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# Asset Allocation
# SAA and TAA Part II
SAA, TAA, Asset Classes, Main Constraints
> Fernando Forcada, CFA
> Selected Topics, February 2025
## 1Asset Classes Definition and
# Classification
Defining correctly the different Asset Classes is an important
step within the SAA process : investors could neglect an opportunity to
diversify if they consider incorrectly either that two investments are
similar or, alternatively that two investments are different
The first approach (1) to defining an asset class is through their
investment attributes . Asset classes are defined as a group of assets
with common characteristics that include :
Sensitivity to the same major economic and/or investment factors
Risk and return characteristics that are similar
A common legal or regulatory structure
This approach is useful in defining asset classes in general terms .
However, some ambiguity remains in terms of the degree to which assets
are influenced by specific economic factors and in the extent to which risk
and return characteristics are similar
## 2(1) F. J. Fabozzi and H. M. Markowitz, The Theory and Practice of Investment Management Asset Classes Definition and
# Classification
A second approach (2) looks beyond specific investment attributes and
considers any group of assets as an asset class, as long as it meets the
following four criteria :
1. An asset class should be relatively independent of other asset classes in the
investors portfolio (mutually exclusive)
2. An asset class should be expected to raise the utility of the investors portfolio
without selection skill on part of the investor
3. An asset class should be comprised of homogeneous investments
4. An asset class should have the capitalization capacity to absorb a meaningful
fraction of the investors portfolio
This approach provides a rigorous framework for defining and evaluating an
asset class in the context of an investors existing portfolio . However, it may
be useful to have a simplified approach for assessing the benefits of an asset
class
The portfolios Sharpe ratio can be used to evaluate whether an asset
class should be considered for inclusion in a portfolio : if the Sharpe ratio of
the new asset class is greater than the Sharpe ratio of the portfolio
multiplied by the correlation between them , then the new asset class
provides an expected benefit to the portfolio
## 3(2) M. Kritzman , Towards Defining an Asset Class Asset Classes Definition and
# Classification
Traditional asset classes include the following :
(Public) Equity :
Domestic common equity vs . International common equity, Developed Market (DM : USA,
Japan, UK, EU ) vs . Emerging Market (EM : Latam , APAC, Eastern Europe )
Large -cap, mid -cap, and small -cap
Fixed Income :
Domestic fixed income vs . International fixed income, Developed Market (DM) vs . Emerging
Market (EM)
Government (Sovereign Core/Non -Core , Semi -Sovereign, Supranational, Govt -
Guaranteed ) vs . Credit (Financials, Corporate, Secured -Unsecured, Senior -Subordinated),
Investment grade vs . High Yield, Private Placements)
Inflation -protected bonds vs . nominal bonds
Short -term - Intermediate -term - Long -term bonds
Cash and cash equivalents or Money Market :
Current account, bank deposits, inverse -repo, commercial paper shorter than x days
Alternative Investments :
Real Estate, Private Equity, Hedge Funds, Commodities but also (commercial/residential)
Mortgages, RE financial leases, Syndicated loans Each of them deserves a specific asset
class . Alternative Investments is not a homogeneous asset class by itself
## 4Example of Asset Allocation
# SAA&TAA
## 5
SAA
Target Lower
Band
Upper
Band
Equity 20% 0% 30%
Domestic 10% 0% 15%
International 10% 0% 15%
Fixed Income 65% 45% 90%
Government 25% 15% 35%
Sovereign 20% 15% 25%
Govt-Guaranteed 5% 0% 10%
Credit 40% 30% 55%
Financials 15% 10% 20%
Corporate 25% 20% 35%
Real Estate 10% 6% 12%
Cash/Money Market 5% 2,5% 15%
TOTAL 100%
TAA Boundaries The Economic Balance Sheet and
# Asset Allocation
An accounting balance sheet reflects a point -in -time snapshot of
an organizations financial condition under certain accounting
principles, showing the assets, liabilities, and owners equity recognized
by accountants
An economic balance sheet includes conventional assets and
liabilities (called financial assets and financial liabilities) as well as
