4. A bank plans to invest $40 \%$ of a certain fund in project $M$ for one year, and the remaining $60 \%$ of the fund in project $N$. It is expected that project $M$ could yield an annual profit of $19 \%$ to $24 \%$, and project $N$ could yield an annual profit of $29 \%$ to $34 \%$. At the end of the year, the bank must recall the funds and pay a certain rebate rate to depositors. To ensure that the bank's annual profit is no less than $10 \%$ and no more than $15 \%$ of the total investment in $M$ and $N$, the minimum rebate rate for depositors is $\qquad$
$4.10 \%$.
Let the total investment of the bank in two projects be $s$. According to the conditions given, the annual profits from investments in $M$ and $N$, denoted as $p_{M}$ and $p_{N}$, respectively, satisfy
$$
\begin{array}{l}
\frac{19}{100} \times \frac{40}{100} s \leqslant p_{M} \leqslant \frac{24}{100} \times \frac{40}{100} s, \\
\frac{29}{100} \times \frac{60}{100} s \leqslant p_{N} \leqslant \frac{34}{100} \times \frac{60}{100} s .
\end{array}
$$

The bank's annual profit $p$ satisfies
$$
\frac{10}{100} s \leqslant p \leqslant \frac{15}{100} s \text {. }
$$

Thus, the percentage of the bank's annual profit that is given back to depositors is
$$
\begin{array}{l}
\frac{p_{M}+p_{N}-p}{s} \times 100 \% . \\
\frac{(1)+(2)-(3)}{s} \text { gives } \frac{10}{100} \leqslant \frac{p_{M}+p_{N}-p}{s} \leqslant \frac{20}{100} .
\end{array}
$$

Therefore, the minimum percentage of the bank's annual profit given back to depositors should be $10 \%$.