Modern portfolio optimization a practical approach using excel solver by JEFF GROVER ANGELINE

IModern Portfolio Optimization:A Practical Approach Using an Excel Solver Single-Index Model JEFF GROVER AND ANGELINE M.. Treasury Bill annual percentage returns, and a market index retu

IModern Portfolio Optimization:

A Practical Approach Using an Excel Solver Single-Index Model

JEFF GROVER AND ANGELINE M LAVIN

JEFF GROVER

is an associate professor at

Indiana Wesleyan

Univer-sity in Marion, IN.

jeff.grover@indwes.edu

A N G E L I N E M L A V I N

is an associate professor at

Beacom School of Business,

University of South Dakota

in Verniillion, SD.

angeline.lavin@u$d.edu

Modern Portfolio Theory (MPT)

is based on the idea that com-binations of assets have the potential to provide better returns with less risk than individual assets

While MPT provides excellent insights into which assets should be included in an investor's optimal portfolio, understanding the under-lying statistical techniques in portfolio opti-mization presents a rigorous challenge This is especially true in determining the standard deviation of a portfolio using the Markowitz methodology The investor's knowledge of the theoretical basis of MPT presents a concep-tual constraint on his/her ability to understand the underlying constructs of MPT, which is required to determine the asset investment strategy that will optimize the portfoho

One solution to overcoming such investor constraints is to provide a rule-based hierar-chal system as a primer to establishing an equiv-alent linear form of the Markowitz [1959]

quadratic objective function which, when opti-mized, affords an optimal solution for an invest-ment portfolio Due to the complexities of calculating the variance-covariance matrix and the computer memory resources required to solve this quadratic objective function, Sharpe [1972] devised a single-index model that nicely replicates the Markowitz function The single-index model allows for the computation of betas, expected returns, and residual variances with a market index return and variance The

solu-tions to this rule-based system, in conjunction with Sharpe's methodology and the Sharpe Ratio, can be simply implemented using the Microsoft Excel Solver function This process provides an excellent example of how the Excel Solver can be used to solve a mathematical optimization problem, which is especially ben-eficial to investors as they face the challenge of determining how to efficiently optimize their mutual fund portfolios This article uses TIAA-CREF mutual fund data as an example to illus-trate the MPT concept as a practical approach

to optimizing an investment portfolio to obtain maximum return with minimal risk However, the model can be applied to any set of mutual fund options to create an optimal fund alloca-tion The process outlined in this article may also prove useful for defined contribution plan providers as they work to implement the pro-visions outlined in the Pension Protection Act passed by the U.S Congress in 2006 The Act calls for greater access to professional advice about retirement for investors and gives workers greater control over how to invest their retire-ment account funds The model presented in this article has the potential to provide useful information to professional advisors as well as individual investors

In 1952, Markowitz first proposed a method by which an investor could optimize the individual components of a portfolio to generate portfolio returns greater than the sum of the returns on the individual portfolio

60

components In 1959 he wrote a textbook on efficient

diversification of investments In a review of this

text-book, Tintner [1959] pointed out that Markowitz's key

concept is the efficient portfoHo "If a portfolio is efficient,

it is impossible to obtain a greater average return without

incurring greater standard deviation; it is impossible to

obtain smaller standard deviation without giving up return

on average" Markowitz [1959, p 22] This optimization

process is a rare economic phenomenon that exists only

when a portfolio is established in return-risk space so that

the optimal mix of the securities results in optimal

port-foho returns Use of the Markowitz methodology, which

is quadratic in nature, requires an understanding of

oper-ations research methodologies Sharpe [1972] introduced

a single-index model that overcame the difficulties

asso-ciated with implementing the quadratic algorithm The

latter methodology suggests a linear framework that

pro-vides a more simplistic method for the practical

imple-mentation of portfolio optimization using the Microsoft

Excel Solver This method can be used to demonstrate the

optimization process for mutual fund investors and enable

them to take control of their future wealth accumulation

Motivation

A search of current software available to perform

portfolio optimizations returns a limited number of cost

effective alternatives The method developed in this article

yields a user-fi^iendly mechanism with which investors can

perform the portfolio optimization process using Microsoft

Solver in its Excel application The intent is to have the

investor download mutual fund daily closing prices, the

13-week U.S Treasury Bill annual percentage returns, and

a market index return, and then have the Excel

applica-tion calculate the optimal portfolio combinaapplica-tion of these

funds The goal of this article is to provide a rigorous

approach to the theoretical application of this

optimiza-tion process using techniques proposed by Sharpe [1972]

