Question

Do bonds reduce the overall risk of an investment portfolio? Let x be a random variable representing annual percent return for the Vanguard Total Stock Index (all Stocks). Let y be a random variable representing annual return for the Vanguard Balanced Index (60% stock and 40% bond). For the past several years, assume the following data. x: 14 0 39 25 32 27 28 14 14 15 y: 6 2 29 17 26 17 17 2 3 5 Compute the coefficient of variation for each fund. Round your answers to the nearest tenth.

Answer 1

COV (all stocks) = 0.55

COV (stocks and bonds) = 0.82

Explanation:

Coefficient of Variation is used to measure variability.

It is defined as the ration of standard deviation and the mean.

It can be used to compare variability of two population or two samples.

Formula:

`\text{Coefficient of Variation} = \displaystyle\frac{\text{Standard Deviation}}{\text{Mean}}`

`\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}`

where
`x_i` are data points,
`\bar{x}` is the mean and n is the number of observations.

`Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}`

x: 14, 0, 39, 25, 32, 27, 28, 14, 14, 15

`Mean = (208)/(10) = 20.8`

`Standard~Deviation = \sqrt{(1169.6)/(9) } = 11.39`

`Coefficient~of~Variation = (11.39)/(20.8) = 0.55`

y = 6, 2, 29, 17, 26, 17, 17, 2, 3, 5

`Mean = (124)/(10) = 12.4`

`Standard~Deviation = \sqrt{(924.4)/(9) } = 10.13`

`Coefficient~of~Variation = (10.13)/(12.4) = 0.82%`

Since coefficient of variation of x is less compared to y, thus it could be said bonds does not reduce overall risk of an investment portfolio.