additional assets and liabilities (known as extended portfolio assets
and liabilities) that are relevant in making asset allocation
decisions but do not appear on conventional balance sheets
For example, in the case of individual investors, extended portfolio assets
include human capital (the present value of future earnings), the present value
of pension income, and the present value of expected inheritances . Likewise, the
present value of future consumption is an extended portfolio liability
## 6Asset Allocation Main
# Constraints
In practice, investors must consider a number of constraints when
modeling and choosing among asset allocation alternatives :
Asset size : the size of the assets portfolio may limit the investment
opportunity set (it may be too small or too large to capture the returns of
certain asset classes or strategies efficiently)
Whereas large asset portfolios may achieve operating efficiencies
(economies of scale), scale may also impose obstacles related to the liquidity
(price impact given large trade sizes) and trading costs of the underlying asset
Investors with larger portfolios can generally consider a broader set of asset
classes and strategies . They are also more likely to have sufficient
governance capacity (sophistication and staff resources) to manage certain
complex investments
## 7Asset Allocation Main
# Constraints
Time horizon is a critical constraint that must be considered in any
asset allocation exercise . A liability to be paid at a given point in the
future or a goal to be funded by a specified date each define the asset
owners horizon
In practice, investors often align lower risk/lower return assets with
short -term goals and liabilities and higher risk/higher return assets
with long -term goals and liabilities (time diversification) . It is generally
believed that longer -horizon goals can tolerate the higher volatility associated
with higher risk/higher return assets (as below average and above average
returns become equal over time) . An assumption of mean -reverting risky
asset returns would support these conventional arguments
## 8Asset Allocation Main
# Constraints
Liquidity : Two dimensions of liquidity must be considered : the liquidity needs of
the investor and the liquidity characteristics of the different asset classes .
Integrating the two dimensions is an essential element of successful investment
planning . Long -term investors can generally exploit illiquidity premiums available
in some asset classes, such as private equity, real estate, and infrastructure
investments
Regulatory and other external constraints : Local regulations can have a material
effect on an investors asset allocation . For example, some regulators do not allow
certain investments, such as commodities and hedge funds, to cover reserves . In
addition, regulators often discourage holding certain asset classes by imposing
significant statutory capital requirements on them .
Also, multinational companies may have a global SAA target that is implemented
locally , which adds significant complexity and governance issues (given its strategic
importance, SAA might require the approval of the local Board of Directors,
Investment Committee) . Other constraints are :
Tax rules
Accounting treatment
ESG considerations
Cultural or religious factors 9Exercise Liquidity and Asset
# Allocation
The timing of payouts for property and casualty insurers is unpredictable
in comparison with the timing of payouts for life insurance companies .
Therefore, in general, property and casualty insurers have :
A. lower liquidity needs than life insurance companies .
B. greater liquidity needs than life insurance companies .
C. a higher return objective than life insurance companies .
Solu tion :
B is correct . The unpredictable nature of property and casualty (P&C)
claims forces P&C insurers to allocate a substantial proportion of their
investments into liquid, short maturity assets . This need for liquidity also
forces P&C companies to accept investments with relatively low expected
returns . Liquidity is of less concern to life insurance companies given the
greater predictability of life insurance payouts
## 10 Exercise Liquidity and Asset
# Allocation
The timing of payouts for property and casualty insurers is unpredictable
in comparison with the timing of payouts for life insurance companies .
Therefore, in general, property and casualty insurers have :