Literature Review

The decision of how to allocate capital across

dif-ferent asset classes is a key issue that all investors face

Markowitz is widely considered to be the father of MPT

His early work was the first to frame portfolio choice as

a mean-variance (M-V) optimization problem that yields

an efficient frontier of potential investment choices for a

rational investor His critical insight was the realization

that the co-movement of assets with each other is more important than the individual security characteristics when forming portfolios of assets According to Elton and Gruber [1997], who provide an excellent overview of portfolio theory, the M-V framework has remained the cornerstone of MPT

The development of M-V portfolio theory created

a need to estimate model inputs such as correlation coef-ficients, and index models became the principal tool for estimating covariance Sharpe [1967] popularized the market model, a variant of the single-index model, for evaluating the risk and return characteristics of portfolios Two of the most basic questions in portfoho analysis are the number and specific types of securities to include

in the portfolio Mao [1970] was the first to use theoret-ical analysis, in the form of a heuristic switching algo-rithm, to solve this optimization problem However, Mao's procedure could not be used for negative or zero beta securities, it did not determine the optimal allocation among the various securities, and it would potentially require enormous amounts of computer time Sharpe [1972] proposed a slight modification to Mao's method-ology that would allow for negative betas and for the selection of securities as well as the allocation of funds among them Sharpe and Stone [1973] extended Sharpe's study to include non-market risk, which was a limitation

of Sharpe's single-index model, and captured the essence

of portfolio selection as a linear program

Elton, Gruber, and Padberg [1976] set out to oper-ationalize MPT, which although meant to be a practical tool, had primarily developed as a normative, theoretical construct They suggested that the implementation dif-ficulties lay in: 1) estimating the correlation matrices, 2) the time and costs in generating efficient quadratic portfolios, and 3) educating portfolio managers to relate

to risk-return tradeoffs expressed as covariances and stan-dard deviations They continued to expand the body of knowledge and proposed a constant correlation model Bawa, Elton, and Gruber [1979] extended their study and introduced simple procedures for multivariate portfoho optimization Chen [1983] challenged the Bawa et al [1979] study's limitation of standard error estimation tech-niques, but Alexander [1985] later refuted these claims and supported the Elton et al [1976] study Kwan [1984] further extended MPT by including the three existing portfolio optimization techniques—the single, multiple and correlation index models—in a single common algo-rithm Green and Burton [1992] offered a solution to

