Markov's trilemma3 asset allocation model

CORRELATIONAsset weights exp return std .dev Sharpe ratio GMGM 0.333333 21.30% 37.60% 0.38 GM 1MRK 0.333333 41.30% 30.10% 1.14 MRK 0.12GE 0.333333 41.00% 24.00% 1.42 GE 0.26

10.333333 0.333333 0.333333

COVARIANCE Risk free rate 7% GMExpected return 34.53% GM 0.141376variance 0.045223 MRK 0.013581Exp st dev 21.27% GE 0.023462Sharpe 1.294728

CORRELATIONMRK GE

0.12 0.261 0.3

0.3 1

COVARIANCE MRK GE0.013581 0.0234620.090601 0.0216720.021672 0.0576

Markov's trilemma3 asset allocation model

CORRELATIONAsset weights exp return std .dev Sharpe ratio GMGM 0 21.30% 37.60% 0.38 GM 1MRK 0.346438 41.30% 30.10% 1.14 MRK 0.12GE 0.653562 41.00% 24.00% 1.42 GE 0.26

10 0.346438 0.653562

COVARIANCE Risk free rate 7% GMExpected return 41.10% GM 0.141376variance 0.045291 MRK 0.013581Exp st dev 21.28% GE 0.023462Sharpe 1.602498

CORRELATIONMRK GE

0.12 0.261 0.3

0.3 1

COVARIANCE MRK GE0.013581 0.0234620.090601 0.0216720.021672 0.0576

Markov's trilemma3 asset allocation model

CORRELATIONAsset weights exp return std .dev Sharpe ratio GmGM -0.008004 21.30% 37.60% 0.38 Gm 1MRK 0.348443 41.30% 30.10% 1.14 MRK 0.12GE 0.659561 41.00% 24.00% 1.42 GE 0.26

1-0.008004 0.348443 0.659561

COVARIANCE Risk free rate 7% GmExpected return 41.26% Gm 0.141376variance 0.045704 MRK 0.013581Exp st dev 0.213785 GE 0.023462Sharpe 1.602646

The Sharpe ratio will not change significantly because the amount of shorting of the GM stock is very little.

CORRELATIONMRK GE

0.12 0.261 0.3

0.3 1

COVARIANCE MRK GE0.013581 0.0234620.090601 0.0216720.021672 0.0576

The Sharpe ratio will not change significantly because the amount of shorting of the GM stock is very little.

Markov's trilemma3 asset allocation model

CORRELATIONAsset weights exp return std .dev Sharpe ratio GMGM -0.613597 21.30% 37.60% 0.38 GM 1MRK 0.213352 41.30% 30.10% 1.14 MRK 0.12GE 1.400245 41.00% 24.00% 1.42 GE 0.8

1-0.613597 0.213352 1.400245

COVARIANCE Risk free rate 7% GMExpected return 53.15% GM 0.141376variance 0.055628 MRK 0.013581Exp st dev 0.235856 GE 0.072192Sharpe 1.956782

The GM returns are lower for high std. dev as compared to other stocks and therefore, including GM in the portfolio and going long will lower the sharpe ratio.The best option to maximize the sharpe ratio is to short the GM stock and buy more of GE as GE returns are higher for lesser risk.The positive correlation between the securities can be exploited to increase the sharpe ratio only if the shorting is allowed in the portfolio

CORRELATIONMRK GE

0.12 0.81 0.3

0.3 1

COVARIANCE MRK GE0.013581 0.0721920.090601 0.0216720.021672 0.0576

The GM returns are lower for high std. dev as compared to other stocks and therefore, including GM in the portfolio and going long will lower the sharpe ratio.The best option to maximize the sharpe ratio is to short the GM stock and buy more of GE as GE returns are higher for lesser risk.The positive correlation between the securities can be exploited to increase the sharpe ratio only if the shorting is allowed in the portfolio

Markov's trilemma3 asset allocation model

CORRELATIONAsset weights exp return std .dev Sharpe ratio GMGM 0.408906 21.30% 37.60% 0.38 GM 1MRK -0.131178 41.30% 30.10% 1.14 MRK 0.12GE 0.722273 41.00% 24.00% 1.42 GE -0.8

10.408906 -0.131178 0.722273

COVARIANCE Risk free rate 7% GMExpected return 32.91% GM 0.141376variance 0.00704 MRK 0.013581Exp st dev 8.39% GE -0.072192Sharpe 3.087443

The sharpe ratio has increased significantly because of the -ve correlation between the GE and GM stocks.The -ve correlation between the stocks has reduced the overall risk in the portfolio and therefore the std dev. This led to an increase in the sharpe ratio.

