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] D = 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. In this section we cover compound interest and continuously compounded interest. Use the compound interest formula to solve the following. Example: If a $500 certificate of deposit earns 4 1/4% compounded monthly then how much will be accumulated at the end of a 3 year period?
Answer: At the end of 3 years the amount is $576.86.
Example: A certain investment earns 8 3/4% compounded quarterly. If $10,000 is invested for 5 years, how much will be in the account at the end of that time period?
Answer: At the end of 5 years the account have $15,415.42 in it. The basic idea is to first determine the given information then substitute the appropriate values into the formula and evaluate. To avoid round-off error, use the calculator and round-off only once as the last step.
One important application is to determine the doubling time. How long does it take for the principal in an account earning compound interest to double? Example: How long does it take to double $1000 at an annual interest rate of 6.35% compounded monthly?
Answer: The account will double in approximately 10.9 years. The key step in this process is to apply the common logarithm to both sides so that we can apply the power rule and solve for time t. Use the calculator in the last step and round-off only once. Example: How long will it take $30,000 to accumulate to $110,000 in a trust that earns a 10% return compounded semiannually?
Answer: Approximately 13.3 years. Example: How long will it take our money to triple in a bank account with an annual interest rate of 8.45% compounded annually?
Answer: Approximately 13.5 years to triple. Make a note that doubling or tripling time is independent of the principal. In the previous problem, notice that the principal was not given and that the variable P cancelled. Use the continuously compounding interest formula to solve the following. Example: If a $500 certificate of deposit earns 4 1/4% annual interest compounded continuously then how much will be accumulated at the end of a 3 year period?
Answer: the amount at the end of 3 years will be $576.99. Example: A certain investment earns 8 3/4% compounded continuously. If $10,000 dollars is invested, how much will be in the account after 5 years?
Answer: The amount at the end of five years will be $15,488.30. The previous two examples are the same examples that we started this chapter with. This allows us to compare the accumulated amounts to that of regular compound interest.
As we can see, continuous compounding is better, but not by much. Instead of buying a new car for say $20,000, let us invest in the future of our family. If we invest the $20,000 at 6% annual interest compounded continuously for say, two generations or 100 years, then how much will our family have accumulated in that time? The answer is over 8 million dollars. One can only wonder actually how much that would be worth in a century.
Given continuously compounding interest, we are often asked to find the doubling time. Instead of taking the common log of both sides it will be easier take the natural log of both sides, otherwise the steps are the same. Example: How long does it take to double $1000 at an annual interest rate of 6.35% compounded continuously?
Answer: The account will double in about 10.9 years. The key step in this process is to apply the natural logarithm to both sides so that we can apply the power rule and solve for t. Use the calculator in the last step and round-off only once. Example: How long will it take $30,000 to accumulate to $110,000 in a trust that earns a 10% annual return compounded continuously?
Answer: Approximately 13 years. Example: How long will it take our money to triple in a bank account with an annual interest rate of 8.45% compounded continuously?
Answer: Approximately 13 years. YouTube Video: --- How do you calculate compounded quarterly?Cq = P [ (1+r)4*n – 1 ]
The quarterly compounding formula is taken from the compounding formula. The only difference is that the rate of interest is raised 4*2 to reflect the quarterly computation of the interest.
How can calculate compound interest in PHP?php function compound_interest($principal, $year, $interest){ $converted_interest = $interest / 100; $completed_interest = $converted_interest + 1; $exponent = pow($completed_interest,$year); $final = $principal * $exponent; return number_format((float)$final, 2, ',', '.
How much is $1000 worth at the end of 2 years if the interest rate of 6% is compounded daily?Compound interest formulas
Hence, if a two-year savings account containing $1,000 pays a 6% interest rate compounded daily, it will grow to $1,127.49 at the end of two years.
How many years will it take to double in value if it earns 5% compounded annually?Using the rule of 72, you would estimate that an investment with a 5% compound interest rate would double in 14 years (72/5).
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What is the value of an investment of php 3,500 after 2 years if it earns 1.5% compounded quarterly?
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