A. lower liquidity needs than life insurance companies .
B. greater liquidity needs than life insurance companies .
C. a higher return objective than life insurance companies .
Solu tion :
B is correct . The unpredictable nature of property and casualty (P&C)
claims forces P&C insurers to allocate a substantial proportion of their
investments into liquid, short maturity assets . This need for liquidity also
forces P&C companies to accept investments with relatively low expected
returns . Liquidity is of less concern to life insurance companies given the
greater predictability of life insurance payouts
## 11 Exercise External Constraints
# and Asset Allocation
A multinational corporation headquartered in Mexico has acquired a
former competitor in the United States . It will maintain both the US
pension plan with US $250 million in assets and the Mexican pension
plan with MXN $18 ,600 million in assets ( US $1 billion) . Both plans are
95 % funded and have similar liability profiles . The Mexican pension trust
has an asset allocation policy of 30 % equities (10 % invested in the
Mexican equity market and 20 % in equity markets outside Mexico), 10 %
hedge funds, 10 % private equity, and 50 % bonds . The treasurer has
proposed that the company adopt a consistent asset allocation policy
across all of the companys pension plans worldwide
Critique the treasurers proposal
## 12 Exercise External Constraints
# and Asset Allocation
Solution :
The treasurers proposal fails to consider the relative asset size of the two
pension plans as well as the likelihood that plans in different jurisdictions
may be subject to different funding, regulatory, and financial reporting
requirements
The US pension plan may be unable to effectively access certain alternative
asset classes, such as private equity, infrastructure, and hedge funds . Although
economies of scale may be realized if management of the pension assets is
consolidated under one team, the legal and regulatory differences of the
markets in which they operate mean that the asset allocation policy must
be customized to each plan
## 13 Asset Allocation and Rebalancing
>
Rebalancing policy : The set of rules that guide the process of restoring
the portfolios original exposures to systematic risk factors
>
Rebalancing is the discipline of adjusting portfolio weights to more
closely align with the strategic asset allocation . Rebalancing is a key
part of the monitoring and feedback step of the portfolio construction,
monitoring, and revision process
>
Normal changes in asset prices cause the portfolio asset mix to deviate from
target weights . Ordinary price changes cause the assets with a high
forecast return to grow faster than the portfolio as a whole . Because
high -return assets are typically also higher risk, in the absence of
rebalancing, overall portfolio risk rises
>
Taken to the extreme, never rebalancing allows the high -return (and
presumably higher -risk) assets to grow and dominate the portfolio :
portfolio risk rises and concentrates . Example : a Buy -and -Hold strategy
(passive approach) where no rebalancing is required
## 14 Asset Allocation and Rebalancing
Examples of rebalancing approaches (see also Excel file) :
Constant Mix strategies maintain an exposure to a risky asset (i .e. stocks) that
is a constant mix of the portfolio value (i .e. 60 /40 ). Tolerance for risk varies
proportionately with investors wealth .
Rebalancing to a constant mix requires selling stocks as they rise and buying
stocks as they fall . Because this rebalancing is countercyclical , it is fundamentally
a contrarian investment approach . Behavioral finance tells us that such
contrarianism will be uncomfortable ; no one likes to sell the most recently best -
performing part of the portfolio to buy the worst
Constant Proportion strategies (i .e. CPPI ) take the following form : Risky
Asset (stocks) Value = m x (Assets Floor) , where m is a fixed multiplier
greater than 1, and floor is the level below which we do not want the portfolio
value to fall
If we think of the difference between assets and the floor as a cushion, then the decision
rule is simply to keep the exposure to risky assets (stocks) a constant multiple of the
cushion . A CPPI strategy sells stocks as they fall and buys stocks as they rise
## 15 Asset Allocation Different
# Approaches
We can identify three broad approaches to SAA : (1) asset -only , (2)
liability -relative , and (3) goals -based :
Asset -only Approach focuses solely on the asset side of the investors balance
sheet . The goal is to create the most efficient mix of asset classes in the absence
of any liabilities (using MVO and/or other optimization techniques)
Liability -relative Approach : ALM -driven investors also take an
asset/liability management approach to SAA, which involves adopting the
optimal asset allocation in relationship to funding liabilities . This
approach typically results in a higher allocation to fixed -income instruments
than an Asset -only approach (fixed -income instruments have prespecified
interest and principal payments that are well suited to offsetting future
obligations . Moreover, ALM -driven investors need to get exposure to assets with
interest rate (IR) sensitivity, given that liabilities are also sensitive to IR .
