Unconstrained Optimization

Data

All the analysis below applies to the data set,

• data/spx_returns_weekly.xlsx

• The file has weekly returns.

• For annualization, use 52 periods per year.

Consider only the following 10 stocks…

As well as the ETF,

Data Processing

	name	mkt cap (billions $)
ticker
AAPL	Apple Inc	4,631
NVDA	NVIDIA Corp	5,105
MSFT	Microsoft Corp	2,861
GOOGL	Alphabet Inc	4,336
AMZN	Amazon.com Inc	2,639
META	Meta Platforms Inc	1,699
TSLA	Tesla Inc	1,531
AVGO	Broadcom Inc	1,903
BRK/B	Berkshire Hathaway Inc	1,064
LLY	Eli Lilly & Co	1,119

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1. Risk Statistics

1.1.

Display a table with the following metrics for each of the return series.

• mean (annualized)

• volatility (annualized)

• Sharpe ratio (annualized)

• skewness

• kurtosis

• maximum drawdown

Note

We have total returns, and Sharpe ratio is technically defined for excess returns. Don’t worry about the difference. (Or subtract SHV if you prefer.)

1.2.

As a standalone investment, which is most attractive? And least? Justify your answer.

1.3.

For each investment, estimate a regression against SPY. Report the

• alpha (annualized as a mean)

• beta

• info ratio

• r-squared

Based on this table, which investment seems most attractive relative to holding SPY?

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Solution 1.1.

	Mean	Vol	Sharpe	Min	Max	skewness	kurtosis	max drawdown
AAPL	24.9%	27.8%	0.895	-17.5%	14.7%	-0.182	1.805	-34.6%
NVDA	64.5%	45.4%	1.420	-20.1%	30.2%	0.351	1.492	-65.9%
MSFT	23.5%	23.8%	0.989	-13.5%	14.0%	0.133	1.685	-35.1%
GOOGL	27.0%	28.7%	0.939	-12.0%	25.8%	0.533	3.104	-41.9%
AMZN	26.1%	30.5%	0.858	-14.5%	16.3%	-0.087	1.342	-54.8%
META	23.2%	35.7%	0.649	-23.7%	24.5%	0.037	3.490	-76.0%
TSLA	43.7%	58.0%	0.755	-25.9%	33.3%	0.559	1.634	-72.2%
AVGO	39.4%	38.3%	1.030	-18.3%	25.2%	0.643	3.272	-40.0%
BRK/B	13.5%	18.8%	0.716	-13.4%	9.8%	-0.186	2.534	-26.5%
LLY	30.1%	29.8%	1.011	-17.9%	17.5%	0.138	1.976	-34.5%
SPY	14.5%	16.9%	0.859	-14.5%	12.1%	-0.581	5.967	-31.8%

Solution 1.2.

As a standalone investment, NVDA has

• the best Sharpe, which is the best vol-adjusted return.

• large positive skewness which is attractive.

• large kurtosis, which would be bad with negative skewness but is appealing with positive skewness.

• a large max drawdown.

If worried about the max drawdown, BRK/B may be a good choice.

• smallest max drawdown

• second-lowest Sharpe.

Solution 1.3.

	alpha	Beta (SPY)	r-squared	Treynor Ratio	Info Ratio
AAPL	8.6%	1.116	46.2%	22.3%	0.425
NVDA	39.6%	1.718	40.8%	37.6%	1.133
MSFT	8.8%	1.016	52.2%	23.2%	0.534
GOOGL	11.0%	1.104	42.2%	24.4%	0.502
AMZN	10.1%	1.107	37.7%	23.6%	0.419
META	6.0%	1.182	31.3%	19.6%	0.204
TSLA	18.4%	1.749	26.0%	25.0%	0.368
AVGO	18.7%	1.423	39.5%	27.7%	0.630
BRK/B	2.1%	0.782	49.3%	17.2%	0.159
LLY	21.3%	0.606	11.8%	49.8%	0.762
SPY	-0.0%	1.000	100.0%	14.5%	-

Based on this table, NVDA is the most attractive. It not only has (by far) the highest alpha, but also the highest Info Ratio (which is the risk-adjusted alpha.)

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2. Portfolio Allocation

2.1.

Display the correlation matrix of the returns.

• Based on this information, which investment do you anticipate will get extra weight in the portfolio, beyond what it would merit for its mean return?

• Report the maximally correlated assets and the minimally correlated assets.

2.2.

Calculate the weights of the mean-variance optimized portfolio, also called the tangency portfolio.

• Display a table indexed by each investment, with the optimal weights in one column and the Sharpe ratios in another column.

• Do the investments with the best Sharpe ratios tend to get the biggest weights?

Note:

To estimate the optimal weights, consider using the provided function below.

2.3.

Report the following performance statistics of the portfolio achieved with the optimized weights calculated above.

• mean

• volatility

• Sharpe

(Annualize all three statistics.)

2.4.

Try dropping the asset which had the biggest short position from the investment set. Re-run the optimization. What do you think of these new weights compared to the original optimized weights?

What is going on?

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Solution 2.1

(<Figure size 1200x1200 with 2 Axes>,
 <Axes: title={'center': 'Correlation matrix (lower triangle)'}>)

Solution 2.2

	optimized weights	Sharpe
BRK/B	169.6%	0.716
LLY	90.5%	1.011
NVDA	84.6%	1.420
GOOGL	55.1%	0.939
AVGO	45.4%	1.030
AAPL	35.3%	0.895
MSFT	30.1%	0.989
TSLA	20.0%	0.755
AMZN	12.9%	0.858
META	10.0%	0.649
SPY	-453.8%	0.859

Solution 2.3

	Mean	Vol	Sharpe	Min	Max
optimized weights	102.01%	52.22%	1.9536	-19.71%	29.63%

Solution 2.4

The optimization is unrealistic in that it prescribes massive positions.

• SPY is short nearly 450%.

• BRK-B is long nearly 170%.

The optimizer is shorting SPY because…

• it has lower mean return than the tech stocks (NVDA)

• it is highly correlated to the tech stocks and can offset much of their risk.

The optimizer is massively long BRK…

• even though BRK has the second-lowest Sharpe ratio of the investments. Only META is lower.

• But BRK is highly correlated to SPY while having relatively low correlation to tech. Thus it lowers total risk.

These weights are much more realistic. They are reasonable magnitudes and don’t include massive short positions.

Without SPY, the correlation matrix has relatively low correlations. Thus, the optimizer doesn’t think it can achieve such balanced (hedged) offsets, so it doesn’t prescribe extremes.

	Mean	Vol	Sharpe	Min	Max
optimized weights	102.0%	52.2%	1.954	-19.7%	29.6%
optimized ex SPY	45.4%	27.2%	1.666	-14.3%	15.7%

	gross mkt weight
optimized weights	1,007.5%
optimized ex SPY	138.5%