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If the assets are perfectly negatively correlated, what is the minimum risk combination of the two assets?

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Question - Consider two assets with the following characteristics and assume that short selling is not allowed:

 

Standard Deviation

Asset 1

3%

Asset 2

6%

(i) If the assets are independent, what is the minimum risk combination of the two assets? If the assets are perfectly negatively correlated, what is the minimum risk combination of the two assets?

Answer - In general, the minimum variance portfolio, of a pair of assets when short sales are not allowed is

X1 = (2σ2 - σ12)/(1σ2 + 2σ2 - 2σ12)

When the assets are independent, ρ equals 0, the minimum-risk combination of two assets can be found by solving X1 = 2σ2/(1σ2 + 2σ2). So, X1 = 36/(36 + 9) = 36/45= 0.8, and since the investment weights must sum to 1, X2 = 1 - X1 = 1 - 36/45 = 9/45 = 0.2.

When the assets are perfectly negatively correlated, ρ equals -1 and we can always find a combination of the two securities that will completely eliminate risk, and we saw that this combination can be found by solving X1 = σ2/(σ1 + σ2). So, X1 = 6/(3 + 6) = 6/9 = 2/3 = 0.66, and since the investment weights must sum to 1, X2 = 1 - X1 = 1 - 6/9 = 3/9 = 1/3 = 0.33.

So a combination of 2/3 invested in security 1 and 1/3 invested in security 2 will completely eliminate risk when ρ equals -1, and σP will equal 0.

(ii) Explain the momentum effect.

Answer - The original empirical work supporting the notion of randomness in stock prices supported the view that the stock market has no memory.

More recent work by Lo and MacKinlay (1999) finds that short-run serial correlations are not zero and that the existence of "too many" successive moves in the same direction enable them to reject the hypothesis that stock prices behave as random walks. There does seem to be some momentum (i.e. the tendency for prices to continue in the same direction, either rising or falling) in short-run stock prices.

Economists and psychologists in the field of behavioural finance (that is, finance from a broader social science perspective including psychology and sociology) find such short-run momentum to be consistent with psychological feedback mechanisms.

For example, Shiller (2000) describes the rise in the U.S. stock market during the late 1990s as the result of psychological contagion leading to irrational exuberance.

As behavioural finance became more prominent as a branch of the study of financial markets, momentum, as opposed to randomness, seemed reasonable to many investigators.

However, there are several factors that should prevent us from interpreting the empirical results reported above as an indication that markets are inefficient.

First, it is important to distinguish statistical significance from economic significance. The statistical dependencies giving rise to momentum are extremely small and are not likely to permit investors to realize excess returns.

Odean (1999) suggests that momentum investors do not realize excess returns. This is so because of the large transactions costs involved in attempting to exploit whatever momentum exists.

Second, while behavioural hypotheses sound plausible enough, the evidence that such effects occur systematically in the stock market is often rather thin.

The key factor is whether any patterns of serial correlation are consistent over time. Momentum strategies, which refer to buying stocks that display positive serial correlation, appeared to produce positive relative returns during some periods of the late 1990s but highly negative relative returns during 2000.

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