SDGB 7844代做、代写R编程、代做R设计、代写ETF留学生

日期：2018-12-04 10:14

SDGB 7844 HW 5: Portfolio Optimization

Instructor: Prof. Nagaraja

Due: 12/6

Submit two files through Blackboard: (a) .Rmd R Markdown file with answers and code

and (b) Word document of knitted R Markdown file. Your file should be named as follows:

“HW5-[Full Name]-[Class Time]” and include those details in the body of your file.

Complete your work individually and comment your code for full credit. For an example

of how to format your homework see the files related to the Lecture 1 Exercises and the

RMarkdown examples on Blackboard. Show all of your code in the knitted Word

document.

Given the uncertainty about the future performance of financial markets, investors typically

diversify their portfolios to improve the quality of their returns. In this assignment, you

will be constructing a portfolio composed of two ETFs (exchange traded funds) that tracks

US equity and fixed income markets. The equity ETF tracks the widely followed S&P 500

while the fixed income ETF tracks long-term US Treasury bonds. You will use optimization

techniques to determine what fraction of your money should be allotted to each asset (i.e.,

portfolio weights).

The Sharpe ratio is a widely used metric to gauge the quality of portfolio returns. It was

developed by William F. Sharpe (a Nobel prize winner) and provides one way to construct

an optimal portfolio. This computation allows investors to determine expected returns while

taking into account a measure of how risky that investment is. Heuristically, it is computed

by taking the expected return of an asset/portfolio, subtracting the risk free rate (i.e., how

much you would make if you simply kept your money in the bank and let it collect interest)

and dividing by the standard deviation of the portfolio returns. The higher the Sharpe ratio

value, the better the investment is considered to be. This formula assumes that returns are

normally distributed (not always the best assumption to make with financial data).

1

The data for the two assets were downloaded from Google Finance (ticker symbols: SPY,

TLT) and the federal funds interest rate, representing the risk free rate, was taken from the

U.S. Federal Reserve Bank website. We will be looking at weekly returns, as opposed to

daily returns, as they are less correlated with each other; monthly returns would be even

less correlated, however, we would need a much longer time period of data to have enough

observations to do a robust analysis. Note that if we used a different time period to compute

our optimal rates, or we used daily instead of weekly returns, we may get very different

results.

1. Upload the data in “asset data.txt” into R and call the tibble data.x. The columns are

defined as follows: date, close.spy is the closing price for the S&P500 ETF, close.tlt

is the closing price for the long term treasury bond ETF, and fed.rate is the federal

funds interest rate in percent form. Look at the data type for the column date; just like

the data types numeric, logical, etc., R has one for date/time data. Extract only the

observations where the federal funds rate is available (so you are left with

weekly data); this is the data you will use for the rest of the analysis. What

is the start date and end date of this reduced data set? Graph the federal funds interest

rate as a time series. Describe what you see in the plot and relate it briefly to the most

recent financial crisis.

2. Now we will split the data into training and test sets. The training data will be used

to compute our portfolio weights and our test set will be used to evaluate our portfolio.

Make two separate tibbles: (a) the training set should contain all observations before

2014 and (b) the test set should contain all observations in 2014. (Install and load the R

package lubridate and use the function year to extract the year from the date column

of your data set data.x.) How many observations are in each subset?

3. The federal funds interest rate is in percent form so convert it to decimal (i.e., fractional)

form. Then, for the S&P 500 and long term treasury bonds ETF assets, compute the

returns using the following formula:

rt =

pt pt11

where rt

is the return at time t, pt

is the asset price at time t, and pt?1 is the asset price

at time t 1 (i.e., the previous period). Add both sets of returns to your training set

tibble. These returns are also called total returns. Construct a single time series plot

with the returns for both assets plotted. Add a dotted, horizontal line at y = 0 to the

plot. Compare the two returns series. What do you see?

4. The Sharpe ratio calculation assumes that returns are normally distributed1

. Construct

1We will continue with our analysis regardless of whether this assumption is satisfied.

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two normal quantile plots, one for training set returns of each asset. Is this assumption

satisfied? Justify your answer.

5. Compute the correlation between the S&P500 and long term treasury bond returns in

the training set and interpret it2

. Now, we will compute a rolling-window correlation as

follows: compute the correlation between the two asset returns only using the first 24

weeks of data (i.e., weeks 2 to 25), next compute the correlation between the two asset

returns for data from week 3 through 26, then week 4 through 27, and so forth.

