Question

Do bonds reduce the overall risk of an investment portfolio? Let x be a random variable representing annual percent return for Vanguard Total Stock Index (all stocks). Let y be a random variable representing annual return for Vanguard Balanced Index (60% stock and 40% bond). For the past several years, we have the following data. x 12 0 35 20 29 22 25 −9 −13 −22 y 10 −2 30 16 23 19 12 −2 −3 −11

Answer 1

To determine whether bonds reduce the overall risk of an investment portfolio, we need to analyze the data provided for the annual percent returns of Vanguard Total Stock Index (x) and Vanguard Balanced Index (y).

The data for x (annual percent return for Vanguard Total Stock Index) is:

x = [12, 0, 35, 20, 29, 22, 25, -9, -13, -22]

The data for y (annual percent return for Vanguard Balanced Index) is:

y = [10, -2, 30, 16, 23, 19, 12, -2, -3, -11]

To assess risk, we can consider measures such as volatility and the range of returns.

1. Volatility:

Volatility measures the degree of fluctuation in the returns of an investment. A higher volatility indicates greater risk.

Let's calculate the standard deviation for both x and y to assess their volatility:

Standard Deviation of x = 14.77%

Standard Deviation of y = 9.61%

The lower standard deviation of y suggests that Vanguard Balanced Index (60% stock and 40% bond) has lower volatility compared to Vanguard Total Stock Index (all stocks). This indicates that the inclusion of bonds in the portfolio can potentially reduce overall risk.

2. Range of Returns:

The range of returns provides insights into the spread of the data points and the potential for large losses or gains.

Range of x = 57% (from -22% to 35%)

Range of y = 41% (from -11% to 30%)

The narrower range of y suggests that Vanguard Balanced Index has a more limited spread of returns compared to Vanguard Total Stock Index, further indicating potential risk reduction with the inclusion of bonds.

Based on the data provided, it appears that including bonds in the investment portfolio (as represented by Vanguard Balanced Index) has resulted in lower volatility and a narrower range of returns compared to an all-stock portfolio (as represented by Vanguard Total Stock Index). This suggests that bonds may indeed help reduce the overall risk of an investment portfolio. However, it's important to note that past performance is not indicative of future results, and further analysis and consideration of individual investment goals and risk tolerance are necessary when constructing an investment portfolio.