extreme weightings caused by a dominant single factor

in equity returns, thus reducing residual risks

Both the theoretical construct of portfolio

opti-mization and its practical apphcation have continued to

evolve since Markowitz first proposed the M-V

frame-work in the 1950s While ever more sophisticated methods

are continually developed, it is important to recognize that

simplicity may be the key to assisting individual investors

who are faced with asset allocation decisions in their

defined contribution retirement plans In fact, a recent

study by Iyengar, Huberman, and Jiang [2004] quoted on

the TIAA-CREF website offers evidence that the

com-plexity of the asset allocation decision often leads employees

to delay savings plan enrollment Rugh [2003] reports that

the majority of TIAA-CREF participants direct at least

half of their contributions to equity and make no changes

in their allocations over the ten to twelve year periods of

time studied He also notes that the average participant

allocation has become more diverse over the past ten years

Data

The TIAA-CREF defined contribution variable

annuity retirement plan flinds were selected for this analysis

because the company is a well-known provider of

low-cost mutual funds as well as a major provider of defined

contribution retirement plans In fact, Rugh [2003] reports

that TIAA-CREF is the largest pension provider in the

U.S., managing $300 billion in total assets for more than

two million individuals The plans, which are provided

through the employer of the respective investor, are

referred to as retirement (or group retirement) annuity

contracts Investment amounts are dependent on

con-tractual agreements between the employee and employer

Typically, contributions are made on a tax-deferred basis,

which means that pre-tax dollars fund the accounts, but

the employees pay taxes on withdrawals Information

about the funds can be obtained from the TIAA-CREF

website, http://tiaa-cref.org/product_profiles/ras.htnil

The purpose of this article is to demonstrate the concepts

of MPT in portfoHo optimization using this TIAA-CREF

fund data However, the model developed in this article

can be applied to optimize any mutual fund portfolio

Tax Implications of Portfolio Rebalancing

The specific example employed in this article assumes

that the fund portfolio being optimized is a tax-deferred

portfoHo of funds inside of a defined contribution retire-ment plan In the context of a tax-deferred or tax-exempt portfolio, there are no tax imphcations associated with portfoHo rebalancing, and periodic asset allocation changes can be made without creating taxable gains The model developed in this article can be used to optimize any mutual fund portfoHo, regardless of whether the portfoho

is held in a tax-deferred, tax-exempt or taxable account

It is important to note that rebalancing a taxable fund portfolio may create tax consequences, which can either

be positive or negative Despite the fact that rebalancing

a taxable fund portfolio has the potential to create taxable gains, periodic portfolio balancing is stiU recommended

In fact, studies have shown that periodic rebalancing can actually improve portfolio returns by forcing investors to sell assets that have appreciated in value and buy assets that have underperformed Most financial advisors rec-ommend that clients review their portfolios at least annu-ally and rebalance as needed upon review This is especiannu-ally true when one considers the effects of inflation on the portfolio Reilly and Brown [2006] suggest that inflation

is the major cause of reductions in investment value

Simple Techniques for Determining an Optimal Portfolio

The portfolio optimization challenge, as suggested

by Mao [1970], is to maximize an objective function such that an investor can create an optimal portfolio providing the maximum return for the minimum risk Mao sug-gested such an algorithm, but it was Hmited by the inability

to solve for a negative beta Although negative betas appear less frequently than positive ones, they do exist, and a mathematical resolution is required to efficiently opti-mize a given sample of securities The proper selection

of the market index is critical to minimizing the number

of securities with negative betas Sharpe [1972] resolved this issue using his single-index methodology The process developed in this article applies Sharpe's methodology to identify undervalued and overvalued securities using the capital asset pricing model (CAPM) as the screening tool for the portfolio optimization process This study wiU consider only undervalued securities as input variables for portfoho optimization After screening out overvalued securities, the Sharpe Ratio is operationalized as an objec-tive function and the portfolio returns and standard devi-ations are used to establish an optimal or tangent portfoHo following the Sharpe [1972] CAPM methodology We

6 2 MODERN PORTFOLIO OPTIMIZATION: A PKJVCTICAL APPROACH USING AN EXCEL SOLVER SINGLE-INDEX MODEL

compute this objective function by maximizing the Sharpe

Ratio This tangent or optimal portfolio, i.e., the

port-folio with minimum risk and maximum return, will be

a linear combination of the portfolios along the trace of

the efficient frontier of portfoHo combinations We defme

the Sharpe Ratio in Equation (1) as:

(1)

Equation (4) computes a portfolio standard devia-tion using the Sharpe method This funcdevia-tion minimizes the square of the sum of products of the security betas and their respective weights summed with the security residual variances

(4)

The Sharpe Ratio, which measures the excess return

per unit of total risk, is the return of the portfolio minus

the risk-free rate divided by the standard deviation of the

portfoho We calculate the tangent portfolio using the

Sharpe Ratio as the maximization objective function,

given in Equation (2), which takes the form:

e =

Maximize

(2)

The constraints for this function include the

following:

2 X >0, for i = 1, n.

The first constraint requires the sum of all weighted

coefficients to be equal to one and the second requires

these weights to be greater than or equal to zero because

short-selling of securities is not allowed

Equation (3) gives the portfolio return using the

Sharpe method (Elton, Gruber, Brown, and Goetzmann

[2003]) The portfoHo return is the sum of products of the

security alphas and their respective weights, plus the sum

of products of the security betas and their respective

weights with the product of the market index return

(3)

where the security weights are designated by X., the

portfolio alpha is computed as OIp = S^^j^,-«, the

port-folio beta is computed as Pp = Z^jX p., and the average

return on the market index is designated as R^.

where (J^^ is computed as O~ - p (7^, is the residual

vari-ance of the portfolio

Substituting in the classical Sharpe Ratio, as

pre-sented in Equation (2), for Rp and Op, we present our

modified Sharpe Ratio as:

where R^is the risk-free rate of return as determined by the 13-week U.S Treasury-biU

