On October 23rd, 2014, we updated our
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Managing the Equity Portfolio
Structuring a Stock Portfolio
The Portfolio Objective
Active vs. Passive Management
What's Wrong with Buy and Hold?
The Costs of Revision
Constant Proportion Rebalancing
Constant Beta Rebalancing
Dollar Cost Averaging
Writing Options to Generate Income
Improving on the Market
Writing Covered Calls for Downside Protection
Using Index Options
Using Index Futures Contracts
Setting objectives is a critical, and sometimes difficult, part of the process of portfolio construction.
Problems in setting objectives include confusion over the meaning of terms and different attitudes
toward relative risk.
The four traditional portfolio objectives are stability of principal, income, growth of income, and
capital appreciation. Whatever the portfolio objective, the ultimate goal is the maximization of
expected utility. Utility can come from income, from capital appreciation, from the fun of playing the
game, and from other intangible sources.
Portfolio management strategies can be either passive or active. Buy and hold is the most obvious
of the passive strategies. A naive strategy is one that requires little analysis; this characteristic does not
imply that such a strategy is a poor one.
Rebalancing involves adjusting the relative proportion held of each portfolio component according
to some target proportion. This strategy may require “selling winners and buying losers.” Popular
methods of rebalancing include maintaining a constant proportional dollar investment in the portfolio
components, maintaining a constant beta, or matching the performance of some stock market index
Dollar cost averaging is a useful method of periodically adding to a long-term investment portfolio
without the need for extensive security analysis. It involves investing the same dollar amount in the
same security at regular intervals.
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The most common use of stock options by both individuals and institutions is writing covered calls.
Writing deep-in-the-money calls can be an effective way to sell stock at a slightly higher than current
price, while writing in-the-money puts facilitates the acquisition of stock at a lower than market price.
Both techniques are called improving on the market.
Portfolio protection involves adding components to a portfolio with the intent of ensuring that the
value of the portfolio will not fall below a predetermined “floor” value. This protection is commonly
sought via index options or stock index futures contracts.
Sophisticated hedging with options requires some familiarity with principles of option pricing.
Delta from the Black-Scholes option pricing model is useful in this regard. This statistic measures how
an option premium changes as the stock price changes.
A protective put is a long put position held in conjunction with a long position in the underlying
stock. It is similar to an insurance policy on stock. The investor selects the deductible and the policy
term, and pays a premium for the insurance.
Stock index futures can be used to reduce systematic risk associated with a well-diversified stock
portfolio. When hedging this way, it is necessary to know the portfolio value, its beta, and the value of
the chosen futures contract.
ANSWERS TO END OF CHAPTER QUESTIONS AND PROBLEMS
1. This is true to some extent. There are several reasons why a portfolio might periodically require
revision. One such reason is that a portfolio should be constructed with a particular investment
objective in mind. Sometimes this objective changes, requiring a change in the portfolio’s asset
allocation. Also, if a security substantially rises in value, it might constitute such a large
proportion of the portfolio that diversification suffers. Another reason why revision is
occasionally required is that over time, cash balances will build up as dividends and interest are
credited to the account. These amounts should eventually be invested in a manner consistent with
the portfolio objective.
You can argue that there is much to be said for a buy and hold strategy that requires little periodic
attention. This is true, but even in this case cash will accumulate and require reinvestment.
2. For the reasons cited in the answer to question 1, it is occasionally necessary to tinker with all
portfolios, even those managed with a buy and hold philosophy.
3. There are trading fees associated with buying and selling, plus there is a requirement for
managerial time in making the decisions as to what to buy or sell. The most significant trading
fees are often taxes, because a realized capital gain is taxable that year.
4. A correlation coefficient of 0.96 shows a strong relationship. Adding additional securities would
probably increase this value even further and reduce the tracking error. An important question is
whether the added benefit of doing so is worth the additional cost in trading fees.
5. There is no clear-cut portfolio turnover rate that is prima facie evidence of churning. We can say
that every change in a portfolio’s components should have a well-conceived rationale behind it.
