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Black and Litterman recommend to combine the investor's view with the market equilibrium, as follows.

The expected return vector μis assumed to have a probability distribution that is the product of two multivariate normal distributions. The first distribution represents the returns at market equilibrium, with meanπ and the covariance matrix τQ, where τ is a small constant ad Q = (σ_ij) denotes the covariance matrix of asset returns (Note that the factor τ should be small since the variance τσ_i² of the random variable μ_i is typically much smaller than the variance σ_i² of the underlying asset returns). The second distribution represents the investor's view about the μ_i's. These views are expressed as

P_μ = q + ε

where P is a k×n matrix and q is a k-dimensional vector that are provided by the investor and ε is a normally distributed random vector with mean 0 and diagonal covariance matrix Ω (the stronger the investor's view, the smaller the corresponding ω_i

The resulting distribution for μ is a multivariate normal distribution with mean
＿
μ=[(τQ)^-1+P^TΩ^-1P]^-1[(τQ)^-1π+P^TΩ^-1q].

                                             ＿
Black and Litterman use μ as the vector of expected returns in the Markowitz model.

Example: Let us illustrate the Black-Litterman approach on the same example used for Mean Variance method. The expected returns on Stocks, Bonds and Money Market were computed to be

	Stocks	Bonds	MM
Market Rate of Return	10.73%	7.37%	6.27%

This is what we use for the vector π representing market equilibrium. We need to choose the value of the small constant τ. We take τ=0.1. We have two views that we would like to incorporate into the model. First, we hold on a strong view that the Money Market rate will be 2% next year. Second, we also hold the view that S&P500 will outperform 10-year Treasury Bonds by 5% but we are not as confident about this view. These two views are expressed as follows

μ_M = 0.02 strong view: ω₁ = 0.00001
μ_S - μ_B = 0.05 weaker view: ω₂ = 0.001

Thus P = 0 0 1 q = 0.02 and Ω = 0.00001 0
1 -1 0, 0.05 0 0.001
                                                    ＿
Applying the earlier formula to compute μ, we get

	Stocks	Bonds	MM
Mean Rate of Return	11.77%	7.51%	2.34%

We solve the same QP except for the modified expected return constraint:

min 0.02778x²_S + 2×0.00387x_Sx_B + 2×0.00021x_Sx_M + 0.01112x²_B - 2×0.00020x_Bx_M + 0.00115x²_M + 0.1177x_S + 0.0751x_B + 0.0234x_M
0.1177x_S + 0.0751x_B + 0.0234x_M ≧ R
x_S + x_B + x_{M = 1}
x_S, x_B, x_{M ≧ 0}




Solving for R=4.0% to R =11.5% with increments of 0.5% we now get the optimal portfolios and the efficient frontier.

[Quote]:Optimization Methods in Finance

We apply Markowitz's MVO model to the problem of constructing a portfolio of US stocks, bonds and cash. We use historical data for the returns of these three asset classes: The S&P 500 index for the returns on stocks, the 10-year Treasury bond index for the returns on bonds, and we assume that the cash is invested in a money market account whose return is the 1-day federal fund rate. The times series for the "Total  Return" are given below for each asset between 1960 and 2003.