Since the goal is to maximize return while mini-mizing risk, we use the Sharpe method for computing returns and standard deviation to determine the global minimum variance portfoHo We operationaHze the Sharpe Ratio presented in Equation (2) with the maximization function presented in Equation (5)

THE SINGLE-INDEX MODEL

While many investors will be most interested in the outcome of the optimization process described in this section, we suspect that some will be interested in the actual process and how it is achieved using the Microsoft Excel Solver Therefore, this article both reports the out-come and details the process

The method presented and demonstrated in this article gives the optimal procedure for selecting a port-foHo when the single-index model is accepted as the best w^ay to forecast the covariance structure of returns The input variables for the objective function are the calcu-lated betas, residual variances, and alphas for the indi-vidual securities and the market index The optimization process, which utilizes five years of monthly return data from December 2000 to December 2005, is outlined in steps below so that it can be replicated by interested

investors using readily available mutual fund, market index,

and risk-free rate datạ

The optimization process uses an eight-step method,

which is conducted as follows:

Step 1: Compute natural log returns ofa market

index and the selected set of mutual funds (either

TIAA-CREF funds or another group of mutual funds) using

end-of-month closing prices, and normalize the returns

of the 13-week ỤS Treasury-bill into monthly returns

Step 2: Compute security averages, variances,

covari-ances, and betas as follows:

ạ Compute averages as "TT

h Compute Betas as ^' ~ ^ , computed as the

covari-ance of security i returns with the returns ofa market

index divided by the population variance of the

market index

c Compute the market variance as <T^ = "' ^——

d Compute the cc^yariances ofjthe individual mutual

funds as G = '"'"" '.,'—-——

Step 3: Use the CAPM methodology to compute

excess returns, identify individual fund valuations and

retain those that are undervalued Funds that are

over-valued should not enter into the optimization process

unless short-selling is aHowed Since most mutual funds

do not aHow short-selling due to legal constraints, this

study does not include this option We will evaluate this

process by computing the CAPM required or expected

return, the excess returns, and the valuation of the

indi-vidual mutual funds as follows:

ạ Compute the CAPM as i^^/iPM ~ Rf'^ (^m "~ Rf)

p., where Rr is derived from 60 months of returns

using the 13-week ỤS Treasury-biỤ The market

index average return is derived from the average of

60 monthly returns, and /? is derived from Step 2b

b Compute excess returns, (R — R^^^pj^), where R is

computed as in Step 2ạ

c Determine individual fund valuations as either under,

over, or fairly valued using the CAPM methodologỵ

Thus, if the excess return is greater than zero, the

fund is selected for inclusion in the optimal

port-folio mix

Step 4: Compute the individual mutual fund alphas

and residual variances as foHows:

ạ Compute alphas as ạ = R - p R^^, where R ^^ is the average return on the market index and p and R.

are the beta and average return on the Jth fund

b Compute residual variances as G^ — Ộ — P^G^ ,

which is the variance of the fund minus the product

of the beta squared of the individual fund and the variance of the market index

Step 5: Set up the Microsoft Excel Solver by:

ạ Establishing the objective function to maximize the modified Sharpe Ratio as presented in Equation (5)

b Equally weighting the starting values for X and

establishing the foHowing initial constraints:

ị The weights are greater than or equal to zero,

iị The sum of the starting values of X is equal to onẹ

Step 6: Compute the optimal or tangent portfolio

by using Tangent Estimates, Forward Derivatives, and Newton Search for the Solver Options, and the default values for Max Time, Iterations, Tolerance, and Conver-gence in the Solver dialog box

Step 7: EstabHsh the portfolio return, beta, and

stan-dard deviation as presented in Equations (3) and (4)

Step 8: Optimize the mutual fund portfoHo by

max-imizing Equation (5) and deterinine the optimum or tan-gent portfolio by using the computed weights of the undervalued mutual funds

