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2023 Case 5 Investment Planning

FINANCE 6080 Case #5 Investment Planning

If you have a 401(k) or other qualified plan through your workplace or someone
you know, please use the investment options available within the retirement plan
as if they were available to Ned and Nelly Normal. Based on those options, how
would you structure their portfolio. Support your recommendation.

If you do not have a 401(k) plan or other qualified plan available to you, please
make a recommendation for the Normal’s from one of the 4 options below.
Recommend what you feel is the best option for their 401(k) allocation. Support
your recommendation.

Option 1. Vanguard S&P 500

Option 2. Vanguard Target Retirement 2045

Option 3. 10% Vanguard 500 Index

10% Vanguard Value Index
10% Vanguard Small Cap Index
10% Vanguard Small Cap Value Index
10% Vanguard REIT Index
10% Vanguard Total International Index
40% Vanguard Total Bond Market Index

Option 4. 54% Vanguard Total Stock Market Index Fund

27% Vanguard Total International Stock Index Fund
6% Vanguard REIT Index Fund
3% Precious Metals
10% Total Bond Market Index Fund

Instruction and Background x
Need Delievery:

Review the Case 5 PDF and attach a one-page written executive summary as if you were a professional adviser writing to the client. Research all the issues/needs/questions the clients have and summarize the pertinent assumptions, case facts, and findings. This executive summary should include specific and actionable recommendations.

We need data or evidence to support the recommendations!!!!

Background info:

Ned and Nelly Normal

Ned is a pilot and Nelly a school teacher. They have 3 children—all daughters. Kris is 23 and single,

graduated from college and working. Olivia is 20 and half way through college, studying engineering.

Ellie is 13, an 8th grader. Ned will be required to retire at 65 and Nelly plans to work through 65, as well.

Both are currently 50 years old. They would like to maintain their same lifestyle in retirement.

They have the following assets:

His Checking $11,000 Her Checking $13,500

His Savings $10,700 Her Savings $18,000

Employee Stock Purchase Plan $38,600 401(k) at D Airlines $828,932

Rental Property $400,000 Primary Residence $675,000

Whole Life Policy $500,000 Death Benefit

They have the following debts:

Chase Visa—generally about $11,000 and paid in full each month

Residential Mortgage $270,755—pay $1,575 per month at 3.5% for 30 years.

Rental Mortgage–original loan was July 2017 for $336,000. Rate is 4.85%, 30 year loan, they pay $2,225

per month.

New Windows–$16,960 original loan amount was $23,104 with a 66 month amortization at 5.99%, pmt

is about 411.75 per month.

Solar and Roof HELOC –$38,000 for 240 months at 10.5%, payments start next month, but they have

forgotten how much it was exactly

Car Loan–$27,911 balance, originally $35,150 for 72 months at 3.49% with a payment of $541.00

Income includes:

His Salary $295,000 Her Salary $65,000

Military Pension $4,000 per month starting in 5 years Rental Income $24,000

They spend pretty much what they make, but have been able to save about $3,000 per month over the

past year in conjunction with Nelly re-entering the work force. His union has negotiated that his

employer contributes 15% of his wage into his 401(k) up to the IRS limit each year. He also contributes

$4,000 a year into the Employee Stock Purchase Plan.

Morningstar_Methodology_MPT_Statistics

Modern Portfolio Theory (MPT)
Statistics

Morningstar Methodology Paper
October 29, 2015

Morningstar, Inc. Reproduction or transcription by any means, in whole or in part, without the prior

written consent of Morningstar, Inc., is prohibited.

Morningstar MPT Statistics| October 29th, 2015

© 2015 Morningstar, Inc. All rights reserved. The information in this document is the property of Morningstar, Inc. Reproduction or transcription by any means,
in whole or part, without the prior written consent of Morningstar, Inc., is prohibited. 2

Introduction

Modern Portfolio Theory Statistics (MPT statistics) are based on the Capital Asset Pricing
Model (CAPM) of expected returns developed by Nobel laureate William Sharpe and others in
the early 1960s. The CAPM is based on Modern Portfolio Theory (MPT) developed in the 1950s
by Sharpe’s teacher and co-laureate Harry Markowitz.

