# Annuity problem, calculating the accumulated value.?

There was an interesting problem that I would like to have some input from people who knows a bit of finance.The following is the situation.>Smith loans $\\$10,000$for$i=5\\%$for$10$years. There are two schemes in reinvestment. A: Smith receives a level annual payment of >$$\\frac{10,000}{a_\\bar{10\\rceil}}=1295.05$$each year, which is reinvested in a bank annually with$i_2=3\\%$>or B: Smith receives$5\\%$of the principal each year, and receives the principal at the end of the$10$th year, reinvesting$\\$500$ yearly in the same bank.What's really bugging me is the fact that if one wants to make investments, the larger the principal is the more money you get at the end, assuming that the person is investing the money for the same amount of time and interest rate.Algebraically, I see that scheme A will end up having a less rate of return. However, according to my argument this is counter intuitive because Smith is receiving more money under scheme A, so why would he lose money?