THE OPTIMAL TIAA-CREF PORTFOLIO

PRACTICUM IN EXCEL

This section reports the specific findings developed using the TIAA-CREF defined contribution variable annuity retirement plan funds as a practicum in evalu-ating the eight-step optimization process outlined in the previous section End of month closing prices of the TIAA-CREF funds and the RusseU 3000 market index, RUA, from December 2000 through December 2005, are used in the optimization The TIAA-CREF closing prices were downloaded from www.tiaa-cref.org, the RUA prices from http://financẹyahoọcom, and the US Treasury-bill returns from www.treasurydirect.gov

Step 1: The monthly natural log returns of RUA

and TIAA-CREF funds were computed and reported in Exhibit 1

6 4 MODERN PORTFOLIO OPTIMIZATION: A PRJ\CTICAL APPROACH USING AN EXCEL SOLVER SINGLE-INDEX MODEL

E X H I B I T 1

Monthly Percentage Returns Computed from Monthly Fund Closing Prices

Date

1/25/01

2/22/01

3/29/01

4/26/01

5/31/01

6/28/01

7/26/01

8/30/01

9/27/01

10/25/01

11/29/01

12/27/01

1/31/02

2/28/02

3/28/02

4/25/02

5/30/02

mim

7/25/02

8/29/02

9/26/02

10/31/02

11/29/02

12/26/02

1/30/03

2/27/03

3/27/03

4/24/03

5/29/03

6/26/03

7/31/03

8/28/03

9/25/03

10/30/03

11/28/03

12/26/03

1/29/04

2/26/04

3/25/04

4/29/04

5/27/04

^RUA 1.30 -7.96 -8.98 7.33 2.12 -2.05 -2.36 -5.95 -11.17 8.02 3.86 1.85 -2.10 -2.22 4.20 -4.33 -2.64 -7.52 -16.27 8.54 -7.12 3.12 5.69 -4.85 -5.09 -0.95 3.51 4.81 4.88 3.78 1.10 1.52 0.03 4.70 1.51 8.52 -2.08 0.94 -2.98 0.17 0.63

CREF Bond 0.76 0.53 1.23 0.22 0.21 0.59 1.35 1.83 1.23 0.93 -0.48 -1.16 1.25 1.04 -1.72 1.95 0.79 0.66 1.36 1.50 1.25 -0.12 -0.05 1.76 0.48 1.42 -0.37 0.97 2.38 -0.31 -3.56 0.73 1.98 -0.50 0.47 1.30 0.29 0.97 1.25 -2.99 -0.06

CREF Equity 1.31 -7.82 -8.85 7.37 2.25 -2.02 -2.27 -5.79 -11.06 8.02 3.97 1.93 -2.03 -2.09 4.23 -4.27 -2.48 -7.42 -16.15 8.67 -6.97 3.23 5.82 -4.74 -4.91 -0.78 3.59 4.87 5.01 3.86 1.14 1.63 0.14 4.80 1.63 3.06 3.57 1.07 -2.89 0.28 0.77

CREF Global 0.16 -8.94 -8.09 6.51 -0.31 -4.12 -3.07 -3.96 -11.38 7.57 2.59 0.99 -3.61 -1.71 4.83 -2.70 -1.48 -7.51 -13.79 5.57 -7.54 3.41 4.96 -4.34 -4.37 -1.35 2.51 5.45 5.45 2.65 1.29 2.00 2.10 5.40 1.49 3.91 3.10 1.66 -2.08 0.52 0.48

CREF Growth 3.92 -16.51 -16.19 10.53 1.38 -2.21 -4.24 -8.80 -12.86 11.81 5.28 0.78 -3.18 -5.20 3.76 -6.83 -4.53 -10.29 -15.64 9.49 -6.49 3.82 5.46 -6.26 -4.69 0.35 4.63 4.65 3.61 3.51 0.51 1.69 0.49 4.12 1.59 1.56 3.74 0.24 -2.98 0.90 0.77

CREF Bond 1.71 1.35 1.08 1.37 0.93 0.23 0.22 1.04 0.68 0.70 -0.96 -0.70 0.32 1.48 -0.73 2.64 1.71 1.34 1.95 2.97 1.96 -2.04 -0.14 2.47 1.47 3.25 -1.95 -0.16 5.19 -0.92 -5.37 1.81 2.60 0.83 0.50 1.53 0.15 2.08 2.64 -5.74 2.38