Year	Stocks	Bonds	MM
1960	20.2553	262.935	100.00
1961	25.6860	268.730	102.33
1962	23.4297	284.090	105.33
1963	28.7463	289.162	108.89
1964	33.4484	299.894	113.08
1965	37.5813	302.695	117.97
1966	33.7839	318.197	124.34
1967	41.8725	309.103	129.94
1968	46.4795	316.051	137.77
1969	42.5448	298.249	150.12
1970	44.2212	354.671	157.48
1971	50.5451	394.532	164.00
1972	60.1461	403.942	172.74
1973	51.3114	417.252	189.93
1974	37.7306	433.927	206.13
1975	51.7772	457.885	216.85
1976	64.1659	529.141	226.93
1977	59.5739	531.144	241.82
1978	63.4884	524.435	266.07
1979	75.3032	531.040	302.74
1980	99.7795	517.860	359.96
1981	94.8671	538.769	404.48
1982	115.308	777.332	440.68
1983	141.316	787.357	482.42
1984	150.181	907.712	522.84
1985	197.829	1200.63	566.08
1986	234.755	1469.45	605.20
1987	247.755	1424.91	646.17
1988	288.116	1522.40	702.77
1989	379.409	1804.63	762.16
1990	367.636	1944.25	817.87
1991	479.633	2320.64	854.10
1992	516.178	2490.97	879.04
1993	568.202	2816.40	905.06
1994	575.705	2610.12	954.39
1995	792.042	3287.27	1007.84
1996	973.897	3291.58	1061.15
1997	1298.82	3687.33	1119.51
1998	1670.01	4220.24	1171.91
1999	2021.40	3903.32	1234.02
2000	1837.36	4575,33	1313.00
2001	1618.98	4827.26	1336.89
2002	1261.18	5558.40	1353.47
2003	1622.94	5588.19	1366.73

Let I_it denote the above "Total Return" for asset i = 1,2,3 and t = 0, ... T, where t = 0 corresponds to
1960 and t = T to 2003. For each asset i, we can convert the raw data I_it, t=0,...,T, into rates of returns r_it, t=1,...,T, using the formula.

                                                        r_it = I_i,t - I_{i, t-1} / I_{i, t-1}

Year	Stocks	Bonds	MM
1961	26.81	2.20	2.33
1962	-8.78	5.72	2.93
1963	22.69	1.79	3.38
1964	16.36	3.71	3.85
1965	12.36	0.93	4.32
1966	-10.10	5.12	5.40
1967	23.94	-2.86	4.51
1968	11.00	2.25	6.02
1969	-8.47	-5.63	8.97
1970	3.94	18.92	4.90
1971	14.30	11.24	4.14
1972	18.99	2.39	5.33
1973	-14.69	3.29	9.95
1974	-26.47	4.00	8.53
1975	37.23	5.52	5.20
1976	23.93	15.56	4.65
1977	-7.16	0.38	6.65
1978	6.57	-1.26	10.03
1979	18.61	-1.26	13.78
1980	32.50	-2.48	18.90
1981	-4.92	4.04	12.37
1982	21.55	44.28	8.95
1983	22.56	1.29	9.47
1984	6.27	15.29	8.38
1985	31.17	32.27	8.27
1986	18.67	22.39	6.91
1987	5.25	-3.03	6.77
1988	16.61	6.84	8.76
1989	31.69	18.54	8.45
1990	-3.10	7.74	7.31
1991	30.46	19.36	4.43
1992	7.62	7.34	2.92
1993	10.08	13.06	2.96
1994	1.32	-7.32	5.45
1995	37.58	25.94	5.60
1996	22.96	0.13	5.29
1997	33.36	12.02	5.50
1998	28.58	14.45	4.68
1999	21.04	-7.51	5.30
2000	-9.10	17.22	6.40
2001	-11.89	5.51	1.82
2002	-22.10	15.15	1.24
2003	28.68	0.54	0.98

Let R_i denote the random rate of return of asset i. From the above historical data, we can compute the arithmetric mean rate of return for each asset:

_             T
r_i = 1/T SUM r_it
                t=1

which gives

	Stocks	Bonds	MM
Arithmetric mean	12.06%	7.85%	6.32%

Because the rates of return are multiplicative over time, we prefer t use the geometric mean instead of the arithmetric mean. The geometric mean is the constant yearly rate of return that needs to be applied in years t = 0 through t = T-1 in order to get the compounded Total Return I_iT
, starting from I_i0. The formula for the geometric mean is:
         
ui = ( Π  (1+r_it) )^1/T -1
       t=1...T

We get the following results.

