How is interest that is continuously compounded different from interest that is compounded every half year? e?
968929
The shorter the interest term the faster it grows as the principal grows more quickly so you get more interest on your interest\n\nFor example lets look at the interest gained over 1 year on an interest rate of 12% per annum\n\nYearly: Well at the end of the year A = 1.12*P\nSo interest gained = 1.12*P - P = 0.12*P\n\nHalf Yearly: Well at the end of the year A = (1.06)*(1.06)*P (because the interest is applied each 6 months\n= (1.06)² * P\n= 1.1236*P\nSo interest gained = 1.1236*P - P = 0.1236*P > 0.12P\n\nQuarterly: Well at the end of the year A = (1.03)*(1.03)*(1.03)*(1.03)*P \n= (1.03)^4 * P\n≈ 1.1255P\nSo interest gained = 1.1255*P - P = 0.1255*P > 0.1236P\n\nMonthly: Well at the end of the year A = (1.01)*(1.01)* ... *(1.01)*(1.01)*P \n= (1.01)^12 * P\n≈ 1.1268P\nSo interest gained = 1.1268*P - P = 0.1268*P > 0.1255P\n\nDaily: Well at the end of the year A = (1.0003288)*(1.0003288)* ... *(1.0003288)*P \n= (1.0003288)^365 * P\n≈ 1.1275P\nSo interest gained = 1.1275*P - P = 0.1275*P > 0.1288P\n\nNow while it may not sound like much over 1 year over 20 years it can become significant\n\nJust quickly over 20 years\n\nYearly: Interest = ((1.12)^20 -1)*P = 8.6463*P\nHalf Yearly: Interest = ((1.06)^40 - 1)*P = 9.2857*P\nQuarterly: Interest = ((1.03)^80 - 1)*P = 9.6409*P\nMonthly: Interest = ((1.01)^240 - 1)*P = 9.8926*P\nDaily: Interest (using 365.25 days per year to account for leap years)= ((1.0003285)^365.25*20 - 1)*P = 10.0154*P
1095361
First, I should not that the "yield" I have been reporting is the interest earned.  Add $10,000 for the total amount in the account at the end of the investment period.\n\nSecond, you wrote the formula incorrectly.  The formula for continuous compounding has to include the interest rate, so\nA = Pe^(rt)\n\nGiven the information from the previous question ($10,000 at 10% for 2 years), continuous compounding would yield $2,214.03.\n\nIf we know that the bank returned $15,000 on a $10,000 deposit at 10%, the length of time must have been 4.05 years.\n\nIn 6.93 years, a deposit will have doubled.\n\nI apologize for the delay on responding to these questions--I have been on vacation.
1168169
Goose is right. The formula is:\n\nFV = P x e^(r x y) \n\nwhere FV = Future Value; P is your principal, r is the annual interest rate and y is the number of years of continuous compounding and e = lim(n-->inf) of (1+1/n)^n\n= 2.7182818.(about)\nHere's link which explains it quite well and also has a periodic/continuous calculator to fool around with.\n\nhttp://www.moneychimp.com/articles/finworks/continuous_compounding.htm
830956
Its really not that complicated…  Look first there is APR and then there is APY.  APR means Annual Percent Rate.  And APY means Annual Percent Yield.  \nIn your example of 4% this is the annual rate, your monthly rate would be 4% / 12.  12 being 12 months in one year.\nAPY means your APR plus any compounded interest that was earned and added to your initial investment.  The APY should always be higher than the APR.  \n\nI hope this helped…
78911
If it was compounded annually:\nToday - $5,397.31\n6 Years ago - $3,401.22\n\nSee the link below to calculte using other options.
337558
It depends if it's simple interest or compound (sometimes called effective interest).\n\nI'm also assuming that the 6.5% is per year.\n\nThe accumulation factor formula (ie what we multiply our initial amount by) for simple interest is:\n\n(1 + ni)\n\nSo if we double our initial amount then this factor must be 2.\n\n1 + 0.065n = 2\n\nThis gives\n\n15.38 years (I am assuming that you receive this interest daily - and not at the end of each year or anything like that)\n\nThe accumulation factor formula (ie what we multiply our initial amount by) for compound interest is:\n\n(1 + i)^n\n\nSo if we double our initial amount then this factor must be 2.\n\n1.065^n = 2\n\nTo solve this we need to take logs:\n\nln (1.065^n) = ln 2\n\nWe then use the rule that ln (a^b) = b ln a\n\nn ln 1.065 = ln 2\n\nSo:\n\nn = ln 2 / ln 1.065 = 11.007 years\n\n(Again I'm assuming that you receive this interest daily - and not at the end of each year or anything like that)\n\n(PS you can use other logs as well for this - it doesn't make a difference here)
95230
The compound interest formula is an exponential function:\n\nA=P(1+(r/n))^(n*t) where\nA=final amount\nP=principal or beginning amount\nr=interest rate (as a decimal)\nn=number of compounding per year\nt=time in years\n\nThe other exponential formula for compounding interest is used for continuously compounding interest or when (n*t) is equal to infinity. It is also the equation used when charting the growth or decay of bacteria.\n\nA=P*e^(r*t).\n\nPlease check the second source for info on how this equation relates to bacteria growth/decay.