In the terminology of another Nobel laureate, the late Milton Friedman, MPT is a normative
theory, meaning that it is a prescription for how investors ought to behave. In contrast, the
CAPM is a positive theory in that it meant to be a description of how investors do behave. The
CAPM is based on MPT in that it assumes the all investors follow the prescriptions of MPT.

The CAPM separates the excess return (i.e., total return minus the return on a risk-free
security) of each security into two components: systematic excess return and unsystematic (or
idiosyncratic) return. Systematic excess return is directly proportional to the excess return of
the market portfolio. The ratio of the excess return of the market to the systematic excess
return of the security in question is the security’s “beta.” Beta measures how sensitive the
excess return on a security is to the excess return of the market as a whole.

One of the main implications of the CAPM is that the expected excess return on a security is
directly proportional to systematic risk as measured by beta and is not related to any other
variable. This means that there are no rewards for taking on unsystematic risk. In an efficient
market in which the CAPM holds, the only way to obtain an expected return above that of the
market portfolio is to take on a beta above one.

In the 1970’s, Michael Jensen proposed a performance measure for actively managed funds
that is based on the CAPM called Jensen’s alpha or simply alpha. The idea of alpha is that a
manager should not receive credit for achieving above-market performance by taking on
systematic risk as measured by beta. Alpha is the average excess return that a portfolio
achieves above and beyond that could have been obtained from position in the market portfolio,
levered or de-levered so as the have the same beta as the fund.

Morningstar MPT Statistics| October 29th, 2015

Introduction (continued)

Strictly speaking, the CAPM cannot be applied in the “real world” because returns on the
market portfolio are unobservable. To make alpha and beta practical measures at first, broad
stock market indexes were used as proxies for the market portfolio. As funds became more
specialized, more narrow benchmarks were developed to track returns on the more narrowly
defined sources of systematic risk and return. Today it is common practice to measure alpha
and beta using a narrowly defined benchmark that is chosen to represent the main source of
the systematic risk of the fund being analyzed.

In addition to alpha and beta, a third MPT statistic is R-squared. R-squared measures the
strength of the relationship between excess returns on the benchmark and excess returns on
the fund being analyzed.

MPT statistics are calculated from a comparison of a fund’s excess returns and the
benchmark’s excess returns. Unless a time horizon is specified, Morningstar’s MPT statistics
are based on three years of monthly returns. Morningstar calculates three sets of MPT
statistics for each fund, using a standard set of benchmarks for each asset group; using a
standard set of benchmarks for each Morningstar Category; and using the index from the fund’s
prospectus. Morningstar also calculates “best-fit” MPT statistics, which are based on the index
that has the highest R-squared with the portfolio in question. For best-fit MPT statistics,
Morningstar compares the portfolio to dozens of different indexes to find the best-fit.

The broad asset class, category, prospectus and best-fit results can all be useful to investors.
The broad index R-squared or category index R-squared can help investors diversify their
portfolios. For example, an investor who already owns a fund with a very high correlation (and
thus high R-squared) with the S&P 500 might not choose to buy another fund that correlates
closely to that index. The best-fit MPT statistics can help investors compare two similar funds.
For example, if two funds have the same best-fit index, an investor can evaluate the risk and
excess returns for those funds by comparing their best-fit betas and alphas.

Morningstar MPT Statistics| October 29th, 2015

Methodology

Morningstar calculates a fund’s alpha, beta, and R-squared statistics by running least-squares
regression of the fund’s excess return over a risk-free rate compared with the excess returns of
the index that Morningstar has selected as the index for the fund’s broad asset class or the
fund’s category index.

