Suppose two people, person $A$ and person $B$ each make deposits of $50,000$ to two funds, fund $A$ and fund $B$ (i.e. person $A$ to fund $A$ and person $B$ to fund $B$). Fund $A$ uses a compound accumulation while fund $B$ uses simple accumulation with annual effective interest rates of $5$% and $8$% respectively. If the forces of interest on these two funds are equal $k$ years into the future, compute the difference between these two funds. Which fund is larger?
My attempt:
The accumulation functions for compound and simple are $a(t) = 1.05^t$ and $b(t) = 1 + 0.08t$ respectively. The force of interest $F(t)$ and $G(t)$ are:
$F(t) = \frac{a'(t)}{a(t)} = \ln(1.05)$,
$G(t) = \frac{b'(t)}{b(t)} = \frac{0.08}{1+0.08t}$
At time $k$, $F(k) = G(k)$. So
$\ln(1.05) = \frac{0.08}{1+0.08k}$
This gives $k = 7.995934$.
At $k = 7.995934$
At this value of $k$ we have
$A(t) = 50000 \cdot 1.05^{7.995934} = 73858.11984$ and $B(t) = 50000 \cdot (1 + 0.08 \cdot 7.995934) = 81983.73726$.
Obviously fund $B$ is larger and the difference is $81983.73726 - 73858.11984 = 8125.617416$
Is this correct? My biggest issue here is the value of $k$. Should I be using $k = 7.995934$ because that is the value in the fund or should I round it down to $k = 7$ because interest is paid annually and has not been paid yet? Any assistance is appreciated.