Given these formulas for calculating interest (yearly):
Simple = Pr(days/365)
Compounded = P((1+r)^(days/365)-1)
P = Principal r = Interest rate
For an investment of 50000 and a yearly interest rate of we can plot the difference between these formulas:
During the first year, simple yields better than compounded. How can I explain this in a simple way? My friend can't accept the math and insists that compounded "by definition" always should be better than simple
Compound interest is often described this way: with simple interest you earn interest only on the principal, but with compound interest you earn interest on the principal and also earn interest on the interest previously accumulated.
If we take that as the definition of compound interest, then by definition you cannot earn less with compound interest than with simple interest (unless the interest you earn is negative, in which case compounding causes you to lose your money faster).
In your example, you start with a principal of $50000$. The interest is $10\%$ compounded once a year. That means until the second year, the two accounts do exactly the same thing. Both have the same principal. Both have the same interest rate. During the first year, neither one applies the interest rate to anything but the principal.
Now what you'll find if you look for compound interest examples is that it's very difficult to find one that shows partial interest accrued for part of a compounding period. They show the value after $1$ period, then $2$ periods, then $3$ periods. That's because the model of compounding at discrete intervals is a sequence, not a continuous function. If anything is said about what kind of number the exponent $k$ in the formula $$ P_k = P_0 (1 + r)^k $$ is, it is said that $k$ is a non-negative integer.
So if you want to graph the value of an account earning compound interest, what would the graph look like? One way the graph might look is like this:
(from https://commons.wikimedia.org/wiki/File:Compound_Interest_with_Varying_Frequencies.svg).
Anything other than continuous compounding is plotted in the form of stair steps: the balance in the account stays the same during a single compounding period, at the end of which the interest is credited and the balance instantly increases to a new, higher level, at which it then stays until the end of the next compounding period.
You could also use a bar graph, but that would only show the value exactly at the end of each period and wouldn't say anything about what happens during the compounding period.
So what does happen during the compounding period? Nothing, if we're looking at the balance on a bank statement. If the bank is willing to give pro-rated interest for money withdrawn before the end of a compounding period, they might prorate it linearly, and you might want to plot that as a continuous (piecewise linear) function showing the closeout value of the account. During the first year, this function would be the same for both accounts.
I think it would be an interesting challenge to find a bank that would apply the formula $P(1+r)^t$ literally for a non-integer value of $t.$ I would be interested in evidence of such a practice if you can find it. I would be especially interested to know what happened after the regulatory authorities found out about it, because the bank would effectively be compounding interest continuously at a lower nominal rate than the advertised rate.