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I. Three general rules for combining assets into portfolios
 

Consider two portfolios A and B: (E(RA), A ), (E(RB),  B ) Form portfolio P by putting part of your invested wealth in A (wP
A ) and the rest of your invested wealth in B ( wP
 

B ). Example: You have $40,000 to invest and you invest $30,000 in A and the rest in B.
wP A =30,000/40,000=.75, wP B =10,000/40000=.25
Note that wP A+wP B = 1 always.
 
Rule 1: RP =wP ARA +wP BRB
 

Rule 2: E(RP) =wP A E(RA) +wPBE(RB)
 
This follows from the rules on expected values: E(X+Y) = E(X) + E(Y) where X and Y are random variables, E(cX) = c E(X) where c is a constant.
 

Rule 3: Var(RP) = (wP A )2Var(RA) + ( wP B )2Var(RB) +2wP A wP B cov(RA,RB) This follows from the rules on variances: Var(X+Y) = Var(X) + Var(Y) + 2cov (X, Y) where X and Y are random variables, Var(cX) = c2 Var(X) where c is a constant.
 
Var(RP) = ( P )2 I will use this notation interchangeably.
 

II. Combining a risky portfolio (P) with the risk-free asset (F) to form the investment opportunity set of complete portfolios (C).
 

The investment opportunity set is the set of all available
 
portfolios which in this case will be combinations of: (E(RP), P ) and (RF, 0) Notice that the risk-free asset is exactly that—free of risk. Therefore, there is no expectation sign (i.e., E(RF) = RF) and  F = 0.
Let wC P = the proportion of invested wealth invested in the risky portfolio, and wC
F = the proportion of invested wealth invested in the risk-free asset.
 
Again, wC
P+wC
P= 1.
Let’s see how our general rules on combining assets into
portfolios apply to our special case of one risky and one
risk-free asset.
 

1: RC =wC PRP +wC F RF
2: E(RC) = wC PE(RP) +wC F RF
3: Var(RC) =( wC P )2Var(RP) + (wC F )2Var(RF)+2wC P wC F cov(RP,RF) Var(RC) =( wC P )2Var(RP)
 C = wC
P  P
 
Numerical example of possible complete portfolios on the
CAL(P)
 
Given: (E(RP) = .10,  P =.26) and RF = .05
 
What is the E(RC) and  C for the following complete
portfolios?
 
1) 100% in P
2) 100% in F
3) 50% in P, 50% in F
4) 75% in P, 25% in F
5) 130% in P, -30% in F
 
1) wC
P= 1, wC
F =0
2) wC
P= 0 wC
F =1
3) wC
P= .5 wC
F =.5
4) wC
P= .75 wC
F =.25
5) wC
P= 1.3 wC
F = -.3
 
Supplemental notes on Risk Aversion and Capital
 
cation to Risky Assets (5.5 in text)
Capital Allocation across Risky and Risk‐free
Assets
 
• Investors can construct portfolios by allocating capital
• across different asset classes.
• The most basic asset allocation choice:
• Choose the fraction of the portfolio invested in risk‐free
• assets versus the portion invested in the risky assets Consequence: different risk and return trade‐off
 

The Risk‐Free Asset
 
• Only the government can issue default‐free
bonds.
• Risk‐free in real terms only if price indexed and
• maturity equal to investor’s holding period.
• T‐bills viewed as “the” risk‐free asset
• Money market funds also considered risk‐free in
practice
 

4Graph for Capital Allocation Line
 
The Sharpe (Reward‐to‐Volatility) Ratio
 
• Commonly used to rank portfolios in terms of risk return
trade‐off:
• Warning:
• A valid statistics only for ranking portfolios
• Not valid for ranking individual assets.
 
You can read more about our case study assignment help services here.
 

 

Some Observations about CAL
 
• CAL depicts all the risk‐return combinations available
• to investors.
• The slope of CAL equals the increase in the expected
• return of the complete portfolio per unit of additional
 
• standard deviation.
• Sharpe Ratio
• The Sharpe ratio is the same for all complete portfolios
that plotted on the CAL.
 

