Travis Wainman
Michelle Chau Pham
Big
Modern Portfolio Theory (MPT)
- Based on the idea that combinations of assets have the
potential to provide better returns with less risk than
individual assets.
- Using Markowitz’s efficient portfolio and Sharpe’s
single-index model.
- This article uses TIAA-CREF mutual fund data as an
example to illustrate the MPT concept.
Data
 Singapore equity historical prices Jan 2003 – Nov 2009
 Included only the securities that survived throughout
the entire period
Sharpe Ratio
 We compute the objective function by maximizing the
Sharpe Ratio, the portfolio with minimum risk and
maximum return.
 We define the Sharpe Ratio in Equation (1) as:
(1)
 It measures the excess return per one unit of risk.
 We also use the Sharpe Ratio to calculate the tangent
portfolio.
Constraints
 There are 2 main constraints which are:
and
The sum of all weighted
coefficients to be equal
to one
The weight for each
individual asset has
to be greater than or
equal to zero
because shortselling of securities
is not allowed.
Calculation
 Step1: Compute percentage of monthly returns for the TSI
Index and for each security
 Step2: Compute security average returns, variances,
covariances (between the security and index), and betas.

Compute average as

Compute Betas as
the covariance of security i’s returns with the returns of a market
index divided by the population variance of the market index.

Calculation (con’t)
 Compute the market variance as
 Compute the covariances of the individual securities as
Calculation
(con’t)
 Step3: Use the CAPM methodology to compute excess
returns, identify individual fund valuations and retain
those that are undervalued.

Compute the CAPM as
 Compute excess returns,

Determine individual fund valuations as either under, over, or fairly valued using
the CAPM methodology. Thus, if the excess return is greater than zero, the fund is
selected for inclusion in the optimal portfolio mix.
• Step4: Compute the individual mutual fund alphas and
residual variances.
•
•
Compute alphas as
Compute residual variances as
Calculation
(con’t)
 Step5: Set up the Microsoft Excel Solver by:

Establishing the objective function to maximize the modified Sharpe Ratio as
presented in the equation.

Equally weighting the starting values for Xi and establishing constraints.
• Step6: The solver was then loaded with the following
inputs:
•
•
•
•
“Set Target Cell:” Input the Tangent or Sharpe Ratio in the cell.
“Equal to: ‘Max’”.
“By Changing Cells” Input Cell Range
“Subject to the Constraints:”
• Set a cell of total weight equal to 1, which is the sum of individual weights,
which satisfied the first constraint in the model.
• Set each cell of individual weight >=0, which satisfied the second constraint in
the model.
Calculation (con’t)
 Step7: Compute the porfolio return, beta, and standard
deviations.
 Step8: The TIAA-CREF portfolio was optimized by
maximizing Sharpe Ratio using the computed weights of
the undervalued TIAA-CREF funds.
Results
 The proportional investment percentages of the selected
undervalued funds in the tangent portfolio suggest that the
TIAA-CREF investor should purchages 87.76% of the TIAA
Real estate fund, and 12.24% of CREF Bond fund, and zero
invest in CREF Inflation-Linked Bond fund.
 Even though the CREF Social Choice, Stock, Global, and
Equity funds were undervalued according to the CAPM
results, but the funds did not optimize due to minimal
residual variances.
 However, the past performance is not necessarily indicative
of future performances.
 Therefore, investors should re-optimize their portfolios
quarterly, at a miminum.