Someone should be able to answer why a security was sold or bought. Evidence of churning often
begins with unusual portfolio activity coupled with sketchy reasons for the activity.
Chapter 18: Managing the Equity Portfolio 63
6. This is possible, but it would be very expensive to do so. Such a portfolio would require revision
much more often than a less-constrained portfolio. In practice, it would be very unusual to
manage a portfolio this way.
7. This is because over time dividends accumulate in the portfolio, and cash has a beta of zero.
Shares Total Shares Portfolio
Date Share Price Purchased with Owned Value
Jan $10.00 10.000 10.000 $100.00
Feb $9.00 11.111 21.111 $190.00
Mar $10.00 10.000 31.111 $311.11
Apr $11.00 9.091 40.202 $402.02
May $10.00 10.000 50.202 $502.02
This investor put $100 into a mutual fund five times, for a total investment of $500.00. The
investment began and ended the period at the same price. The terminal value of $502.02 is greater
than the amount invested.
(Note that dollar cost averaging will not produce a profit if the security rises and then falls to the
original level; the fluctuation must be both up and down.)
9. While there are many possible solutions, most will involve selling some of the NMB and perhaps
some of the PPM to buy additional shares of NRW.
10. a. The portfolio beta is about 1.08, as shown in the worksheet below.
Stock Beta Value Contribution to
JUI 1.07 $11,750 0.1071
LLO 0.92 9,500 0.0745
KI 1.10 12,000 0.1125
NMB 1.22 17,875 0.1858
ERW 1.10 8,875 0.0832
OP 0.88 10,625 0.0797
XXC 1.00 12,455 0.1061
PPM 1.03 13,800 0.1211
PPU 1.22 9,500 0.0987
WQE 1.14 11,000 0.1068
Total $117,380 1.0755
b. You could sell some of a low-beta stock (like OP) and buy shares in a higher beta stock
(like PPU). (Note that buying PPU is probably preferable to buying more NMB. Although NMB
and PPU have the same beta, NMB already constitutes a substantial portion of the portfolio;
adding to the PPU position would give better diversification.)
64 Chapter 18: Managing the Equity Portfolio
11. The beta of a portfolio is a weighted average of the component betas, so we can solve for the
β + ( 0) = 110
( 250000 + 10000) ( 250000 + 10000)
Solving, the equity beta is 1.14. To reduce the portfolio beta to 0.95, solve for the proportion x
to be invested in equities as follows.
1.14 x = 0.95
x = 0.83
With 83% in equities, 17% will be in cash.
12. The increase in the S&P index is a gain of 5.9%. This rise, coupled with the security’s beta,
indicates a likely new value for each security. (Note that you should not multiply beta by 1.059;
rather, the expression for the new value is [(.059(beta))+1][old value].
Stock Beta Initial Value Value after 5.9%
JUI 1.07 $11,750 $12,492
LLO 0.92 9,500 $10,016
KI 1.10 12,000 $12,779
NMB 1.22 17,875 $19,162
ERW 1.10 8,875 $9,451
OP 0.88 10,625 $11,177
XXC 1.00 12,455 $13,190
PPM 1.03 13,800 $14,639
PPU 1.22 9,500 $10,184
WQE 1.14 11,000 $11,740
Total $117,380 $124,830.00
Date Fund A Fund A Return Fund B Fund B Return
JAN $12.00 11.53
FEB 11.47 -0.0442 11.22 -0.0269
MAR 12.21 0.0928 11.98 0.0677
APR 12.33 0.0098 12.05 0.0058
MAY 12.44 0.0089 12.61 0.0465
JUN 13.17 0.0587 12.93 0.0254
JUL 12.76 -0.0311 13.08 0.0116
Variance 0.0027 0.0011
b. In Fund A, you accumulate 141.961 shares worth $12.76 each, for a total value of $1,811.42.
In Fund B, you accumulate 143.854 shares worth $13.08, for a total value of $1,881.61. The
table below shows the details.
