Download - Investment Principle (Demo in EIU)
-
8/10/2019 Investment Principle (Demo in EIU)
1/26
-
8/10/2019 Investment Principle (Demo in EIU)
2/26
AGENDA
CAPITAL ALLOCATIONBETWEEN RISK AND RISK-
FREE ASSET
CAPITAL ALLOCATIONBETWEEN TWO RISKY
ASSETS
-
8/10/2019 Investment Principle (Demo in EIU)
3/26
ASSET ALLOCATION
PROCESS OF CAPITAL ALLOCATION
RISK-FREE ASSET PROPERTIES OF THE COMBINED PORTFOLIO
THE CAPITAL ALLOCATION LINE
OPTIMAL PORTFOLIO
CAPITAL ALLOCATION BETWEEN RISK
AND RISK-FREE ASSET
-
8/10/2019 Investment Principle (Demo in EIU)
4/26
ASSET ALLOCATION
Capital allocation is the choice of the proportion
of portfolio.
Example: You have $10,000 to invest. Following
your brokers recommendation you have therisky asset P as 60% stock and as 40% bonds(recall that even government bonds are risky,unless held to maturity). When you choose y
(risky assets proportion) = 0.6, the combinedportfolio C is:
-
8/10/2019 Investment Principle (Demo in EIU)
5/26
With y = 0.8, the combined portfolio C would be?
You make yourself to figure it out.
ASSET ALLOCATION
-
8/10/2019 Investment Principle (Demo in EIU)
6/26
Examine risk/return tradeoff
Demonstrate how different degrees of risk aversionwill affect allocations between risky and risk free
asset. Consider the optimal risky portfolio as given and
analyze the allocation decision between the riskyasset and the risk-free asset (T-bills)
Rate of return:
PROCESS OF CAPITAL ALLOCATION
-
8/10/2019 Investment Principle (Demo in EIU)
7/26
Technically, the risk-free asset is default-free andwithout inflation risk (a price-indexed default-freebond)
In practice, Treasury bills come closest, because:- Short term means little interest-rate or inflation risk
- Default risk is practically zero, since the governmentwould no default
RISK-FREE ASSET
-
8/10/2019 Investment Principle (Demo in EIU)
8/26
PROPERTIES OF THE COMBINED PORTFOLIO
Notation:
rf= rate of return on the risk-free asset
rp= rate of return on the risky asset
rc= rate of return on the complete portfolio(including both the risk-free asset and the risky
asset)
y = proportion of the investment budget to be placed
in the risky asset p= standard deviation of the return on the risky
asset
c= standard deviation of the return on the completeportfolio
-
8/10/2019 Investment Principle (Demo in EIU)
9/26
Expect rate of return:
Variance:
Standard Deviation:
PROPERTIES OF THE COMBINED PORTFOLIO
-
8/10/2019 Investment Principle (Demo in EIU)
10/26
Solve for y:
Replace the equation for the expected rate ofreturn:
This defines a line in the mean-variance space the capital allocation line (CAL)
PROPERTIES OF THE COMBINED PORTFOLIO
-
8/10/2019 Investment Principle (Demo in EIU)
11/26
The CAL gives the trade-off between risk andreturn or the CAL describes all risk-returncombinations available to investor.
It allows to see what expected return on thecombined portfolio (E[Rc]) is attainable for a givenlevel of risk (c).
is the Reward-to-Variability ratio of P
is the risk premium on P
CAPITAL ALLOCATION LINE
-
8/10/2019 Investment Principle (Demo in EIU)
12/26
CAPITAL ALLOCATION LINE
-
8/10/2019 Investment Principle (Demo in EIU)
13/26
Example: Rf = 7%
E (Rp) = 15%
p = 22% Risk premium = 15% - 7% = 8%
Rate of return of portfolio:
Standard Deviation for portfolio:
CAPITAL ALLOCATION LINE
-
8/10/2019 Investment Principle (Demo in EIU)
14/26
CAPITAL ALLOCATION LINE
-
8/10/2019 Investment Principle (Demo in EIU)
15/26
We have shown how to develop the CAL, the graphof all feasible riskreturn combinations availablefrom different asset allocation choices. The investorconfronting the CAL now must choose one optimalportfolio, C, from the set of feasible choices.
Preview formula of Utility:
U = R(rc) 1/2A
where A is the coefficient of risk aversion and 1/2 is ascale factor.
OPTIMAL PORTFOLIO
-
8/10/2019 Investment Principle (Demo in EIU)
16/26
The investor attempts to maximize utility, U, bychoosing the best allocation to the risky asset, y.
We derive the formula:
the solution is given by the first-order constraint
solving for y gives the optimal choice of investment inthe risky portfolio
OPTIMAL PORTFOLIO
-
8/10/2019 Investment Principle (Demo in EIU)
17/26
Going back to our previous example the optimalssolution for an investor with a coefficient of riskaversion A = 4, 4 is:
n other words, this particular investor will invest41% of the investment budget in the risky asset and
59% in the risk-free asset which Utility ismaximized.
Notice that 3.28/9.02 = .36, which is the reward-to-variability ratio assumed for this problem.
OPTIMAL PORTFOLIO
-
8/10/2019 Investment Principle (Demo in EIU)
18/26
You manage a risky portfolio with an expected rate of return of 18% and astandard deviation of 28%. The T-bill rate is 8%.
1. Your client chooses to invest 70% of a portfolio in your fund and 30% in aT-bill money market fund. What is the expected value and standarddeviation of the rate of return on his portfolio?
2. Suppose that your risky portfolio includes the following investments in thegiven proportions:
Stock A: 25%
Stock B: 32%
Stock C: 43%
What are the investment proportions of your clients overall portfolio,including the position in T-bills?
3. Your clients degree of risk aversion is A = 3.5. What proportion, y, of the
total investment should be invested in your fund? What is the expectedvalue and standard deviation of the rate of return on your clients
optimized portfolio
PREVIEW EXERCISE
-
8/10/2019 Investment Principle (Demo in EIU)
19/26
A PORTFOLIO WITH TWO RISKY ASSETS
OPTIMAL PORTFOLIO
CAPITAL ALLOCATION OF TWO RISKY
ASSETS
-
8/10/2019 Investment Principle (Demo in EIU)
20/26
The Return of portfolio:
R1is the return and w is the weight of asset 1R2is the return and (1-w) is the weight of asset 2
The weights of the portfolio sum to 1.
The variance of the portfolio:
A PORTFOLIO WITH TWO RISKY ASSETS
-
8/10/2019 Investment Principle (Demo in EIU)
21/26
A PORTFOLIO WITH TWO RISKY ASSETS
Recalling that:
The variance of the return on a portfolio of tworisky assets can be expressed as a function of thecorrelation coefficient:
-
8/10/2019 Investment Principle (Demo in EIU)
22/26
There are 4 cases of correlation coefficient:
Perfect positive correlation
A PORTFOLIO WITH TWO RISKY ASSETS
-
8/10/2019 Investment Principle (Demo in EIU)
23/26
Perfect negative correlation,
We make first order constraint, we conclude:
A PORTFOLIO WITH TWO RISKY ASSETS
-
8/10/2019 Investment Principle (Demo in EIU)
24/26
No correlation,
A PORTFOLIO WITH TWO RISKY ASSETS
-
8/10/2019 Investment Principle (Demo in EIU)
25/26
A PORTFOLIO WITH TWO RISKY ASSETS
-
8/10/2019 Investment Principle (Demo in EIU)
26/26
THANKSFORLISTENING