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3 Responses
Use 1000*(1 + 0.06/12)^(24). “6% APR compounded monthly” means a nominal annual rate of 6% with interest credited each month, so the periodic rate is 0.06/12 = 0.5% per month. Each month you multiply the current balance by 1.005, so the interest from one month becomes part of the base for the next month. The formula 1000*(1 + 0.06*2) just adds 12% of the original principal over two years; it treats interest as linear and never lets interest earn interest, so it misses the snowball effect entirely.
A quick check: start at 1000. After one month at 0.5% you have 1000*1.005 = 1005. After two months you have 1005*1.005 = 1010.025 (the second month earns 5.025 because it also earns on last month’s 5). Continuing for 24 months gives 1000*(1.005)^24 ≈ 1127.16. By contrast, simple interest gives 1000*(1 + 0.12) = 1120. If you compound annually at 6% you get 1000*(1.06)^2 = 1123.60, which is slightly different because you only let the balance snowball once per year instead of every month; it’s even a tiny overestimate compared to monthly compounding.
Use 1000*(1+0.06/12)^(24): each month you earn 0.5% on whatever the balance has already grown to, so the same little monthly “brick” is stacked on an ever-taller tower (that’s the interest-on-interest), whereas 1000*(1+0.06*2) is simple interest with no compounding and 1000*(1+0.06)^2 compounds yearly, not monthly. Want to peek at the effective annual rate from monthly compounding, (1+0.06/12)^12 − 1 ≈ 6.17%, and see how it makes the 2‑year total a smidge bigger?
Use 1000*(1 + 0.06/12)^(24). “6% APR compounded monthly” means the nominal annual rate is 6%, broken into a monthly rate of 0.06/12 = 0.5%, and that rate is applied to the current balance each month, so the base keeps growing. The formula 1000*(1 + 0.06*2) is simple interest: it adds 6% of the original $1000 each year and never lets interest earn interest, so it misses the snowball effect. And 1000*(1 + 0.06)^2 would be right only if 6% were an effective once-per-year rate; monthly compounding at the same nominal 6% actually gives a bit more because you compound within the year. Quick example: start at 1000. After 1 month you have 1000*1.005 = 1005. In month 2, you earn 0.5% on 1005, so it becomes 1005*1.005 = 1010.025-notice the $5.025 is slightly more than the first $5, that’s interest-on-interest. Carrying this through 24 months gives 1000*(1.005)^24 ≈ $1127.16, compared with $1120 for simple interest (1000*(1 + 0.12)) and $1123.60 for annual compounding at an effective 6% (1000*(1.06)^2).