bond does go down, but
the coupon could |
| be
reinvested at the higher interest rate. These two effects cancel each other
out at the end of |
| the
duration of a bond. The duration of a bond is the average time in which a
bond is repaid. |
| For
example, if a bond has a maturity of 3 years with face value $1,000 and an
annual coupon |
| payment
of $100 and a yield of 10%, the duration is 2.735 years. Thus, one way of
protecting |
| against
fluctuation of interest rates is to have a portfolio whose duration is equal to one's own |
| investment
time horizon. A portfolio of bonds has a duration that is the weighted
average of |
| the duration of the individual
bonds. |
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| In
model Bond1 we assume we know the duration of the bonds available and want to
maximize |
| the
yield of the portfolio while keeping the duration equal to a given investment
time horizon. |
| This
technique is known as bond portfolio 'immunization.' Bond2 is another immunization |
| model,
but this time the duration for each bond is calculated from the face value,
maturity, |
| annual coupon and
yield to maturity of each bond. |
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| In
model Bond3 we look at another common way to protect against interest rate
fluctuations. |
| When
we acquire a portfolio of bonds, we can exactly calculate the cash flow that
arrives from |
| this
portfolio. This consists of the coupons plus the face values of the bonds
that mature. |
| This
also works the other way around. If we know we need to have certain amounts
of cash |
| available
at certain periods, we can put together a portfolio that does exactly this.
The Solver |
| chooses that portfolio that covers the required amounts in
each year and costs the least. |
| Bond4
is another example of this technique called 'Exact Matching.' This time we
allow |
| excess funds in a certain period to be reinvested to meet
requirements in a future period. |
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