	Stocks	Bonds	MM
Geometric mean	10.73%	7.37%	6.27%

We also compute the covariance matrix:

                                                 _      _
cov(Ri, Rj) = (1/T)  SUM (r_it - r)(r_jt - r).
                              t=1...T

Covariance	Stocks	Bonds	MM
Stocks	0.02778	0.00387	0.00021
Bonds	0.00387	0.01112	-0.00020
MM	0.00021	-0.00020	0.00115

Although not needed to slolve the Markowitz model, it is interesting to compute the volatility of the rate of return on each asset σ_i =sqrt(cov(Ri, R,)):

	Stocks	Bonds	MM
Volatility	16.67%	10.55%	3.40%

and the correlation matrix p_ij = cov(R_i, R_j) / σ_iσj

Correlation	Stocks	Bonds	MM
Stocks	1	0.2199	0.0366
Bonds	0.2199	1	-0.0545
MM	0.0366	-0.0545	1

Setting up the QP for portfolio optimization

min 0.02778x² _S + 2×0.00387x_Sx_B + 2×0.00387x_Sx_M + 0.01112x_²B - 2×0.00020x_Bx_M + 0.00115x_²M + 0.1073x_S + 0.0737x_B + 0.0627x_M
0.1073x_S + 0.0737x_B + 0.0627x_M ≧ R
x_S + x_B + x_M = 1
x_S, x_B, x_M ≧ 0

and solving it for R=6.5% to R=10.5% with increments of 0.5% we get the optimal portfolios shown below and the corresponding variance. The optimal allocations on the efficient frontier are also depicted in the graph below.

Rate of Return R	Variance	Stocks	Bonds	MM
0.065	0.0010	0.03	0.10	0.87
0.070	0.0014	0.13	0.12	0.75
0.075	0.0026	0.24	0.14	0.62
0.080	0.0044	0.35	0.16	0.49
0.085	0.0070	0.45	0.18	0.37
0.090	0.0102	0.56	0.20	0.24
0.095	0.0142	0.67	0.22	0.11
0.100	0.0189	0.78	0.22	0
0.105	0.0246	0.93	0.07	0

Based on the first two columns of table above, the graph plots the maximum expected rate of return R of a portfolio as a function of its volatility (standard deviation). This curve is called the effecient frontier. Every possible portfolio of Stocks/Bonds/MM is represented by a point lying on or below the efficient frontier in the expected return/standard deviation plane.

[Quote]:Optimization Methods in Finance

In Part 1 of this series we show three methods for calculating the value at risk (VAR or VaR) of a single stock investment. Here in part 2 we explain how to convert one VAR of one time period into the equivalent VAR for a different time period and show you how to use VAR to estimate the downside risk of a single stock investment.
 
Converting One Time Period to Another
In Part 1, we calculate VAR for the Nasdaq 100 index (ticker: QQQ) and establish that VAR answers a three-part question: "What is the worst loss that I can expect during a specified time period with a certain confidence level?"

Since the time period is a variable, different calculations may specify different time periods - there is no "correct" time period. Commercial banks, for example, typically calculate a daily VAR, asking themselves how much they can lose in a day; pension funds, on the other hand, often calculate a monthly VAR.

To recap briefly, let's look again at our calculations of three VARs in part 1 using three different methods for the same "QQQ" investment:

* We do not need a standard deviation for neither the historical method (because it just re-orders returns lowest-to-highest) or the Monte Carlo simulation (because it produces the final results for us).

Because of the time variable, users of VAR need to know how to convert one time period to another, and they can do so by relying on a classic idea in finance: the standard deviation of stock returns tends to increase with the square root of time. If the standard deviation of daily returns is 2.64% and there are 20 trading days in a month (T = 20), then the monthly standard deviation is represented by the following:


To "scale" the daily standard deviation to a monthly standard deviation, we multiply it not by 20 but by the square root of 20. Similarly, if we want to scale the daily standard deviation to an annual standard deviation, we multiply the daily standard deviation by the square root of 250 (assuming 250 trading days in a year). Had we calculated a monthly standard deviation (which would be done by using month-to-month returns), we could convert to an annual standard deviation by multiplying the monthly standard deviation by the square root of 12.