Chapter 18: Managing the Equity Portfolio 65
Date Fund A Shares Shares Held Fund B Shares Shares
Purchased Purchased Held
JAN $12.00 20.833 20.833 $11.53 21.683 21.683
FEB 11.47 21.796 42.629 11.22 22.282 43.965
MAR 12.21 20.475 63.104 11.98 20.868 64.833
APR 12.33 20.276 83.380 12.05 20.747 85.580
MAY 12.44 20.096 103.476 12.61 19.826 105.406
JUN 13.17 18.983 122.369 12.93 19.335 124.741
JUL 12.76 19.592 141.961 13.08 19.113 143.854
14. Because an equal dollar amount is going into each fund, the size of the investment does not
Date Fund A Return Fund B Return Average
Jan $12.00 $11.53
Feb 11.47 -0.0442 11.22 -0.0269 -0.0356
Mar 12.21 0.0645 11.98 0.0677 0.0661
Apr 12.33 0.0098 12.05 0.0058 0.0078
May 12.44 0.0089 12.61 0.0465 0.0277
Jun 13.17 0.0587 12.93 0.0254 0.0421
Jul 12.76 -0.0311 13.08 0.0116 -0.0098
The variance of the portfolio returns (the far right column) is 0.0013 (using n-1 weighting).
15. Writing covered calls makes sense if the portfolio manager anticipates a relatively flat or slow-
rising market in the near term, or if the manager believes that option premiums are unusually high
and represent an investment opportunity in their own right.
16. Writing index options eliminates the potential for portfolio disruption because of option exercise.
It also avoids any adverse tax consequences from the sale of security positions containing large
unrealized capital gains.
17. This could makes sense, although it is important to recognize that writing the index calls caps the
upside appreciation potential for the portfolio. Investors often call this package of options and
stock a fence, collar, or hedge wrapper.
18. Many people believe this. From a purely economic perspective, this is almost certainly true. The
principal disadvantage of writing index calls is the fact that many supervisory bodies and clients
do not understand these options. It may be difficult to ensure that the customer is familiar with
this investment activity before embarking upon it.
19. By writing a deep in the money covered call, it is possible to capture the time value associated
with this option. At the same time, the likelihood is small that the option will not be exercised.
This means the option writer effectively sells the stock at a “better” price than that which
prevailed at the time the option was written. There is the risk, however, that the stock might fall
substantially, in which case the outright sale of the stock would have been preferable.
20. If the plan is to acquire stock, the investor would want to write a put with a good chance of being
exercised. Consequently, you would write an in-the-money put.
66 Chapter 18: Managing the Equity Portfolio
21. 6 1/8
73 7/8 80
22. a. You paid $27 per share and effectively sold at $33 per share. The holding period return is
$6/$27 = 22.2%
b. Solve the following equation for R:
27000 = +
(1 + R 365) 61 (1 + R 365) 79
R is approximately 94.85%.
23. The formula is Portfolio value × Beta = Option contract value × delta × # contracts
# contracts = ($2.5 million × 1.12)/(57500 × .223) = 218
24. The formula is:
# contracts = beta × (dollar value of portfolio)/(dollar value of futures contract)
= 1.12 × $2.5 million/(1439.40 × $250) = 7.78 8 contracts (to sell)
15% of index value .15 x 540 x 100 x N 8100.00N
market value of options 1.125 x 100 x N + 112.50N
out of the money amount (540.00 - 550.00) x 100 x N - 1000N
$10 million = $7212.50N
N = 1386 contracts
26. The index options you wrote are in-the-money. Your account will be charged 50 × 100 × (560.00
− 550.00) = $50,000. However, the portfolio will increase in value. The gain in the index was
(560 − 540)/540 = 3.7%. Assuming a portfolio beta of 1.0, the portfolio should gain 3.7% x $10
Chapter 18: Managing the Equity Portfolio 67
million = $370,000. You also get to keep the premium from writing the options. This amount is
50 × 100 × $1.125 = $5625. The net gain is $5625 + $370000 − $50000 = $325,625.