CORRELATIONMRK GE

0.12 -0.81 0.3

0.3 1

COVARIANCE MRK GE0.013581 -0.0721920.090601 0.0216720.021672 0.0576

The sharpe ratio has increased significantly because of the -ve correlation between the GE and GM stocks.The -ve correlation between the stocks has reduced the overall risk in the portfolio and therefore the std dev. This led to an increase in the sharpe ratio.

Markov's trilemma3 asset allocation model

CORRELATIONAsset weights exp return std .dev Sharpe ratio GMGM 0.077475 21.30% 37.60% 0.38 GM 1MRK 0.643009 41.30% 30.10% 1.14 MRK 0.12GE 0.279515 30.00% 30.00% 0.77 GE 0.26

10.077475 0.643009 0.279515

COVARIANCE Risk free rate 7% GMExpected return 36.59% GM 0.141376variance 0.057701 MRK 0.013581Exp st dev 0.240211 GE 0.029328Sharpe 1.231915

The sharpe ratio has fallen.The MRK weight in the portfolio will increase as the stock returns are higher for same risk when compared to the GE stock.Again, the weight of GM is very low as the stock returns are lower with higher risk.

CORRELATIONMRK GE

0.12 0.261 0.3

0.3 1

COVARIANCE MRK GE0.013581 0.0293280.090601 0.02709

0.02709 0.09

The MRK weight in the portfolio will increase as the stock returns are higher for same risk when compared to the GE stock.

Markov's trilemma3 asset allocation model

CORRELATIONAsset weights exp return std .dev Sharpe ratio GMGM 0 21.30% 37.60% 0.38 GM 1MRK 0.346438 41.30% 30.10% 1.14 MRK 0.12GE 0.653562 41.00% 24.00% 1.42 GE 0.26

10 0.346438 0.653562

COVARIANCE Risk free rate 7% GMExpected return 41.10% GM 0.141376variance 0.045291 MRK 0.013581Exp st dev 21.28% GE 0.023462Sharpe 1.602498

If no shorting is allowed, the investor will not include GM in its portfolio as it will induce lesser returns for higher risk in the portfolio.The portfolio's sharpe ratio will have no change as compared to the base case

CORRELATIONMRK GE

0.12 0.261 0.3

0.3 1

COVARIANCE MRK GE0.013581 0.0234620.090601 0.0216720.021672 0.0576

If no shorting is allowed, the investor will not include GM in its portfolio as it will induce lesser returns for higher risk in the portfolio.

Markov's trilemma3 asset allocation model

CORRELATIONAsset weights exp return std .dev Sharpe ratio GMGM 0 21.30% 37.60% 0.38 GM 1MRK 0.346438 41.30% 30.10% 1.14 MRK 0.12GE 0.653562 41.00% 24.00% 1.42 GE 0.8

10 0.346438 0.653562

COVARIANCE Risk free rate 7% GMExpected return 41.10% GM 0.141376variance 0.045291 MRK 0.013581Exp st dev 21.28% GE 0.072192Sharpe 1.602498

Even if the correlation between GM and GE increases to 0.8, the Sharpe ratio will not increase as the investor will not include GM in his portfolio without shorting.As discussed earlier, positive correlation between the securities can be leveraged only if one of the security is shorted.

CORRELATIONMRK GE

0.12 0.81 0.3

0.3 1

COVARIANCE MRK GE0.013581 0.0721920.090601 0.0216720.021672 0.0576

Even if the correlation between GM and GE increases to 0.8, the Sharpe ratio will not increase as the investor will not include GM in his portfolio without shorting.As discussed earlier, positive correlation between the securities can be leveraged only if one of the security is shorted.

Markov's trilemma3 asset allocation model

CORRELATIONAsset weights exp return std .dev Sharpe ratio GMGM 0.359558 21.30% 37.60% 0.38 GM 1MRK 0 41.30% 30.10% 1.14 MRK 0.12GE 0.640442 41.00% 24.00% 1.42 GE -0.8

10.359558 0 0.640442

COVARIANCE Risk free rate 7% GMExpected return 33.92% GM 0.141376variance 0.008655 MRK 0.013581Exp st dev 9.30% GE -0.072192Sharpe 2.893308

In this case, the high -ve correlation between the GM and GE securities can be used effectively to reduce the risk in the portfolio.As a result, Sharpe ratio will increase as the risk in the portfolio is reduced as a result of diversification.

