An insurer enters into a four-year contract today. The contract requires the insured to deposit $500$ into a fund that earns an annual effective rate of $5.0\%$, and from which all claims will be paid. The insurer expects that $100$ in claims will be paid at the end of each year, for the next four years. At the end of the fourth year, after all claims are paid, the insurer is required to return $75\%$ of the remaining fund balance to the insured. To issue this policy, the insurer incurs $100$ in expenses today. It also collects a fee of $125$ at the end of two years. Calculate the insurer’s yield rate.
My solution: Let the annual effective yield rate be $i$. Then
$$500(1+i)^4 = 100 s_{4\rceil 5\%} + 125 (1+i)^2 -100(1+i)^4 -0.75(500(1.05)^4 - 100 s_{4\rceil 5\%})$$
This leads to $(1+i)^2 = 0.817$, which is not correct. Could someone point out where I have gone wrong and how my mistake can be corrected? Am I misinterpreting yield rate and interest rate in this problem?
$$0 = 125 (1+i)^2 -100(1+i)^4 +0.25(500(1.05)^4 - 100 s_{4\rceil 5\%})$$
We only keep $0.25$ of the fund balance.
Also, the $5\%$ of the effective rate is applied to the deposit, not the yield rate.