6Levered Complete Portfolios
 
• These are the portfolios to the right of P on the CAL.
• Weight in risky portfolio P > 1
 
• borrow a proportion of at the risk‐free rate and invest all
• funds into P
• Expected return and risk for levered portfolios:

This should be done in groups of up to 4 students (3 is ideal). Please put all group member names on the first page. You should have the same groups for the problem sets and the group project. Please turn in a neat, typed, easy to follow set of answers. When I ask for explanations or comments, only brief explanations are necessary. This problem
set is worth 40 points total, with the points listed in parentheses.
 

Data380ps1.XLS is available from Canvas. This spreadsheet contains monthly returns for ten Vanguard index funds and five actively managed portfolios from January 2011 to December 2015. The units are in decimals. If the return is -0.00123, this means -0.123%. For your convenience, here are some examples of useful Excel commands (An = sign
always precedes a command in Excel)
 
=A2*B2 Multiplies cell A2 and B2
=A2^.2 Calculates the Fifth Root of A2(Raises A2 to the 1/5 power)
=1+A2 Adds 1 to A2
 
=AVERAGE(A2:A10) Calculates the sample mean of rows A2 through A10
=STDEV(A2:A10) Calculates the sample standard deviation (A2-A10)
=PRODUCT(A1:A10) Multiplies all the cells A1-A10
 

For the ten Vanguard index funds and for each actively managed mystery portfolio (i.e., portfolio A, B, C, D, E). Note the full names of the Vanguard index funds are included in the second row below the ticker symbols
period? Why or why not?
 

1. (6) Calculate the returns requested in i) and ii) below for all 15 of the funds.
 
i) The net holding period return for an investor who bought the fund on January 1,
2011 and sold it on December 31, 2015.
 
ii) The annualized return (also known as the geometric average annual return)
over the 5 year period from January 2011 to December 2015.
period? Why or why not?
 

2. (8) Complete this exercise for only one fund of your choice and for the Vanguard
Index 500 (on the same graph)
 
You purchased $10,000 worth of the fund on January 1, 2011. Calculate and plot the dollar value of your investment over the period. (Don’t just calculate the annual return for the five years and plot five points. You should basically have a continuous line connecting 60 data points. For an example see the graph “Growth of Hypothetical
$10,000” available at Also see sample calculations in first lecture notes and second daily assignment.) Include
your graph in your completed problem set answers.
 
period? Why or why not?
 

3. (3) Calculate and report the annualized sample mean return and sample standard deviation over the 5 year sample period. In this case you should annualize the mean and standard deviation of the monthly returns by multiplying the monthly mean return by 12 and the monthly standard deviation by the square root of 12. Again label your answers
with the Index or portfolio name.
 
period? Why or why not?
 

4. (3) Calculate and report the pairwise correlations for all 15 of the funds. Include labels. (This can all be done in one step using all the monthly returns. Go to Tools/Data Analysis/Correlation and choose all the data. If Data Analysis does not appear under the Tools menu, this means that you must add it in under “Add Ins.” It is called “Analysis Toolpack”. Then you need to copy over the fund labels to replace the generic labels.)
 
period? Why or why not?
 

5. (8) Using the historical annualized mean and standard deviation in number 3, and the pairwise correlation from number 4, graph the investment opportunity set of portfolios combining the Vanguard Short Term Bond index and the Vanguard Small Cap Stock index on an expected return-standard deviation graph like the one on in Figure 6.4 in the text (in this example the relevant opportunity set curve is the one associated with the correlation equal to 0.2). You should be using Excel to make this graph. Note that you will be using historical data on these two portfolios to obtain estimates of expected return, expected standard deviation, and correlation.
 
period? Why or why not?
 

6. (3) Assume that monthly returns on these 15 portfolios are drawn from normal distributions. What is the worst return you’re likely to see in a one year holding period with 95% confidence for each index and actively managed fund?
 
period? Why or why not?
 

7. (6) Although I have not told you what type of actively managed fund portfolio A through E in the spreadsheet are, you should be able to tell with all the statistics you generated. Determine what type of fund (e.g., large cap domestic stocks…) they are and briefly describe how you arrived at your answer. What do you suspect about Portfolio E and how did you arrive at this?
 
period? Why or why not?
 

8. (3) Compare statistics for the Vanguard 500 Index (e.g., using estimates produced for questions above) from other historical periods listed in your text page 138 in Table 5.4.List out all the statistics in Table 5.4 and include estimates from 2011 to 2015. Note that the return on t-bills over this period is effectively 0. Does 2011 to 2015 look like a typical .
 
period? Why or why not?
 
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