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VIPSolutions · Accepted Answer

## Objective

Determine if a **30% loss** next year would be outside the **95% confidence interval** for a portfolio with:
- Mean return ($\mu$) = **11.9%**
- Standard deviation ($\sigma$) = **19.2%**
- Confidence level = **95%**
- Time period = **1 year**

## Step 1: Formula for 95% Prediction Interval

For a normal distribution, the **95% prediction interval** for next year's return is:

$$
\text{Prediction Interval} = \mu \pm z \cdot \sigma
$$

Where:
- $\mu$ = mean return
- $\sigma$ = standard deviation
- $z$ = z-score for 95% confidence ($z = 1.96$ for 2-tailed)

## Step 2: Calculate the Low End of the Interval

$$
\text{Low end} = \mu - z \cdot \sigma
$$

Plug in the values ($\mu = 11.9\%$, $z = 1.96$, $\sigma = 19.2\%$):

$$
\text{Low end} = 11.9\% - (1.96 \times 19.2\%)
$$

$$
1.96 \times 19.2\% = 37.632\%
$$

$$
\text{Low end} = 11.9\% - 37.632\%
$$

$$
\text{Low end} = -25.732\%
$$

Rounded to one decimal place:

$$
\text{Low end} \approx -25.7\%
$$

## Step 3: Compare to 30% Loss

- A 30% loss = **-30%**
- The low end of the 95% prediction interval = **-25.7%**
- Since $-30\%$ is **less than** $-25.7\%$, a 30% loss is **outside** the 95% interval.

## Step 4: Select the Correct Answer

**Option A** is correct:
> Yes, you can be confident that the portfolio will not lose more than 30% of its value next year. This is because the low end of the prediction interval is less than -30%.

---

## Final Answer

**The low end of the 95% prediction interval is $-25.7\%$.**  
A 30% loss would be outside the 95% confidence interval, so you can be confident that the portfolio will not lose more than 30% of its value next year.

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