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I'm trying to figure out the total APY of \$6,600 continuously compounded at 12.93% APY, and 4.1% APY in loyalty rewards. Every five days I cash in my rewards for \$5.50, and compound that. I'm also interested in the total rate if rewards were compounded daily. Is the total APY 17.03%, or much higher? How should I be writing the formula?

What I've done is calculate the yearly return on \$6,600 for 12.93% APY as \$911.01. To do this I used formula $P(t) = P_{0}e^{rt} $ I also calculated the 4.1% rewards APY if compounded daily by $P = C(1 +r/365)^{365t}$ this gives me \$276.21. When I add both values together and calculate the change with $\dfrac{a_{1} - a_{0}}{ a_{0}}$ or $\dfrac{1,187.22}{6,600}$ I get a total APY of 17.99%. This is obviously smaller than what the combined APY should be since the continuously compounded rate needs to be modeled with daily deposits, which I did not do. How can I model daily compounds of the rewards rate into the compound rate on the initial investment?

ZeroPhase
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    Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please [edit] the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – Shaun Aug 19 '23 at 17:01
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    Here's a MathJax tutorial :) – Shaun Aug 19 '23 at 17:01
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    @Shaun Added the work I've done. Not sure how to make this more generic. I'm trying to understand how to model the impact of compounding loyalty rewards paying one rate, while the initial investment earns a separate rate. loyalty rewards have a value, which I cash in to deposit more money into the investment pool. I'm trying to figure out the total APY from doing this from daily deposits to deposits every five days. – ZeroPhase Aug 19 '23 at 17:27

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