Compound Interest: The future value (FV) of an investment of present value (PV) dollars earning interest at an annual rate of r compounded m times per year for a period of t years is: Show
FV = PV(1 + r/m)mtor FV = PV(1 + i)n where i = r/m is the interest per compounding period and n = mt is the number of compounding periods. One may solve for the present value PV to obtain: PV = FV/(1 + r/m)mt Numerical Example: For 4-year investment of $20,000 earning 8.5% per year, with interest re-invested each month, the future value is FV = PV(1 + r/m)mt = 20,000(1 + 0.085/12)(12)(4) = $28,065.30 Notice that the interest earned is $28,065.30 - $20,000 = $8,065.30 -- considerably more than the corresponding simple interest. Effective Interest Rate: If money is invested at an annual rate r, compounded m times per year, the effective interest rate is: reff = (1 + r/m)m - 1. This is the interest rate that would give the same yield if compounded only once per year. In this context r is also called the nominal rate, and is often denoted as rnom. Numerical Example: A CD paying 9.8% compounded monthly has a nominal rate of rnom = 0.098, and an effective rate of: r eff =(1 + rnom /m)m = (1 + 0.098/12)12 - 1 = 0.1025. Thus, we get an effective interest rate of 10.25%, since the compounding makes the CD paying 9.8% compounded monthly really pay 10.25% interest over the course of the year. Mortgage Payments Components: Let where P = principal, r = interest rate per period, n = number of periods, k = number of payments, R = monthly payment, and D = debt balance after K payments, then R = P � r / [1 - (1 + r)-n] andD = P � (1 + r)k - R � [(1 + r)k - 1)/r] Accelerating Mortgage Payments Components: Suppose one decides to pay more than the monthly payment, the question is how many months will it take until the mortgage is paid off? The answer is, the rounded-up, where: n = log[x / (x � P � r)] / log (1 + r) where Log is the logarithm in any base, say 10, or e.Future Value (FV) of an Annuity Components: Ler where R = payment, r = rate of interest, and n = number of payments, then FV = [ R(1 + r)n - 1 ] / r Future Value for an Increasing Annuity: It is an increasing annuity is an investment that is earning interest, and into which regular payments of a fixed amount are made. Suppose one makes a payment of R at the end of each compounding period into an investment with a present value of PV, paying interest at an annual rate of r compounded m times per year, then the future value after t years will be FV = PV(1 + i)n + [ R ( (1 + i)n - 1 ) ] / i where i = r/m is the interest paid each period and n = m � t is the total number of periods. Numerical Example: You deposit $100 per month into an account that now contains $5,000 and earns 5% interest per year compounded monthly. After 10 years, the amount of money in the account is: FV = PV(1 + i)n + [ R(1 + i)n - 1 ] / i =
Value of a Bond: V is the sum of the value of the dividends and the final payment. You may like to perform some sensitivity analysis for the "what-if" scenarios by entering different numerical value(s), to make your "good" strategic decision. Replace the existing numerical example, with your own case-information, and then click one the Calculate. How many compounding periods are there in a 6-year investment that compounds quarterly? (1T) a) 6 b) 12 c) 24 d) 48 2. Calculate how much interest is eamed and the value of the investment if S5000 is invested at ,59 per year; simple interest for 26 weeks. (4 K) Raymond invested S400 at 5% compounded monthly for 3 years. What will the investment be worth at the end of the 3-year term? (4 A) Rami plans to g0 on a cruise years from now: He will need 57500 at Ihat time. What principal should Rami invest now at 8.4% per year compounded monthly to obtain the required amount? (4 A) 5.Henry deposited S100 at the end of each month for 5 years at 4% per year . compounded monthly: Calculate the amount of the annuity: (4 T) 1, Hudson wants t0 withdraw S700 at the end of each month for 8 months. His bank account earns 5.4% per year compounded monthly How much must he deposit in his bank account today to pay for the withdrawals? (3A) Bank deposits, over time, usually have compound interest . That is, interest is computed on an account such as a savings account or a checking account and the interest is added to the account. Because the interest is added to the account (the alternative would be to mail the interest to the customer), the interest itself earns interest during the next time period for computing interest. This is what is meant when it is said that the interest compounds . See Salas and Hille, page 448-449. The time interval between the occasions at which interest is added to the account is called the compounding period . The chart below describes some of the common compounding periods:
The interest rate, together with the compounding period and the balance in the account, determines how much interest is added in each compounding period. The basic formula is this: the interest to be added = (interest rate for one period)*(balance at the beginning of the period).Generally, regardless of the compounding period, the interest rate is given as an ANNUAL RATE (sometimes called the nominal rate) labeled with an r. Here is how the interest rate for one period is computed from the nominal rate and the compounding period:
If we put these two formulas together we get
Interest Rate For One Period with various periods and a nominal annual rate of 6% per year
1. "Nominal" in ordinary English can indicate something formal, in name only, but not quite reality and perhaps something that needs further description. It fits well here, because the effect of compounding is a real rate of interest slightly higher than the nominal rate of interest. Click here to return to the first use of
the word "nominal". What Happens To An Account With Compounded Interest And No Withdrawals?