280771
the rule says that\n\nif the interest rate is x %, then your investment would approximately double in 72/x years.\n\nfor example, if the interest rate is 9%, it would take approximately 8 years to double the value of your initial investment.\n\nThis rule is more accurate if the interest rate is small. You might prove it mathematically. It's very easy to prove it.
749623
Simple Interest\nA = P *r * t\n\nCompound Interest (monthly)\nA = P(1 + r/12)^(12t)\n\nA1 = P(1 + .075/12)^(12*30)\nA2 = A1 * .06 * 1\n\nP(1 + .075/12)^(12*30) * .06 * 1 >= 12000\nP >= 12000 / ((1 + .075/12)^(12*30) * .06)\nP >= $21,227.97
149317
There really are no calculations because they are not charging you ANY interest at all (probably for a certain amount of time only, right?) You will only be paying the Principle during that time, which means you are making a payment that actually reduces the total cost by the exact amount you pay.  WHen interest begins accruing, then each payment you make will be partially going to the product cost (prinicple) and partially to the interest cost (the creditor's profit for lending the money).  There is no better interest rate than 0%.  So, good going!
873237
well, you didn't mention the compounding factor, so....\n\nsimple interest\nA sub t is the value (amount) after "t" time\nP is the initial amount invested\nr is the interest rate\nt is time ( in years)\n\n    A sub t  = P ( 1 + r) ^ t\n\nfor compound interest rate\nn is the number of times that the interest is compounded per year\n\n    A sub t = P ( 1 + r/n) ^ (n*t)\n\n\nnow, you have another problem...\nsome CDs only pay interest yearly....\nsome quarterly, some monthly...\nsince your interest rate is different from your APR, \nI'll assume a monthly compounding\n\n     Amt = $2000 ( 1 + (.0489/12)) ^ (12* (11/12))\n\nnote: the  time is 11/12 because interest is figured on a yearly basis and 11 months is 11/12 of a year\n\n  if my calculator does not deceive me\nyou should get   $2091.50
82296
Interest is taxable on the federal return.  Its not taxable on your state return.
240247
You must add one important detail: is it 4.8% per annum (yearly) or monthly?\n \nIf it's per annum:\n1000 * [1 + 0.048/12]^months\n \nIf it's per month:\n1000 * [1 + 0.048]^months\n \nTo find out how much after 1yr and after 2yrs, subtitute months by 12 and 24.\n \n \n\nIn Reply to the Mar 6, 2006 addition\n  \nLynn, you did say "compounded monthly" and people will almost always take this as "per annun with monthly componding" and almost always, it is. But it's not necessarilly so. It could be 4.8% per month with monthly compounding, for example. Ok, I'm kind of pushing it, but it's still true. You should be aware of these trick questions.\n \nAlso be aware that there are countless ways to calculate interest. I've worked with finance all my life and am a PhD student now and still see new ways once in a while. There are several things that may affect the result and the way to work thru the problem. The kind of interest you are learning now is a very basic form and for now you should only be aware that the way the interest rate is expressed must refer to a period. This period has no relation to the compounding period. Usually, it will be per annum but doesn't have to. For example, my bank informs me of the overdraft interest rates as a monthly rate, it is compounded daily but in a completely diferent way than the one you're learning. So keep in mind that you might be asked a question in which the rate is (for example) 5% per semester with semi-monthly compounding, over a period of one year. That is a legitimate question and will result in more than 10% per year. Good luck.
549854
the rate is the stated rate such as 4%.  The yield is higher rate because of compounding.  Let's say $100 @ 4% per year is $4.00 income. The same 4% rate compounded monthly is higher because the intrest income is added at the end of each month. Let's say $100 @ 4% monthly is 1st month is $100 x 4% = $4.00 / 12 months is $100.33 at the end of the first month so month 2 is $100.33 x 4% = / 12 months is $100.66. and so on...  So yield is interest on top of interest.
341197
well if make 7% every year on your money you double your money in 7 years.  28 years at that rate.  Typically just having a balanced portfolio gets that for you.
1072634
There are 2 types of interests, simple and compound. My assumption is you are talking about simple interest. In simple interest, the interest remains the same over the time period, in compound interest, the interest increases. \n\n For simple interest, this are the things you need.\n I = the interest.\n r= rate (in percentage)\n t= time frame (mostly per annum)\n p= principal amount\n\nthe formula is\n I = (p x rx t) / 100 \n solving for r gives you this results,\n\nr = (100 x I) / (p x t) \n\n This is for general Maths. For specific fields, you'll have to check    for it in that field, or you'll have to mention it.\n \n Take care.
377198
Formula:\n\n\n\nP is the principal (the initial amount you borrow or deposit) \n\nr is the annual rate of interest (percentage)\n\nn is the number of years the amount is deposited or borrowed for.\n\nA is the amount of money accumulated after n years, including interest. \n\nWhen the interest is compounded once a year: \n\nA = P(1 + r)n \n\n\nHowever, if you borrow for 5 years the formula will look like: \n\nA = P(1 + r)5\n\nThis formula applies to both money invested and money borrowed.\n\nFrequent Compounding of Interest:\n\nWhat if interest is paid more frequently? \nHere are a few examples of the formula: \n\nAnnually = P × (1 + r) = (annual compounding) \n\nQuarterly = P (1 + r/4)4 = (quarterly compounding)\n\nMonthly = P (1 + r/12)12 = (monthly compounding)\n\n\nApprox. $716.43 I believe