CREF Money 0.50 0.43 0.50 0.37 0.41 0.29 0.27 0.33 0.28 0.25 0.20 0.14 0.15 0.11 0.07 0.17 0.16 0.13 0.12 0.14 0.11 0.15 0.11 0.08 0.09 0.06 0.06 0.06 0.08 0.06 0.05 0.04 0.06 0.07 0.05 0.06 0.07 0.06 0.06 0.04 0.05

CREF Social 1.21 -4.85 -4.36 3.97 1.15 -1.34 -0.25 -2.70 -5.30 5.10 1.71 0.44 -0.76 -0.84 1.71 -1.59 -1.09 -4.16 -8.92 5.85 -3.42 2.46 3.28 -1.92 -2.53 -0.02 1.79 3.51 4.38 2.02 -0.66 1.16 0.91 2.90 1.07 2.18 2.48 1.03 -0.86 -1.14 0.55

CREF Stock 1.02 -7.70 -8.72 7.14 1.29 -2.71 -2.51 -4.69 -11.29 7.62 3.47 1.70 -2.59 -1.60 4.47 -3.39 -1.88 -7.74 -14.80 7.70 -7.38 3.40 5.60 -4.54 -4.59 -0.99 3.05 5.00 5.48 3.51 1.31 1.74 1.01 4.88 1.46 3.50 3.56 1.12 -2.61 0.30 0.55

TIAA Real Estate 0.51 0.29 0.65 0.76 0.88 0.69 0.51 0.77 0.15 0.21 0.88 -0.04 -0.13 0.40 0.27 0.49 0.10 0.58 -0.48 0.96 0.36 -0.27 0.46 0.29 0.50 0.36 0.69 0.49 0.58 0.50 1.00 0.48 1.07 0.68 0.26 0.58 0.80 0.34 0.93 0.08 0.83 SUMMER 2007

E X H I B I T

Date

6/24/04

7/29/04

8/26/04

9/30/04

10/28/04

11/26/04

12/30/04

1/27/05

2/24/05

3/31/05

4/28/05

5/26/05

6/30/05

7/28/05

8/25/05

9/29/05

10/27/05

11/25/05

12/29/05

1 (Continued)

^RUA 1.63 -3.93 0.41 1.38 1.32 5.15 2.88 -3.58 2.27 -1.47 -3.41 4.99 0.21 4.58 -2.61 1.32 -4.33 7.59 -1.00

CREF Bond 0.05 0.75 1.79 0.76 0.57 -0.15 0.35 0.56 0.01 -0.92 1.44 0.53 0.97 -0.55 0.32 -0.27 -0.99 0.68 0.68

CREF Equity 1.72 -3.80 0.53 1.50 1.40 5.27 3.02 -3.52 2.41 -1.37 -3.31 5.11 0.32 4.66 -2.49 1.45 -4.26 7.74 -0.85

CREF Global 2.11 -3.92 0.47 1.29 2.36 5.30 3.21 -2.75 2.73 -1.21 -3.15 2.94 0.70 3.64 0.13 3.12 -4.62 6.04 2.18

CREF Growth 0.63 -6.17 0.63 1.03 1.43 4.22 3.27 -4.82 1.47 -2.07 -2.53 6.17 -0.94 5.51 -2.60 1.51 -2.72 8.74 -2.00

CREF IL Bond -0.63 0.65 2.31 1.04 0.69 0.55 0.97 -0.24 0.55 -0.50 2.04 -0.16 1.01 -1.85 0.84 1.16 -1.71 0.63 0.79

CREF Money 0.05 0.10 0.09 0.12 0.11 0.12 0.17 0.15 0.15 0.20 0.19 0.21 0.26 0.23 0.24 0.31 0.26 0.29 0.35

CREF Social 1.01 -2.04 1.05 0.99 0.99 3.17 2.10 -1.90 1.12 -1.16 -1.48 3.36 0.64 3.10 -1.27 0.82 -2.59 4.89 -0.37

CREF Stock 1.84 -3.76 0.48 1.75 1.94 5.44 3.27 -3.19 2.48 -1.26 -3.30 4.27 0.54 4.34 -1.37 2.06 -4.32 7.03 0.42

TIAA Real Estate 1.14 1.41 0.92 2.26 0.93 0.80 1.56 0.05 0.60 0.89 1.15 1.23 1.90 1.02 0.74 1.95 1.23 1.01 1.32

Notes:

1) CRJSF IL-Bond is an itiflcition linked bond fund.

