The Cox-Ingersoll-Ross (CIR) model describes the dynamics of the short-term interest rate through the following stochastic differential equation:

$dr = k(\theta - r)dt + \sigma \sqrt{r} dw$

Where:

• $r$ = current short rate
• $k$ = strength of mean reversion
• $\theta$ = long-run average interest rate
• $\sigma$ = volatility parameter
• $dt$ = time step
• $dw$ = random stochastic component (Wiener process increment)

Given the parameters:

• $r(0) = 2.11\% = 0.0211$
• $k = 0.57$
• $\theta = 7.64\% = 0.0764$
• $dt = 1/12$
• $\sigma = 3.17\% = 0.0317$
• $dw = 0.160$

We calculate the change in the rate, $dr$:

1. Drift Component (Deterministic Part): $k(\theta - r)dt = 0.57 \times (0.0764 - 0.0211) \times (1/12)$ $= 0.57 \times 0.0553 \times 0.083333 \approx 0.00262675$

2. Volatility Component (Stochastic Part): $\sigma \sqrt{r} dw = 0.0317 \times \sqrt{0.0211} \times 0.160$ $= 0.0317 \times 0.145258 \times 0.160 \approx 0.00073675$

3. Total Change ($dr$): $dr \approx 0.00262675 + 0.00073675 = 0.0033635$

4. New Short-Rate ($r(1/12)$): $r(1/12) = r(0) + dr = 0.0211 + 0.0033635 = 0.0244635$

Converting back to a percentage, we get $2.446%$.