#### A. Household Risk Exposures in Sweden

A weakness of this literature is that it cannot directly measure households' risk exposures. Surveys do not identify individual stocks or mutual funds, and brokerage accounts do not reveal total portfolios. In joint research with Laurent Calvet and Paolo Sodini (Calvet et al. (2006)), we use Swedish data to look more directly at the idiosyncratic risk in Swedish household portfolios. We adopt the perspective that systematic risk is compensated and idiosyncratic risk is not, so that taking idiosyncratic risk is an investment mistake. Since the time dimension of our data set is short, we do not attempt to measure the performance of Swedish household portfolios directly.

The Swedish data appear to be broadly consistent with U.S. data as regards asset allocation: At the aggregate level, real estate accounts for over 70% of household assets, bank deposits and money market funds for 11%, directly held stocks and mutual funds for 6% each, and bonds, derivatives, and capital insurance products for the remainder. At the end of 2002, 62% of households participated in financial markets by holding financial assets other than bank deposits and money market funds.

We construct a sample of 100,000 households and measure the composition of their portfolios at the end of 2002, down to the level of individual stocks and mutual funds. We calculate the risk properties of these portfolios by estimating a variance–covariance matrix **Σ** for the returns of all stocks and mutual funds held by Swedish households. Then, if a household *h* has portfolio weight vector **ω**_{h}, the variance of its portfolio return is estimated as **ω**′_{h}**Σω**_{h}. This procedure captures the risk in household portfolios at a point in time; it does not track the trading decisions of households within the year.

The median household in our sample has a risky portfolio standard deviation of about 20%. Part of this standard deviation comes from exposure to systematic risk in the world equity market, and part comes from unsystematic risk. As a measure of systematic risk, we calculate the standard deviation of the fitted value in a regression of each household's portfolio return on the dollar excess return of the MSCI All World Index. For the median household, this systematic standard deviation is slightly less than the standard deviation of the residual, a measure of unsystematic risk, implying that more than half of the median Swedish household's portfolio variance is idiosyncratic.

Although Swedish households can obtain the dollar excess return on international stocks by hedging their currency exposure when they invest internationally, this may be an unrealistic benchmark given that international equity funds widely marketed in Sweden are not currency hedged. If we repeat the above exercise with the Swedish Krona excess return on the MSCI index, we find that slightly less than half of the median household's portfolio variance is idiosyncratic.

While the median standard deviation of the risky portfolio is about 20%, there is wide variation in this number across households. Some households take low risk and hold primarily bond funds; others take high risk. The 95^{th} percentile of the risky portfolio standard deviation is about 50%, and the 99^{th} percentile is 70%. Portfolios with this level of risk tend to have betas above one, but they also have extremely high shares of idiosyncratic as opposed to systematic risk.

The above analysis treats all assets in the portfolio equally, whether they are stocks or mutual funds. An alternative approach is to assume that mutual funds are fully diversified, with zero idiosyncratic risk. Let *D*_{h} denote the share of directly held stocks in the risky portfolio, and let *C*_{sh}=*C*_{ah}/*D*^{2}_{h} denote the concentration of the stock portion of the portfolio. Then

- (3)

where *s* subscripts denote the characteristics of the directly held stocks in the portfolio. This alternative decomposition attributes idiosyncratic risk to a high share of stocks rather than mutual funds in the portfolio, volatile stocks, a concentrated stock portfolio, and correlated stocks.

In the Swedish data, we find that portfolios with high idiosyncratic risk tend to have high shares of directly owned stocks, and the directly owned portfolios tend to be concentrated in one or two volatile stocks. Concentration, however, can be a misleading statistic; many portfolios with low idiosyncratic risk also contain one or two directly owned stocks, but these portfolios are dominated by mutual funds and contain only a small share of directly owned stocks. This pattern illustrates the danger of looking only at the number of directly held stocks in a portfolio without considering the broader context within which those stocks are held. Correlation across stocks in the portfolio contributes very little to the cross-sectional risk pattern in Swedish portfolios.

In order to evaluate the consequences of underdiversification for household welfare, we assume that mean returns on stocks and mutual funds obey an international asset pricing model (either the CAPM or the three-factor Fama–French (1993) model, estimated in dollars). This assumption avoids the difficult task of estimating average returns on individual stocks and mutual funds from short historical time series, while enabling us to plot Swedish household portfolios on a mean-standard deviation diagram. By assumption, all portfolios must fall below the efficient frontier, which in the case of the international CAPM is a straight line connecting the riskless rate to the currency-hedged return on the MSCI world index. We find that many household portfolios come close to the Sharpe ratio of the unhedged world index (which we estimate at 35%), but almost none attain the efficient Sharpe ratio of the currency-hedged world index, which we estimate at 45%. The Swedish domestic equity index, with an estimated Sharpe ratio of 27%, lies within the middle of the distribution of household portfolios.