2) ^RUA is the Rtdssell 3000 Index.

3) These returns do not iticlude distributed dividends.

4) End of month closing price security data for the variable annuities were obtained from http: / /u>u'w.tiaacref.org/product_profiles/ras.html and the end of month closing fund data for ^RUAfrom http://fmance.yahoo.com.

Step 2: Fund averages, variances, covariances, and

betas were computed and are reported in Exhibit 2

Step 3: Fund valuations were identified using the

CAPM, and undervalued funds were retained and are

reported in Exhibit 2

Step 4: The TIAA-CREF fund alphas and residual

variances were computed and are reported in Exhibit 3

Step S: The Excel Solver is set up as follows:

a The objective function is established to maximize the

modified Sharpe Ratio presented in Equation 5

b The starting values for X are equally weighted, and

the following initial constraints were established:

i The weights are set to values greater than or equal

to zero

ii The sum of the starting values of X is equal

to one

These values are reported in Exhibit 3

Step 6: The Solver was then loaded with the

fol-lowing inputs:

a "Set Target Cell:" Input the Tangent or Sharpe Ratio in Cell F17

b "Equal to: 'Max'"

c "By Changing Cells" Input Cell Range E6:E12

d "Subject to the Constraints:"

i Set Cell El5 = 1 This sums the individual weights to 1 or 100%, satisfying the first con-straint to Equation (2)

66

E X H I B I T 2

Monthly Data to Determine Undervaluation or Overvaluation of Funds

Notes:

Fund i

CREF Bond

CREF Equity

CREF Global

CREF Growth

CREF IL-Bond

CREF Money

CREF Social

CREF Stock

TIAA Real Estate

R.

1.161 -1.290 0.466 -0.053 23.237 23.228 0.087 0.958 20.280 21.158 0.080 0.872 36.357 27.776 -0.459 1.145 3.123 -1.280 0.666 -0.053 0.014 -0.101 0.168 -0.004 7.520 12.979 0.277 0.535 21.559 22.301 0.147 0.920 0.261 0.601 0.693 0.025

Valuation 0.199

-0.009 0.009 -0.048 0.199 0.189 0.078 -0.001 0.183

0.267 0.096 0.072 -0.411 0.467 -0.021 0.199 0.148 0.510

Undervalued Undervalued Undervalued Overvalued Undervalued Overvalued Undervalued Undervalued Undervalued

8%, Rj^^ — 0 0 i 8 % , C^— 24.251 The monthly average risk-free rate was computed by dividing each monthly U.S Treasury-bill annual investment rate percentage by 72 to convert the APR to monthly returns.

2) The Russell 3000 Index (^RUA) was used as the market index fund This is the index that TIAA-CREF uses to evaluate and benchmark their funds 3) CREF IL-Bond is an inflation linked bond fund.

ii Set Range E6:E12 > = 0, which satisfies the

second constraint to Equation (2)

iii.Used Tangent Estimates, Forward Derivatives,

and Newton Search for the Solver Options

and the default values for Max Time,

Itera-tions, Tolerance, and Convergence in the

Solver dialog box

Note: These weights would need to be adjusted

if future undervalued securities change These

values are reported in Exhibit 3

Step 7: The portfolio return, beta, and standard

deviation as presented in Equations (3) and (4) were

com-puted and are reported in Exhibit 4

Step 8: The TIAA-CREF portfolio was optimized

by maximizing Equation (5) and determining the

optimum or tangent portfolio using the computed weights

of the undervalued TIAA-CREF funds These values are reported in Exhibit 4

EXCEL SOLVER PRACTICUM RESULTS

The proportional investment percentages of the selected undervalued funds in the tangent portfolio using the hybrid Sharpe [1972] approach outlined in this article suggests that the TIAA-CREF investor should purchase 81.484% of the TIAA Real Estate fund, 11.658% of the

C R E F Bond fund, and 6.859 % of the C R E F Inflation-Linked Bond fund Even though the CREF Social Choice, Stock, Global, and Equity funds were undervalued according to the CAPM results, the funds did not opti-mize due to minimal residual variances The suggested allocation of funds optimizes the return and minimizes the risk ofa retirement portfoho of TIAA-CREF funds This allocation strategy is the outcome of the historical