The Morningstar broad asset class indexes for the US are as follows:

Broad Asset Class Broad Asset Class Index
U.S. Stocks S&P 500
International Stocks MSCI ACWI Ex USA
Balanced Morningstar Moderate Target Risk
Alternative ML USD LIBOR 3 Mon
Taxable Bonds BarCap US Aggregate Bond
Municipal Bonds BarCap Municipal

Morningstar’s editorial team assigns the category index. The chosen index represents the best
fit (has the highest correlation to funds in the category) of those indexes that have at least
three years worth of return history available (for newer categories) and at least 10 years worth
of return history available (for existing categories).

The category index assignments for Europe/Asia can be found in the methodology document
Morningstar Category Definitions Europe and Asia. The category index assignments for the US
can be found in the methodology document Morningstar Category Classifications.

The prospectus index is the index (a.k.a. benchmark) that the fund defines in its prospectus. A
high correlation to this index suggests that the fund has chosen an appropriate index against
which to evaluate its performance. A low correlation may indicate that the fund chose an index
that may not be very representative of the fund’s investment style.

The calculations are made using the trailing 36-month period.

In least-squares regression there is dependent variable and an independent variable. A least
squares regression can be understood by looking at a scatter plot of the independent variable
on the horizontal axis and the dependent variable on the vertical axis. The regression line is the
straight line that minimizes the sum of the squared vertical distances of each scatter point from
the line.

To estimate the MPT statistics for a fund using its broad asset class index, Morningstar runs a
regression with the monthly excess return on the fund as the dependent variable and the
excess return on the broad asset class index as the independent variable. The slope of the

Morningstar MPT Statistics| October 29th, 2015

Methodology (continued)

resulting regression line is the estimate of beta and the intercept (multiplied by 12 to express it
as an annual figure) is the estimate of alpha.

Y: Portfolio

X: Benchmark Index

Morningstar MPT Statistics| October 29th, 2015

Beta

Beta is a measure of a fund’s sensitivity to movements in the index. By construction, the beta
of the index is 1.00. A fund with a 1.10 beta has tended to have an excess return that is 10%
higher than that of the index in up markets and 10% lower in down markets, holding all other
factors remain constant. A beta of 0.85 would indicate that the fund has performed 15% worse
than the index in up markets and 15% better in down markets. A low beta does not imply that
the fund has a low level of volatility, though; rather, a low beta means only that the fund’s
index-related risk is low. A specialty fund that invests primarily in gold, for example, will usually
have a low beta (and a low R-squared), as its performance is tied more closely to the price of
gold and gold-mining stocks than to the overall stock market. Thus, although the specialty fund
might fluctuate wildly because of rapid changes in gold prices, its beta will be low.

In the examples below, beta is 0.69 for the fund on the left and 1.70 for the fund on the right.

Morningstar MPT Statistics| October 29th, 2015

Beta (continued)

Beta is calculated as:

where:

r = Beta of portfolio r

rbCov = Covariance between the excess returns of the portfolio r and the benchmark b
2
b = Variance of the excess returns of the benchmark

and:

where:
e
iR = Excess return of the portfolio for month i = Ri – RFi, where Ri is the portfolio return for

month i and RFi is the risk-free return for month i
e

R = Average monthly excess return of the portfolio over n periods (simple mean)
e
iB = Excess return of the benchmark for month i = Bi – RFi, where Bi is the benchmark

return for month i and RFi is the risk-free return for month i
e

B = Average monthly excess return of the benchmark index over n periods (simple mean)

n = number of periods (Morningstar typically uses 36 months)

eR is the simple arithmetic average excess return for the portfolio:

The denominator for beta is the variance of the excess returns of the benchmark:

A similar calculation can also be used for the variance of the portfolio, 2

r .
Standard deviation is the square root of variance.

