Question

The Ishares Bond Index fund (TLT) has a mean and annual standard deviation of returns of 5%

and 10%, respectively. What is the 66% confidence interval for the returns on TLT?
A) -7%, 10%
B) 5%, 10%
C) -5%, 15%
D) -10%, 10%

Answer 1

C) (-5%.15%)

Explanation:

1) Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

`\bar X =10\%=0.1` represent the sample mean for the sample

`\mu` population mean (variable of interest)

`\sigma`=5% =0.05 represent the population standard deviation

2) Confidence interval

We assume that the random variable X who represent The Ishares Bond Index fund (TLT) follows this distribution:

`X \sim N(\mu, \sigma=10\%=0.1)`

The confidence interval for the returns on TLT is given by the following formula:

`\bar X \pm z_(\alpha/2)\sigma` (1)

In order to calculate the critical value
`z_(\alpha/2)`. Since the Confidence is 0.66 or 66%, the value of
`\alpha=0.34` and
`\alpha/2 =0.17`, and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-NORM.INV(0.17,0,1)".And we see that
`z_(\alpha/2)=0.95`

Now we have everything in order to replace into formula (1):

`0.05-0.95(0.1)=-0.05`

`0.05+0.95(0.1)=0.15`

So on this case the 66% confidence interval would be given by (-0.05;0.15) and we can convert this into % and wr got (-5%; 15%).