Equity, Real Estate do not have a clear IR sensitivity)
Goals -based Approach (individual investors) : Conceptually, goals -based
approaches are similar to liability -relative asset allocation in viewing risk in
relation to specific needs or objectives associated with different time horizons
and degrees of urgency . Individuals often have multiple goals, therefore, a
goals -based asset allocation process disaggregates the investors portfolio into a
number of sub -portfolios 16 Asset Allocation Optimization
Mean -Variance Optimization (MVO) or Efficient Frontier , developed by
Markowitz, is perhaps the most common approach used in practice to develop
and set asset allocation policy . MVO is also often the basis for more
sophisticated approaches that overcome some of the limitations or
weaknesses of MVO
MVO provides us with a framework for determining how much to allocate to
each asset in order to maximize the expected return of the portfolio for
an expected level of risk . It considers only the expected returns, risks, and
correlations of the asset classes in the opportunity set
Typically, MVO is constrained with these two conditions :
The weights must sum to 1 (100 %)
Only positive weights or allocations (i .e., no negative or short positions)
Principles of MVO :
Correlation : whenever the returns of two assets are not perfectly correlated, the
assets can be combined to form a portfolio whose risk (as measured by standard
deviation or variance) is less than the weighted -average risk of the assets
themselves
Portfolio risk : as one adds assets to the portfolio, one should focus not on the
individual risk characteristics of the additional assets but rather on those assets
effect on the risk characteristics of the entire portfolio
## 17 Example of MVO Utility
# Function (solution in Excel file)
An individual investor with an above -average risk tolerance ( = 2) has an initial cash amount
to invest of Eur 1.2 million and expects to liquidate Eur 60 k in 12 months without affecting the
initial capital
The investor can choose between 3 different SAA choices (see attached Excel file)
Note : Mean variance optimization requires three sets of inputs : returns, risks (standard
deviations), and pair -wise correlations for the assets in the opportunity set . The objective
function is often expressed as follows :
Where
Um = the investors utility for asset mix (allocation) m
Rm = the return for asset mix m
= the investors risk aversion coefficient
2m = the expected variance of return for asset mix m
In words, the objective function says that the value of an asset mix for an investor is equal to
the expected return of the asset mix minus a penalty that is equal to one -half of the expected
variance of the asset mix scaled by the investors risk aversion coefficient (which, unfortunately,
it is rather difficult to estimate . To do so, one should examine both the investors preference for
risk - willingness to take risk - and the investors capacity for taking risk)
## 18 Example of MVO Corner Portfolio
# Theorem (solution in Excel file)
Corner portfolios allow us to create other efficient portfolios . Knowing the
composition of the corner portfolios allows us to compute the weights of any
portfolio on the minimum -variance frontier (1): a positive linear
combination of the corresponding weights in the two adjacent
corner portfolios that bracket it in terms of expected return (or Std Dev of
return)
Example : A US institutional investor is developing a SAA among 6 asset
classes : US equities, international (ex -US) equities, domestic intermediate -
term and long -term bonds, international bonds, and real estate
The investor cannot sell securities short or buy on margin
Risk -free rate : 2.00 %
Capital market expectations (see attached Excel file)
Note : Efficient portfolios make efficient use of risk ; they offer the maximum
expected return for their level of variance or standard deviation of return
## 19 (1) Each portfolio on the minimum -variance frontier (MVF) represents the portfolio with the smallest variance of return for its level of
> expected return. The efficient portion of the MVF is called the efficient frontier (which represents efficient portfolios).
# Asset Allocation Optimization
Criticisms of MVO :
The outputs (asset allocations) are highly sensitive to small changes in the
inputs (particularly, expected returns), which emphasizes the importance of
the inputs quality
The asset allocations tend to be highly concentrated in a subset of the
available asset classes (extreme solutions)
Many investors are concerned about more than the mean and variance of
returns , the focus of MVO
Although the asset allocations may appear diversified across assets, the
sources of risk may not be diversified
Most portfolios exist to pay for a liability and MVO allocations are not directly
connected to what influences the value of the liability
MVO is a single -period framework that does not take account of
trading/rebalancing costs and taxes
## 20 Asset Allocation Optimization
Risk factor optimization is an alternative approach that uses investment
factors rather than traditional asset classes to optimize the Asset Allocation
The objective is to find the optimal allocation to the selected beta
risk factors and then determine which asset classes allow us to achieve this
risk factor allocation
It tries to overcome the issue with portfolios sensitivity to overlapping risk factors