There are several ways to measure portfolio inefficiency within this framework. One is to calculate the percentage difference between a household portfolio's Sharpe ratio *S*_{h} and the Sharpe ratio of a benchmark index *S*_{B}, 1 −*S*_{h}/*S*_{B}. A second approach is to calculate the return lost, at the portfolio's given standard deviation, by the lower Sharpe ratio of the household portfolio. This is *w*_{h}(*S*_{B}σ_{h}−μ_{h}), where *w*_{h} is the portfolio's weight in risky assets, σ_{h} is the standard deviation of the household's risky portfolio return, and μ_{h} is the mean of that return. A third approach is to calculate the utility lost by a household that correctly perceives its own Sharpe ratio, and chooses its risk optimally given its risk aversion, but fails to understand that a higher Sharpe ratio is available by investing efficiently. This utility loss is equivalent to a decrease in the riskless interest rate of *w*_{h}σ_{h}(*S*^{2}_{B}−*S*^{2}_{h})/2*S*_{h}.

According to the first measure of portfolio inefficiency, the median Swedish household gives up slightly more than a third of the maximum available Sharpe ratio if the international CAPM holds, and slightly less than a third if the international Fama–French model holds. The difference is caused by the fact that Swedish household portfolios are tilted towards small stocks and value stocks, which earn higher returns in the Fama–French model than in the CAPM. The Sharpe ratio loss is reduced by more than half if we take as our benchmark the world index in Swedish kronas rather than the currency-hedged world index. The median Swedish household portfolio has a higher Sharpe ratio than the Swedish equity index, reflecting the fact that many Swedish households hold global equity mutual funds.

Reductions in Sharpe ratios have little effect on portfolio returns if households invest conservatively. Return loss, the second measure of portfolio inefficiency, places greater weight on low Sharpe ratios that are accompanied by aggressive investment strategies. If converted to dollars by multiplying by portfolio value, it also places greater weight on large portfolios. The median Swedish household loses not much more than 1% or $100 per year relative to the currency-hedged world index under the CAPM. Relative to the unhedged world index, the median household loses only one-quarter as much. Clearly portfolio underdiversification has only modest effects on the welfare of the median Swedish household.

Once again, however, there is wide variation in these numbers across households. At the right tail of the distribution of return losses, these losses are substantial. The 95^{th} percentile of the return loss, relative to the hedged world index, is almost 5 times greater than the median in percentage units, and over 15 times greater in dollar units. In dollar units, the 95^{th} percentile of the loss is over $2,200 per year relative to the hedged world index, and almost $850 per year relative to the unhedged index.

These numbers suggest that underdiversification is a problem for a minority of households. A natural next question is which households lose the most by inefficient investing. If the CAPM holds, the overall return loss can be written as the product of three household-specific and one market-wide component

- (4)

Here, *w*_{h} is the share of the household's portfolio invested in risky assets, β_{h} is the beta of those risky assets with the benchmark portfolio, (*S*_{B}/*S*_{h}− 1) is a transformed measure of the relative Sharpe ratio, and E *r*^{e}_{m} is the expected excess return on the world market portfolio.

In Calvet et al. (2006) we take logs of both sides of this equation and then regress the log return loss and its three household-specific components onto demographic characteristics of households. We find offsetting effects on return losses. On the one hand, financially sophisticated households with high disposable income, wealth, education, private pension savings, and financial liabilities tend to invest more aggressively. They invest a higher fraction of their wealth in risky assets, and those assets have higher betas. On the other hand, these households also tend to invest more efficiently, consistent with the findings of Goetzmann and Kumar (2004) for U.S. brokerage account data. In the Swedish data we find that the first effect dominates, so financially sophisticated households actually have higher overall return losses.^{21}

These results have two important limitations. First, we assume that mutual fund returns to investors obey the CAPM. If mutual funds hold stocks that obey the CAPM, and if they charge fees to investors, mutual funds will deliver returns with negative alphas reflecting the fee drag. This effect is likely to be significant, as Hortacsu and Syverson (2004) report average fees for equity funds ranging from almost 100 basis points for S&P 500 index funds to over 225 basis points for global and international funds, with wide dispersion across individual funds. A priority for future research is thus to measure the fees charged by each mutual fund available to Swedish investors and the effect of these fees on household portfolio performance.

Second, we treat the financial portfolio in isolation, abstracting from the possibility that financial assets are used to hedge households' labor income risk. Massa and Simonov (2006) explore this issue and find that while investors in general hold stocks that are positively correlated with their labor income, possibly because these stocks are familiar to them, wealthy investors have a greater tendency to pick negatively correlated stocks that can hedge their labor income risk. Massa and Simonov's results are consistent with the theme of this paper that sophisticated households come closer to the investment strategies recommended by standard financial theory.

An important remaining question is to what extent the results for Sweden describe household behavior in other countries. There are several reasons to think that Swedish households may diversify more effectively than households elsewhere. First, Sweden is a country with a well-educated population and an unusually high stock market participation rate. Second, it is a small country, so Swedish investors are used to the idea that they must diversify internationally. Third, Swedish households were exposed to a national financial education campaign in the late 1990s as part of a reform of the pension system.

An additional priority for future research is to try to understand the implications of underdiversification for the wealth distribution. Household underdiversification has the potential to explain the puzzling dispersion of wealth at retirement reported in U.S. data by Venti and Wise (2001). Venti and Wise argue that differences in lifetime earnings or asset allocation do not explain dispersion, and conclude that it must be caused by differences in savings propensities. However, poorly diversified stock investments could also explain a great deal of dispersion.