Over 35 years, there are some remarkable points out of the historical performance of the three assets:
Firstly, the mean (average return) of each individual asset exceeds the risk-free rate of return, in other words, the fund has gained a positive risk premium over the period. That has been an incentive for investing in risky assets.
Secondly, as expected from wealth cumulative wealth analysis, the smallest expected rate of return goes to US bonds and the highest is US large cap stocks. The graph above also contains information about the volatility of each asset class’s return. The order of standard deviation is not the same as the order of expected return. Based on the order of average return, we expect US large cap stocks to have highest standard deviation, followed by International stocks and the least risky asset is US bonds. However, US large cap stocks have smaller standard deviation than International stocks. Because US large cap stocks have higher return and lower volatility than International stocks, US large cap stocks dominate International stocks. We refer this as the unconditional dominance because the expected return is proxied by a long historical average. As an investor, we really care about future returns. The future average return is not necessarily the same as historical average return.
isky assets gained trsification benefits from diversifying out future returns. The future average return Furthermore, the correlation coefficients between any 2 assets are all below 1, which means that they are not perfectly positive correlated so there will be some risk reducing benefits from diversifying across these three risky assets. The combined portfolio of the three assets exposed to less systematic risk than each individual asset did.
Input data | A | B | C | D | 1 | Annualised standard deviation, average return and correlation coefficients | | 2 | of three risky assets, 35 years ending 12/2008 | | 3 | | Mean | Standard Deviation | | 4 | US Large Cap Stocks | 11.0 | 19.0 | | 5 | International Stocks | 10.0 | 23.0 | | 6 | US Bonds | 8.5 | 6.8 | | 7 | | Correlation matrix | | | 8 | | US Large Cap Stocks | International Stocks | US Bonds | 9 | US Large Cap Stocks | 1 | 0.59 | 0.14 | 10 | International Stocks | 0.59 | 1 | 0.09 | 11 | US Bonds | 0.14 | 0.09 | 1 |
Covariance matrix for the three asset classes is below | A | B | C | D | 14 | Covariance matrix: cell formulae | | 15 | | US Large Cap Stocks | International Stocks | US Bonds | 16 | US Large Cap Stocks | =c4*c4*b9 | =c5*c4*c9 | =c6*c4*d9 | 17 | International Stocks | =c4*c5*b10 | =c5*c5*c10 | =c6*c5*d10 | 18 | US Bonds | =c4*c6*b11 | =c5*c6*c11 | =c6*c6*d11 | 19 | | Covariance matrix result | | | 20 | | US Large Cap Stocks | International Stocks | US Bonds | 21 | US Large Cap Stocks | 361.000 | 257.830 | 18.088 | 22 | International Stocks | 257.830 | 529.000 | 14.076 | 23 | US Bonds | 18.088 | 14.076 | 46.240 |
PART C | A | B | C | D | E | 25 | Border-multiplied covariance matrix for the equally weighted risky portfolio and portfolio variance | 26 | | cell formulae | | | 27 | | | US Large Cap Stocks | International Stocks | US Bonds | 28 | | Weights | b29 | b30 | b31 | 29 | US Large Cap Stocks | 0.333 | c29*b30*b22 | d29*b30*c22 | e29*b30*d22 | 30 | International Stocks | 0.333 | c29*b31*b23 | d29*b31*c23 | e29*b31*d23 | 31 | US Bonds | 0.333 | c29*b32*b24 | d29*b32*c24 | e29*b32*d24 | 32 | | sum(b29:b31) | sum(c29:c31) | sum(d29:d31) | sum(e29:e31) | 33 | Portfolio variance | sum(c32:e32) | | | | 34 | Portfolio SD | b33^.5 | | | | 35 | Portfolio mean | b4*b29+b5*b30+b6*b31 | | | | | A | B | C | D | E | 25 | Border-multiplied covariance matrix for the equally weighted risky