(inflation risk, interest -rate risk )
However, it is more difficult to carry out : we must estimate factor sensitivities (great
estimation error )
The risk factors in question are typically similar to the fundamental (or
structural) factors in widely used multi -factor investment models :
Equity risk (global equity markets affect most asset classes)
Interest rate risk (interest rate movements have an effect on the value of most
asset classes and also on the liabilities)
Credit spread risk (credit spread movements also affect most asset classes)
Commodity risk (commodity markets had a low correlation with other risky assets
prior to the 2008 crisis)
Duration (term structure can act as a proxy for inflation risk)
Others : size (large -cap vs . small -cap stocks), valuation (Growth vs . Value),
momentum, liquidity, volatility
## 21 Asset Allocation Liability -
# relative Approach
There are various approaches to liability -relative asset allocation, that is,
optimally allocating assets (across different asset classes) to maximize
return relative to liabilities for the desired level of risk . In other words,
we need to determine how to optimally invest assets in order to pay the
liabilities and to provide a return on Economic Capital :
Surplus optimization is an economic balance sheet approach that finds the most
efficient asset class mix in the presence of liabilities . It involves applying MVO to
an efficient frontier based on the volatility of the surplus . This Liability -
relative optimization is simultaneously concerned with the return of the assets, the
change in value of the liabilities, and how assets and liabilities interact to determine
the overall value of the total portfolio . Surplus optimization exploits natural
hedges that may exist between assets and liabilities as a result of their
systematic risk characteristics
Hedging/return -seeking portfolios approach involves separating assets into
two groups : a hedging portfolio (i .e. Policyholders Portfolio in the case of an
insurance company) and a return -seeking portfolio (i .e. Shareholders Portfolio)
applying different SAAs . It assumes that there is a positive surplus available to
allocate to the return -seeking portfolio . Asset segmentation involves segmenting
general account assets into sub -portfolios associated with specific lines of business or
blocks of liabilities
## 22 Executive Summary
SAA Definition and main characteristics
TAA Definition and main characteristics
Approaches to Asset Allocation (Asset -only / Liability -relative)
Optimization MVO uses and limitations
How to define Asset Classes
Main constraints to SAA
Asset Allocation and Rebalancing
## 23 References (I)
>
Introduction to Asset Allocation . William W. Jennings, CFA, and
Eugene L. Podkaminer , CFA . CFA Institute, 2017
>
Principles of Asset Allocation . Jean L.P. Brunel, CFA, Thomas
M. Idzorek , CFA, and John M. Mulvey, PhD . CFA Institute, 2017
>
Asset Allocation with Real -World Constraints . Peter Mladina ,
Brian J. Murphy, CFA, and Mark Ruloff , FSA, EA, CERA . CFA
Institute, 2017
>
Asset Allocation . William F. Sharpe, Peng Chen, PhD, CFA, Jerald
E. Pinto, PhD, CFA, and Dennis W. McLeavey , CFA . CFA Institute,
2013
>
A Primer on Tactical Asset Allocation Strategy Evolution .
Yesim Tokat , PhD, and Kimberly A. Stockton . Vanguard Investment
Counseling & Research
## 24 References (and II)
>
A Case for Dynamic Asset Allocation for Long Term
Investors . Gabriel Petre (World Bank) . Procedia Economics and
Finance 29 , 2015
>
The Importance of Asset Allocation . Roger G. Ibbotson . Financial
Analysts Journal, Volume 66 , Number 2, 2010
>
Multi -Asset Strategies . The Future of Investment
Management . Larry Cao, CFA . CFA Institute, 2018
>
Dynamic Strategies for Asset Allocation . Andr F. Perold,
Willian F. Sharpe, Financial Analyst Journal, Jan -Feb 1995
>
www .investopedia .com
>
https ://www .slideshare .net/larryshaw/benefits -of -asset -allocation -
rebalancing
## 25 Annex: Asset Allocation
# Optimization
Different ways to address the criticisms of MVO :
Reverse Optimization can be used to estimate expected returns for use in a
forward -looking optimization . Reverse optimization works in the opposite direction
than MVO : it takes as its inputs a set of asset allocation weights that are assumed
to be optimal and, with the additional inputs of covariances and the risk aversion
coefficient, solves for expected returns
The Black Litterman Approach is a model (created by Fischer Black and Robert
Litterman ) that enables investors to combine their unique forecasts of expected
returns with reverse -optimized returns in an elegant manner . The model starts with
excess returns (in excess of the risk -free rate) produced from reverse optimization
and then provides a technique for altering reverse -optimized expected returns in
such a way that they reflect an investors own views . When coupled with MVO, the
resulting Black Litterman expected returns often lead to well -diversified asset
allocations
Monte Carlo Simulations complement MVO by addressing its limitations as a
single -period framework and providing practical guidance to deal with taxes and
rebalancing . Additionally, in the case in which the investors risk tolerance is either
unknown or needs further validation, Monte Carlo simulation can help paint a
realistic picture of potential future outcomes, including the likelihood of meeting