SUMMER 2007

E X H I B I T 3

Solver Optimization Results

1

2

3

4

5

6

7

8

9

10

11

17.

n

14

15

16

17

A

Fund

CREF Bond

CREF Equity

CREF Global

CREF

IL-Bond

CREF Social

CREF Stock

TIAA Real

Estate

CREF Growth

CREF Monev

B Risk-free rate Market Rate of Return

Beta -0.053 0.958 0.872 -0.053 0.535 0.920 0.025 1.145 -0.004 Total

C

0.188

-0.018

Residual Variance 1.092 0.988 1.821 3.056 0.574 1.052 0.247

D

Alpha 0.465 0.104 0.096 0.666 0.286 0.163 0.693

E Market Variance Maximum Weight

Weight 0.117 0.000 0.000 0.069 0.000 0.000 0.815

1.000 Portfolio Totals:

0.665

F

24.251

100.0%

Beta*

-0.006 0.000 0.000 -0.004 0.000 0.000 0.020

0.010

Return 1.077

Residual Vatiance*

0.015 0.000 0.000 0.014 0.000 0.000 0.164

0.193 Sharpe Ratio 0.442

Alpha*

0.054 0.000 0.000 0.046 0.000 0.000 0.565

0.665

Variance 0.196

Valuation Undervalued Undervalued Undervalued Undervalued Undervalued Undervalued Undervalued Overvalued Overvalued

Notes:

1) This exhibit reports the results of the Microsoft Solver optimization calculation which includes the nine TIAA-CREF funds together with the information required to execute the Solver application, the valuation results (overvalued or undervalued) according to the Capital Asset Pricing Model, and the Sharpe Ratio, which was maximized to obtain the tangent or efficient portfolio.

2) Only undervalued funds were used for the optimization process.

3) Beta* was computed as Fund i's beta times its weight, Residual Variance* as the residual variance times its weight, and Alpha* as Alpha times its weight 4) Undervalued funds are those with an actual 60-month average return that exceeded the expected CAPM return.

data period chosen for this analysis The real estate market

demonstrated outstanding performance during the equity

bear market of 2000—2002, and this performance drove

the allocation model, which was based on the historical

return data from December 2000 to December 2005

Indeed, a TIAA-CREF investor who might have

actu-ally employed the optimal mix of the tangent portfolio

in the TIAA Real Estate fimd, the CREF Inflation-Linked Bond fund, and the CREF Bond fund as suggested, would have enjoyed excellent returns from December 2000 to December 2005 However, we caution the reader that past performance is not necessarily indicative of future

E X H I B I T 4

Efficient Frontier Results

Min Variance

Portfolio

•Efficient Portfolio

Return

0.641 0.655 0.660 0.664 0.665 0.667 0.669 0.671 0.673 0.675 0.677 0.678 0.680 0.681 0.683 0.684 0.685 0.686 0.688 0.689 0.690 0.691 0.691

Sharpe Ratio

1.046 1.070 1.076 1.077 1.077 1.077 1.075 1.074 1.071 1.069 1.066 1.064 1.061 1.058 1.055 1.052 1.048 1.045 1.042 1.039 1.036 1.033 1.028

Std Dev

0.433 0.436 0.439 0.442 0.442 0.444 0.447 0.450 0.453 0.455 0.458 0.461 0.463 0.466 0.469 0.471 0.474 0.476 0.479 0.482 0.484 0.487 0.489

CREF Bond

0.188 0.154 0.135 0.119 0.117 0.107 0.096 0.087 0.078 0.070 0.062 0.055 0.049 0.043 0.037 0.031 0.025 0.020 0.015 0.010 0.005 0.000 0.000

Weights CREF IL-Bond

0.067 0.067 0.067 0.068 0.069 0.069 0.070 0.071 0.071 0.072 0.072 0.073 0.073 0.073 0.074 0.074 0.074 0.075 0.075 0.075 0.076 0.075 0.057

TIAA Real Estate

0.728 0.776 0.797 0.812 0.815 0.824 0.834 0.843 0.851 0.859 0.865 0.872 0.878 0.884 0.890 0.895 0.900 0.905 0.910 0.915 0.920 0.925 0.943

Totals

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

SUMMER 2007