**<u>Table 4. Matrix of Correlations</u>**

|Equi|ty Local|Equity Int'l<br>RER|Short<br>rate<br>index|term<br>local (π<br>ed)<br>Short<br>rate<br>index|term<br>Int'l<br>(π<br>ed)<br>Credit<br>local|risk<br>Credit<br>Int'l|Risk|
|---|---|---|---|---|---|---|---|
|Equity Local|1.00|0.55|‐0.32|0.11|‐0.15|‐0.26|‐0.51|
|Equity Int'l|0.55|1.00|‐0.55|0.07|‐0.16|‐0.31|‐0.70|
|RER|‐0.32|‐0.55|1.00|‐0.07|0.32|0.30|0.48|
|Short term rate local (π indexed)|0.11|0.07|‐0.07|1.00|0.33|0.05|0.08|
|Short term rate Int'l (π indexed)|‐0.15|‐0.16|0.32|0.33|1.00|0.19|0.38|
|Credit risk local|‐0.26|‐0.31|0.30|0.05|0.19|1.00|0.43|
|Credit Risk Int'l|‐0.51|‐0.70|0.48|0.08|0.38|0.43|1.00|

Source: Authors’ calculations Each of the parameters deserves to be discussed. The discussion about parameters of domestic securities is more complicated, as they should reflect forward-looking expectations, rather than simple historical averages. A reality check is necessary to compare forward-looking parameters with the ones estimated based on historical information using the methodologies described in the previous section. Some emerging economies have short historical records and past returns typically reflect periods of high illiquidity. As local equity markets develop and foreign investors become interested in local equities, there is a process of return equalization with international markets. While returns tend to converge to international levels, volatility tends to be higher than volatility in international markets, simply because of fewer possibilities for broader diversification. It is not rare to use a local equity premium slightly above the international premium, but with higher volatility than international equity indices. The same discussion could be extended to the cases where instead of investment in local markets, the investments are broadened to emerging economies or sub-regions.

# **_Third Step. Defining investment strategies_**

The starting point of the approximation method is to put in place a set of different plausible investment strategies. We took, as starting point, some of the investment strategies proposed by Viceira (2010) and we add our own.

Figure 6 provides a graphic representation of the investment strategies, which are also explained below:

- a. **Conservative** . The conservative portfolio is a lifestyle fund invested 75 percent in the risk free asset and 25 percent in international fixed income.

- b. **Balanced local** : The balanced local is also a lifestyle fund invested 2/3 risk free assets and 1/3 domestic equity portfolio.

- c. **Balanced International** . This lifestyle portfolio invests 44 percent in risk free assets, 14 percent in international fixed income, 10 percent in local equity and 32 percent in international securities along the lifecycle of individuals.

- d. **Stylized Chilean fund** . This is a lifecycle portfolio that assumes a gradual migration from one fund to another within a 5 year time period (1/5 each year).

- i. up to 31 years of age: 17.5 percent  risk free assets; 55 percent international equity; 20% domestic equity; 7.5 percent international fixed income

- ii. from 31 to 50 years of age: 28 percent risk free assets; 35 percent international equity;25 percent domestic equity; 12 percent international fixed income;

- iii. From 50 to 65 years of age: 56 percent risk free assets; 13 percent international equity; 7 percent domestic equity; 24 percent international fixed income

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