Pj+1=P j + the interest earned by Pj in one compounding period. In words, the balance at the end of a new compounding period is the balance at the end of the preceding period plus the interest that older balance earned during the compounding period. The interest earned is r * (1/K) * Pj,, as described above in the interest calculation for one period. Thus, at the end of the (j+1)th period,
In the last line of the table above, Pj has been factored from the two terms of the previous equality. Here are some examples of the use of this formula, period by period:
For The Saver, There Is An Advantage To Compounding More Frequently. If One Fixes The Nominal Interest Rate And The Total Time The Account Collects Interest, More Frequent Compounding Produces More Interest. In the analysis below,we assume that the total time is a whole number multiple of compounding periods.If one fixes the initial balance (P 0), the nominal interest rate (r) and the duration of the deposit (T, in years) , you earn more interest with more compounding periods per year (K). The number of compounding periods that make up T will be KT. To avoid fractions of compounding periods, which were not analyzed above, assume that K is such that KT is a whole number. Then, by the formula above,P KT = P 0 * (1+r/K)KT.With T and r fixed (not changing) for this discussion, view the right-hand side above as a function of real variable K, say f(K). As long as 1+r/K is positive, this function will have a derivative: (d/dK)[f(K)] = P 0 * (1+r/K)KT * [ T * ln(1+r/K) + K * T* (1/(1+r/K))*(-r/(K2 )) ].This simplifies somewhat:
(d/dK)[f(K)] = P 0 * (1+r/K)KT * T* [ ln(1+r/K) - r/(K+r)) ] It well known that for x in the interval [0,1), we have ln(1+x) >= x - x2/2. If we substitute r/K for x and assume that r>0 and K>r, we find thatln(1+r/K) - r/(K+r)) >= (K-r)r2/(2 K2 (K+r)) > 0 ["ln" refers to the natural logarithm, the log to the base e.] Note that the derivative exists and is positive when P 0 , r, K, and T are all positive and K > r (which are natural assumptions about a savings account!). Since the derivative is positive, the original function f(K) is increasing. Thus, larger values of K make f(K) larger. If we make K larger and also make KT be an integer, then f(K) happens to coincide with P KT . Thus compounding more frequently produces more interest (subject to the assumption that T is a whole number multiple of the compounding period). If T is not a multiple of the compounding period, the conclusion depends strongly on the account's policies on withdrawals in the middle of a compounding period. For example, in some certificates of deposits the bank may charge a substantial penalty for "early" withdrawal. What if we are utterly greedy, and insist that the bank compound our interest continuously?What happens if we make the compounding period a millionth of a second, and ever smaller? Does the amount of interest increase forever without bounds, or do we reach a ceiling (a limit!) as we compound more and more frequently?
g(K) = ln(P0) + (KT) * ln(1+r/K).As K approaches positive infinity, we have a race between two factors because KT is also approaching positive infinity (we assume that T is positive) while r/K approaches 0. As r/K approaches 0, 1+r/K approaches 1 and ln(1+r/K) approaches 0. Thus we seem to have infinity*0 in our limit as K approches positive infinity. Recall that L'Hôspital's rule applies to indeterminate forms 0/0 and infinity/infinity. Rewrite the difficult part of g(K) to take advantage of this rule: g(K) = ln(P0) + ln(1+r/K) / [1/(KT)].Note that 1/(KT) is approaching 0, so that we have the indeterminate form of 0/0. By L'Hôspital's rule, examine the limit of a new ratio which is the ratio of the separate derivatives of the top and bottom of the indeterminate form: {[1/(1+r/K)](-r/(K2)} / {-(KT)-2*T}After simplifying this new ratio, one has [1/(1+r/K)] * (r/T) * [(KT)2] / (K2) = (rT) * [1/(1+r/K)].As K approaches positive infinity, this new ratio approaches (rT) * [1/(1+0)] = rT. Thus, g(K) has the limit ln(P0) + rT as K approaches positive infinity. Because ex is a continuous function, we can apply ex to the function g(K) to get f(K) back AND a limit for f(K) which is e[ln(P0)+rT]=P 0*erT.Thus, compounding faster and faster does have a finite limit; this finite limit defines what economists (and bankers) mean by continuous compounding. If compounding is continuous at a nominal interest rate of r for a duration T (in years) with an beginning balance of P0, the balance at the end is P 0*erT.Your comments and questions are welcome. Please use the email address at www.math.hawaii.edu Edited on September 6, 2006. What is the rate per interest period of 12% compounded quarterly?The correct answer is c) 12.55%.
How do you calculate interest in conversion period J?The nominal rate of interest is the stated yearly rate. The actual rate of interest will depend on the length of the conversion period. For example, if we state that the nominal rate is 6% compounded quarterly, then the conversion period is 3 months and the interest rate is ¼(6%) = 1.5 % for each conversion period.
What is the conversion period for interest compounded quarterly?COMPOUND INTEREST. What nominal rate compounded monthly is equivalent to 12% compounded quarterly?So we take this to the 1/3 and then we subtracted one off again. We do that, we get 3.85 percent is the I'm all right, A phenomenal rate compounded quarterly. That's equivalent to that. 12% compounded monthly.
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If the interest rate is 12% compounded quarterly what is the interest rate per conversion period j
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