2
b

r 
 rbCov







n

1i
rb )])([(

1-n

1 ee
i

ee
i BBRRCov





n

22 )(
1-n

1 ee
ib BB





n

e
i

e R
n

R
1

Morningstar MPT Statistics| October 29th, 2015

Alpha

Alpha measures a fund’s performance after adjusting for the funds systematic risk as measured
by the fund’s beta with respect to the index. An investor could have formed a passive portfolio
with the same beta that of the fund by investing in the index and either borrowing or lending at
the risk-free rate of return. Alpha is the difference between the average excess return on the
fund and the average excess return on the levered or de-levered index portfolio. For example, if
the fund had an average excess return of 6% per year and its beta with respect to the S&P 500
was 0.8 over a period when the S&P 500’s average excess return was 7%, its alpha would be
6% – 0.8*7% = 0.4%.

There are limitations to alpha’s ability to accurately depict a fund’s added or subtracted value. In
some cases, a negative alpha can result from the expenses that are present in the fund figures
but are not present in the figures of the comparison index. The usefulness of alpha is
completely dependent on the accuracy of beta. If the investor accepts beta as a conclusive
definition of risk, a positive alpha would be a conclusive indicator of good fund performance.

where:

M = Monthly measure of alpha
e

R = Average monthly excess return of the portfolio
e

B = Average monthly excess return of the benchmark index

The resulting alpha is in monthly terms, because the average returns for the portfolio and
benchmark were monthly averages. Morningstar then annualizes alpha to put it in annual
terms.1

where:

A = Annualized measure of alpha

1 Prior to 2/28/2005, Morningstar annualized the monthly alpha with a geometric method, (1+α)12-1.

M BR  

MA  12

Morningstar MPT Statistics| October 29th, 2015

R-squared

R-squared is another statistic that is produced by a least-squares regression analysis. R-
squared is a number between 0 and 100% that measures the strength of the relationship
between the dependent and independent variables. An R-squared of 0 means that there is no
relationship between the two variables and an R-squared of 100% means that the relationship
is perfect with every scatter point falling exactly on the regression line. Thus, stock index funds
that track the S&P 500 index will have an R-squared very close to 100%.

A low R-squared indicates that the fund’s movements are not well explained by movements in
the index. An R-squared measure of 35%, for example, means that only 35% of the fund’s
movements can be explained by movements in the index.

R-squared can be used to ascertain the significance of a particular beta estimate. Generally, a
high R-squared will indicate a more reliable beta figure.

R-squared ranges from 0 (perfectly uncorrelated) to 100 (perfectly correlated). Correlation (  )
is the square root of R-squared. R-squared is calculated as follows:

where:

rbCov = Covariance between the excess returns of the portfolio r and the benchmark index b

r = Standard deviation of the excess returns of the portfolio r

b = Standard deviation of the excess returns of the benchmark index b

rb2 )(100


Cov
R 

Morningstar MPT Statistics| October 29th, 2015

Best-Fit Index

Morningstar also shows additional alpha, beta, and R-squared statistics based on a regression
against the best-fit index. The best-fit index for each fund is selected based on the highest R-
squared result from separate regressions on a number of indexes. For example, many high-yield
funds show low R-squared results and thus a low degree of correlation when regressed against
the broad asset class index for the taxable bond funds, the Lehman Brothers Aggregate. These
low R-squared results indicate that the index does not explain well the behavior of most high-
yield funds. Most high-yield funds, however, show significantly higher R-squared results when
regressed against the CSFB High-Yield Bond index.

The broad asset class, category, prospectus, and best-fit results can all be useful to investors.
The broad asset class index or the category index R-squared statistics can help plan the
diversification of a portfolio of funds. For example, if an investor wishes to diversify and already
owns a fund with a very high correlation (and thus a high R-squared) with the S&P 500 or the
FTSE 100, he or she might choose not to buy another fund that correlates closely to that index.
In addition, the best-fit index can be used to compare the betas and alphas of similar funds that
show the same best-fit index. The prospectus index R-squared statistics can help determine
whether the portfolio manager has chosen an appropriate index to be measured against.

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