various goals, the distribution of the portfolios expected value through time, and
potential maximum drawdowns . It is a large -scale sensitivity analysis in which
hundreds or perhaps thousands of variations on baseline capital market
assumptions lead to an equal number of mean variance optimization frontiers
## 26 Annex: Taxes and Asset Allocation
Portfolio theory was developed in a frictionless world . But in the real world, taxes on
income and capital gains can erode the returns achieved by taxable investors . The asset
owner who ignores taxes during the asset allocation process is overlooking an economic
variable that can materially alter the outcome
Some assets are less tax efficient than others because of the character of their
returns the contribution of interest, dividends, and realized or unrealized capital
gains to the total return :
Interest income is taxed typically (but not always) at progressively higher income tax rates
Dividend income and capital gains are taxed typically (but not always) at lower tax rates
than those applied to interest income and earned income (wages and salaries, for example)
Capital losses can be used to offset capital gains (and sometimes income), and long -term
capital gains receive the most favorable tax treatment in many jurisdictions
Note : Different countries may have different tax rules and rates . However, looking across the
major economies, there are some high -level commonalities in how investment returns are taxed
Jurisdictional rules can also affect how the returns of certain assets are taxed : In the
US, for example, the interest income from state and local government bonds is
generally exempt from federal income taxation, while municipal bonds are commonly
tax -free at the federal level but can be taxable at state or local income tax levels
Because after -tax returns volatility is less than pre -tax returns volatility and
asset class correlations remain the same, it takes larger asset class movements to
materially alter the risk profile of the taxable portfolio 27 Annex: Exercise Asset Allocation
# and the Taxable Investor
Sarah Moreau, 45 years old, is a mid -level manager at a consumer products company .
Her investment portfolio consists entirely of tax -deferred retirement savings accounts .
Through careful savings and investments, she is on track to accumulate sufficient
assets to retire at age 60 . Her portfolio is currently allocated as indicated below :
The common stock dividend income strategy focuses on income -oriented, high -
dividend -paying stocks
The common stock total return strategy focuses on stocks that represent good, long -
term opportunities but pay little to no dividend
Note : we will assume that the expected long -term return is equivalent between both
common stock (equity) strategies
Moreau has a high comfort level with this portfolio and the overall level of risk it
entails 28 Annex: Exercise Asset Allocation
# and the Taxable Investor
Moreau has recently inherited additional monies, doubling her investable
assets . She intends to use this new, taxable portfolio to support causes
important to her personally over her lifetime . There is no change in her risk
tolerance . She is interviewing prospective investment managers and has
asked each to recommend an asset allocation strategy for the new portfolio
using the same set of asset classes . She has received the following
recommendations :
Which asset allocation is most appropriate for the new portfolio?
Justify your response 29 Annex: Exercise Asset Allocation
# and the Taxable Investor
Solution :
Recommendation C would be the most appropriate asset allocation for the new
portfolio . The high -yield bond and common stock dividend income strategies are
tax disadvantaged in a taxable portfolio . (Although investment -grade bonds are
also tax disadvantaged, they maintain the role of controlling portfolio risk to
maintain Moreaus risk preference)
Given the lower standard deviation characteristics of after -tax equity returns
when held in the taxable portfolio, a higher allocation to common stocks may be
justified without exceeding Moreaus desired risk level
Recommendations A and B do not consider the negative tax
implications of holding the high -yield and/or common stock dividend
income strategies in a taxable portfolio . Recommendation B also fails to
consider Moreaus overall risk tolerance : The volatility of the common stock
capital gain strategy is lower when held in a taxable portfolio, thus a higher
allocation to this strategy can enhance returns while remaining within Moreaus
overall risk tolerance
## 30 Annex Mitigating Behavioral
# Biases in Asset Allocation
Ivy Lee, the retired founder of a publicly traded company, has two primary
goals for her investment assets :
The first goal is to fund lifetime consumption expenditures of US $1 million
per year for herself and her husband ; this is a goal the Lees want to achieve with a
high degree of certainty
The second goal is to provide an end -of -life gift to Auldberg University
Ivy has a diversified portfolio of stocks and bonds totaling US $5 million and a
sizable position in the stock of the company she founded . The following table
summarizes the facts :
## 31 Annex Practice Problems
## 32 Annex Practice Problems
## 33 Annex Practice Problems
## 34 Annex Practice Problems
## 35 Annex Practice Problems
## 36 Annex Solutions
## 37 Annex Solutions
## 38 Annex Solutions
## 39