A municipal budget allocates $B(t) = 10000(1 + 0.05 \sin(\frac{\pi t}{6}))$ dollars for street repairs, where $t$ is the month of the year. If the city earns interest on unspent funds at a continuous rate of $r = 0.03$, and the total expenditure is the integral of $B(t)$ over the year, what is the net value of the budget at $t=12$ if the initial fund is $F_0 = 120000$?