Applying a VAR Method to a Single Stock
Both the historical and Monte Carlo simulation methods have their advocates; but the historical method requires crunching historical data, and the Monte Carlo simulation method is complex. The easiest method is variance-covariance.

Below we incorporate the time-conversion element into the variance-covariance method for a single stock (or single investment):


Now let's apply these formulas to the QQQ. Recall that the daily standard deviation for the QQQ since inception is 2.64%. But we want to calculate a monthly VAR, and assuming 20 trading days in a month, we multiply by the square root of 20:

* Important Note: These worst losses (-19.5% and -27.5%) are losses below the expected or average return. In this case, we keep it simple by assuming the daily expected return is zero. We rounded down, so the worst loss is also the net loss.

So, with the variance-covariance method, we can say with 95% confidence that we will not lose more than 19.5% in any given month. The QQQ clearly is not the most conservative investment! You may note, however, that the above result is different from the one we got under the Monte Carlo simulation, which said our maximum monthly loss would be 15% (under the same 95% confidence level).

Conclusion
Value at risk is a special type of downside risk measure. Rather than produce a single statistic or express absolute certainty, it makes a probabilistic estimate. With a given confidence level, it asks, "What is our maximum expected loss over a specified time period?" There are three methods by which VAR can be calculated: the historical simulation, the variance-covariance method and the Monte Carlo simulation.

The variance-covariance method is easiest because you need to estimate only two factors: average return and standard deviation. However, it assumes returns are well-behaved according to the symmetrical normal curve and that historical patterns will repeat into the future.

The historical simulation improves on the accuracy of the VAR calculation, but requires more computational data; it also assumes that "past is prologue". The Monte Carlo simulation is complex, but has the advantage of allowing users to tailor ideas about future patterns that depart from historical patterns.
[Quote]:http://www.investopedia.com/articles/04/101304.asp

Value at risk (VAR or sometimes VaR) has been called the "new science of risk management", but you do not need to be a scientist to use VAR. Here, in part 1 of this series, we look at the idea behind VAR and the three basic methods of calculating it. In Part 2, we apply these methods to calculating VAR for a single stock or investment.

The Idea behind VAR
The most popular and traditional measure of risk is volatility. The main problem with volatility, however, is that it does not care about the direction of an investment's movement: a stock can be volatile because it suddenly jumps higher. Of course, investors are not distressed by gains!  (See The Limits and Uses of Volatility.)

For investors, risk is about the odds of losing money, and VAR is based on that common-sense fact. By assuming investors care about the odds of a really big loss, VAR answers the question, "What is my worst-case scenario?" or "How much could I lose in a really bad month?"

Now let's get specific. A VAR statistic has three components: a time period, a confidence level and a loss amount (or loss percentage). Keep these three parts in mind as we give some examples of variations of the question that VAR answers:

• What is the most I can - with a 95% or 99% level of confidence -  expect to lose in dollars over the next month?
• What is the maximum percentage I can - with 95% or 99% confidence - expect to lose over the next year?

You can see how the "VAR question" has three elements: a relatively high level of confidence (typically either 95% or 99%), a time period (a day, a month or a year) and an estimate of investment loss (expressed either in dollar or percentage terms).

Methods of Calculating VAR
Institutional investors use VAR to evaluate portfolio risk, but in this introduction we will use it to evaluate the risk of a single index that trades like a stock: the Nasdaq 100 Index, which trades under the ticker QQQQ. The QQQQ is a very popular index of the largest non-financial stocks that trade on the Nasdaq exchange.

There are three methods of calculating VAR: the historical method, the variance-covariance method and the Monte Carlo simulation.

