If NKE has an annualized average return of 14.64% and an annualized standard deviation of 19.75%, what is it's 95% Confidence interval? If TSLA has an annualized average return of 14.20 and an annualized standard deviation of 41.16% what is it's 95% confidence interval?

To calculate the 95% confidence interval for the returns of **NKE** and **TSLA**, we'll use the formula for the confidence interval of a normally distributed variable:

\[
CI = \text{Mean} \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}
\]

Where:
- **Mean** is the average return (annualized in this case).
- **\(\sigma\)** is the standard deviation (annualized in this case).
- **\(n\)** is the sample size, which in finance is often 1 when we calculate for one period.
- **\(Z_{\alpha/2}\)** is the Z-value corresponding to the desired confidence level. For a 95% confidence interval, \(Z_{\alpha/2} = 1.96\).

Since we are interested in the confidence interval of the return distribution over one year (where \(n = 1\)), the formula simplifies to:

\[
CI = \text{Mean} \pm 1.96 \times \sigma
\]

### NKE (Nike):
- **Mean return** = 14.64%
- **Standard deviation (\(\sigma\))** = 19.75%
- **Confidence interval**: 

\[
CI = 14.64\% \pm 1.96 \times 19.75\%
\]

\[
CI = 14.64\% \pm 38.71\%
\]

So the 95% confidence interval for NKE is:

\[
[-24.07\%, 53.35\%]
\]

### TSLA (Tesla):
- **Mean return** = 14.20%
- **Standard deviation (\(\sigma\))** = 41.16%
- **Confidence interval**: 

\[
CI = 14.20\% \pm 1.96 \times 41.16\%
\]

\[
CI = 14.20\% \pm 80.68\%
\]

So the 95% confidence interval for TSLA is:

\[
[-66.48\%, 94.88\%]
\]

### Conclusion:
- **NKE 95% CI**: \([-24.07\%, 53.35\%]\)
- **TSLA 95% CI**: \([-66.48\%, 94.88\%]\)

Would you like further details or explanations on these results?

### Related Questions:
1. How do changes in standard deviation affect confidence intervals?
2. What is the significance of using a 95% confidence interval?
3. How would the confidence intervals change with a different confidence level, say 99%?
4. Can historical data be used to adjust or refine these estimates over time?
5. How does this analysis relate to investment risk management?

### Tip:
Higher volatility (standard deviation) significantly increases the range of possible returns, making the investment riskier.

If NKE has an annualized average return of 14.64% and an annualized standard deviation of 19.75%, what is its 95% confidence interval? If TSLA has an annualized average return of 14.20% and an annualized standard deviation of 41.16%, what is its 95% confidence interval?

Calculate the 95% confidence interval for NKE and TSLA annualized returns using their average returns and standard deviations. Learn how to apply the formula CI = Mean ± 1.96 * σ and interpret the results. Suitable for students in finance or statistics at the college level.

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