The Cox-Ingersoll-Ross (CIR) model specifies that the change in the short-term rate $dr$ is given by: $dr = k(\theta - r)dt + \sigma \sqrt{r} dw$

Where:

• Current short term rate, $r = 5\% = 0.05$
• Long run value, $\theta = 7.5\% = 0.075$
• Speed of mean reversion, $k = 0.05$
• Volatility, $\sigma = 0.8\% = 0.008$
• Random variable, $dw = 0.3$
• Time step, $dt = 1 \text{ month} = 1 / 12 \text{ year}$

First, we calculate the drift component: $\text{Drift} = 0.05 \times (0.075 - 0.05) \times (1 / 12) = 0.05 \times 0.025 \times 0.08333 = 0.000104$

Next, we calculate the random shock component: $\text{Shock} = 0.008 \times \sqrt{0.05} \times 0.3 = 0.008 \times 0.2236 \times 0.3 = 0.000537$

Adding both components gives the change in the rate: $dr = 0.000104 + 0.000537 = 0.000641 = 0.0641\%$

The new short-term rate is: $r_{new} = 5\% + 0.0641\% = 5.0641\% \approx 5.06\%$