Once you compute the rolling-window correlations, make a time series plot of the rollingwindow

correlation with each point plotted on the last day of the window. Add a

horizontal, dotted, gray line at 0 to your plot. Is the correlation or rolling-window

correlation a better way to describe the relationship between these two assets? Justify

your answer.

6. Compute the Sharpe ratios3

for each asset on the training set as follows:

Step 0. Let rt be the return and yt be the federal funds interest rate for week t = 1, . . . , T.

Step 1. Compute the excess returns, et

, for each week in the data set4

:

et = rt

yt1

52

Excess returns are returns that you earn in excess to the risk free rate.

Step 2. Convert the excess returns into an excess returns index, gt

:

g1 = 100

gt = gt1 × (1 + et)

Step 3. Compute the number of years of data, n, by taking the number of weeks for which

you have returns (i.e., number of observations in your training set minus 1) and

dividing by 52 (since there are 52 weeks in a year); therefore the number of years

of data can be a fractional amount.

Step 4. Compute the compounded annual growth rate, CAGR:

CAGR =



gT

g1

1/n

1

Step 5. Compute the annualized volatility, ν:

ν =

√

52SD[et

]

where SD[et

] is the standard deviation of the excess returns.

2

If you construct a scatterplot of these two particular assets, you will get a linear relationship.

3The Sharpe ratio can be positive or negative.

4We divide the federal funds interest rate by 52, the number of weeks in a year, to turn it into a weekly

rate from an annual rate.

Page 3 of 5

Step 6. Compute the Sharpe Ratio, SR, which is the ratio of the compounded annual

growth rate and the annualized volatility:

SR =

CAGR

ν

Which asset is a better investment? Justify your answer.

7. Write a function which takes the following inputs: (a) a vector of portfolio weights (call

this argument x; weights are between 0 and 1), (b) a vector of returns for asset 1, (c) a

vector of returns for asset 2, and (d) a vector of the corresponding weekly federal funds

interest rates. The function will then do the following: for each weight value in your

vector x, you will compute the Sharpe ratio for the corresponding portfolio. To obtain

the returns for the portfolio, use the following equation5

:

rt,portf olio = xrt,S&P500 + (1 x)rt,treasury

That is, x proportion of the funds will be invested in the S&P500 ETF and (1 x)

proportion of the funds will be invested into the treasury bond ETF. After you compute

the returns for the portfolio, apply the steps in question 6 to get the Sharpe ratio for that

portfolio. Your function should output a vector of Sharpe ratios, one for each portfolio

weight in x.

Use stat function() to plot the function you just wrote. Weights between 0 and 1

should be on the x-axis and the Sharpe ration should be on the y-axis. The training set

data should be used as the input for (b), (c), and (d) above. Do you see a portfolio

weight that produces the maximum Sharpe ratio?

8. Using the training set, use optimize() to determine the optimum weight for each asset

using the function you wrote in question 7; how much of the funds should be allocated

to each asset? What is the Sharpe ratio of the overall portfolio? According to your

analysis, is it better to invest in S&P500 only, long term treasury bonds only, or your

combined portfolio? Justify your answer.

9. For the remainder of this assignment, we will be evaluating our portfolio using the test set

data. We will be comparing three strategies: investing only in the S&P500, investing

only in long term treasury bonds, and investing in the combined portfolio (computed in

question 8).

In your test set, convert the federal funds interest rate from percent to decimal form and

compute the returns series for each of the three assets (see question 3 for more details).

Next, compute the excess returns index for each asset in the test set (as outlined in

question 6). Plot the excess returns index for each asset on the same time series plot.

Add a dotted, horizontal line at y = 100. Describe what you see.

5Note to people with a finance background: these computations correspond to a setting where you are

rebalancing the portfolio daily.

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10. The excess returns index can be interpreted as follows: if you invested in $100 in at time

t = 1, the index value at time T represents how much you have earned in addition to

(i.e., in excess of) the risk free interest rate. If you invested $100 in each asset (portfolio,

all in long term treasury bonds, or all in S&P500) in the first week of January, 2014 ,

how much would you have at the end of the test set period for each asset in addition to

the risk-free interest rate? Did your portfolio perform well in the test set? Justify your

answer.