1. Historical Method
The historical method simply re-organizes actual historical returns, putting them in order from worst to best. It then assumes that history will repeat itself, from a risk perspective.

The QQQ started trading in Mar 1999, and if we calculate each daily return, we produce a rich data set of almost 1,400 points. Let's put them in a histogram that compares the frequency of return "buckets". For example, at the highest point of the histogram (the highest bar), there were more than 250 days when the daily return was between 0% and 1%. At the far right, you can barely see a tiny bar at 13%; it represents the one single day (in Jan 2000) within a period of five-plus years when the daily return for the QQQ was a stunning 12.4%!


Notice the red bars that compose the "left tail" of the histogram. These are the lowest 5% of daily returns (since the returns are ordered from left to right, the worst are always the "left tail"). The red bars run from daily losses of 4% to 8%. Because these are the worst 5% of all daily returns, we can say with 95% confidence that the worst daily loss will not exceed 4%. Put another way, we expect with 95% confidence that our gain will exceed -4%. That is VAR in a nutshell. Let's re-phrase the statistic into both percentage and dollar terms:

• With 95% confidence, we expect that our worst daily loss will not exceed 4%.
• If we invest $100, we are 95% confident that our worst daily loss will not exceed $4 ($100 x -4%).

You can see that VAR indeed allows for an outcome that is worse than a return of -4%. It does not express absolute certainty but instead makes a probabilistic estimate. If we want to increase our confidence, we need only to "move to the left" on the same histogram, to where the first two red bars, at -8% and -7% represent the worst 1% of daily returns:

• With 99% confidence, we expect that the worst daily loss will not exceed 7%.
• Or, if we invest $100, we are 99% confident that our worst daily loss will not exceed $7.

2. The Variance-Covariance Method
This method assumes that stock returns are normally distributed. In other words, it requires that we estimate only two factors - an expected (or average) return and a standard deviation - which allow us to plot a normal distribution curve. Here we plot the normal curve against the same actual return data:


The idea behind the variance-covariance is similar to the ideas behind the historical method - except that we use the familiar curve instead of actual data. The advantage of the normal curve is that we automatically know where the worst 5% and 1% lie on the curve. They are a function of our desired confidence and the standard deviation ():


The blue curve above is based on the actual daily standard deviation of the QQQ, which is 2.64%. The average daily return happened to be fairly close to zero, so we will assume an average return of zero for illustrative purposes. Here are the results of plugging the actual standard deviation into the formulas above:


3. Monte Carlo Simulation
The third method involves developing a model for future stock price returns and running multiple hypothetical trials through the model. A Monte Carlo simulation refers to any method that randomly generates trials, but by itself does not tell us anything about the underlying methodology.

For most users, a Monte Carlo simulation amounts to a "black box" generator of random outcomes. Without going into further details, we ran a Monte Carlo simulation on the QQQ based on its historical trading pattern. In our simulation, 100 trials were conducted. If we ran it again, we would get a different result--although it is highly likely that the differences would be narrow. Here is the result arranged into a histogram (please note that while the previous graphs have shown daily returns, this graph displays monthly returns):


To summarize, we ran 100 hypothetical trials of monthly returns for the QQQ. Among them, two outcomes were between -15% and -20%; and three were between -20% and 25%. That means the worst five outcomes (that is, the worst 5%) were less than -15%. The Monte Carlo simulation therefore leads to the following VAR-type conclusion: with 95% confidence, we do not expect to lose more than 15% during any given month.

Summary
Value at Risk (VAR) calculates the maximum loss expected (or worst case scenario) on an investment, over a given time period and given a specified degree of confidence. We looked at three methods commonly used to calculate VAR. But keep in mind that two of our methods calculated a daily VAR and the third method calculated monthly VAR. In Part 2 of this series we show you how to compare these different time horizons.
[Quote]:http://www.investopedia.com/articles/04/092904.asp