CORRELATIONMRK GE

0.12 -0.81 0.3

0.3 1

COVARIANCE MRK GE0.013581 -0.0721920.090601 0.0216720.021672 0.0576

In this case, the high -ve correlation between the GM and GE securities can be used effectively to reduce the risk in the portfolio.As a result, Sharpe ratio will increase as the risk in the portfolio is reduced as a result of diversification.

Markov's trilemma3 asset allocation model

CORRELATIONAsset weights exp return std .dev Sharpe ratio GMGM 0.052357 21.30% 37.60% 0.38 GM 1MRK 0.447644 41.30% 30.10% 1.14 MRK 0.12GE 0.5 41.00% 24.00% 1.42 GE 0.26

1.0000010.052357 0.447644 0.5

COVARIANCE Risk free rate 7% GMExpected return 40.10% GM 0.141376variance 0.044509 MRK 0.013581Exp st dev 21.10% GE 0.023462Sharpe 1.569069

The returns and sharpe ratio will get negatively affected because of the constraints.Because the innvestor is not allowed to invest more than 50% in any one of the securities, he is bound to invest some amount in other non-high return securities.As a result, the overall returns of the portfolio will be lower and the Sharpe ratio will fall.

CORRELATIONMRK GE

0.12 0.261 0.3

0.3 1

COVARIANCE MRK GE0.013581 0.0234620.090601 0.0216720.021672 0.0576

Because the innvestor is not allowed to invest more than 50% in any one of the securities, he is bound to invest some amount in other non-high return securities.

Markov's trilemma3 asset allocation model

CORRELATIONAsset weights exp return std .dev Sharpe ratio GMGM 0.003085 21.30% 37.60% 0.43 GM 1MRK 0.345455 41.30% 30.10% 1.21 MRK 0.12GE 0.65146 41.00% 24.00% 1.50 GE 0.26

10.003085 0.345455 0.65146

COVARIANCE Risk free rate 5% GMExpected return 41.04% GM 0.141376variance 0.045137 MRK 0.013581Exp st dev 21.25% GE 0.023462Sharpe 1.696499

The decrease in the Risk free rate has increased the sharpe ratio because the portfolio will earn higher returns than expected.The excess return over the risk free rate will lead to an increase in the Sharpe ratio.This type of phenomenon is very rare in the world as the expected or required rates of return will also decline with the decrease in the risk free rate.

CORRELATIONMRK GE

0.12 0.261 0.3

0.3 1

COVARIANCE MRK GE0.013581 0.0234620.090601 0.0216720.021672 0.0576

The decrease in the Risk free rate has increased the sharpe ratio because the portfolio will earn higher returns than expected.

This type of phenomenon is very rare in the world as the expected or required rates of return will also decline with the decrease in the risk free rate.

Markov's trilemma3 asset allocation model

CORRELATIONAsset weights exp return std .dev Sharpe ratio INTELINTEL 0.302423 68.80% 43.80% 1.41 INTEL 1MRK 0.105523 41.30% 30.10% 1.14 MRK 0.52GE 0.592054 41.00% 24.00% 1.42 GE 0.2

10.302423 0.105523 0.592054

COVARIANCE Risk free rate 7% INTELExpected return 49.44% INTEL 0.191844variance 0.053357 MRK 0.068556Exp st dev 23.10% GE 0.021024Sharpe 1.837247

CORRELATIONMRK GE

0.52 0.21 0.3

0.3 1

COVARIANCE MRK GE0.068556 0.0210240.090601 0.0216720.021672 0.0576

Markov's trilemma3 asset allocation model

CORRELATIONAsset weights exp return std .dev Sharpe ratio INTELINTEL 0.302423 68.80% 43.80% 1.41 INTEL 1MRK 0.105523 41.30% 30.10% 1.14 MRK 0.52GE 0.592054 41.00% 24.00% 1.42 GE 0.2

10.302423 0.105523 0.592054

COVARIANCE Risk free rate 7% INTELExpected return 49.44% INTEL 0.191844variance 0.053357 MRK 0.068556Exp st dev 23.10% GE 0.021024Sharpe 1.837247

CORRELATIONMRK GE

0.52 0.21 0.3

0.3 1

COVARIANCE MRK GE0.068556 0.0210240.090601 0.0216720.021672 0.0576