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Financial Models (NEW). Section 5.7. Compound Interest Formula. If P represents the principal investment, r the annual interest rate (as a decimal), t the time in years, and n the frequency of compounding, then the future value is given by the formula: A = P( 1 + r/n) nt. Example.
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Financial Models (NEW) Section 5.7
Compound Interest Formula • If P represents the principal investment, r the annual interest rate (as a decimal), t the time in years, and n the frequency of compounding, then the future value is given by the formula: A = P( 1 + r/n)nt
Example • Suppose you invest $32,000 into a certificate of deposit that has an annual interest rate of 5.2% compounded quarterly for 3 years. • ANSWER: Use the compound interest formula. • A= 32000(1+.052/4)(4)(3) = 32000(1.013)12 = $37,364.86
Continuous Compounded Interest • What would happen if we let the frequency of compounding get very large. That is we would compound not just every hour, or every minute or every second but for every millisecond! • What happens is that the expression (1 +r/n)nt goes to ert. This e is the famous Euler number. It’s value is the irrational number 2.7182818 … • The future value formula is A = Pert.
Example of Continuous Compound interest. • Consider the $32,000 from the earlier example. Now we will invest the money in an account that has 5.2% annual interest compounded continuously for 3 years. What is the future value? • ANSWER: A = 32000e(.052)(3) = $37,402.44 • Note this investment option is only greater by $37.58.
Sometimes we would like to know how long an investment will take to grow to a certain value. • This type of question involves solving an exponential equation. • The technique for solving these types of equations is taking the natural logarithm of both sides of the equation.
Example using log Lets say we want to know how long it will take $32,000 to grow to $50,000 invested in an account that has 5.2% annual interest compounded quarterly. We use the formula A = P(1 + r/n)nt. 50000 = 32000(1 + .052/4)4t Note the unknown is in the exponent. Divide both sides of the equation by 32000, and also simplify the inside of the parentheses. This will give 1.5625 = (1.013)4t . Now take the natural log of both sides. (Must you use ln?) ln(1.5625) = ln((1.013)4t) Thus the equation is now ln(1.5625) = (4t)ln(1.013). Thus t = ln(1.5626)/(4ln(1.013) = 8.64 years or 8 years and 8 months.
From the previous problem was the amount key? NO We can use the technique to solve for time given any amount (how long to double, triple, reach $50,000).
Effective Rate of Return (Interest) Suppose you want to invest and you research three different banks. Bank A offers you 6% compounded daily, Bank B offers 6.02% compounded quarterly, and Bank C offers 5.98% compounded continuously. Which bank gives the best deal? (Best rate of return or “real” interest rate)
When looking at the equations for interest you only need to examine the “growth” factor A = P( 1 + r/n)nt The blue part gives you the real “growth” rate, (remember y = abx ) Or A = Pert , again the blue is the real rate of return.
So from the previous example: Bank A: ( 1 + r/n)n = (1 + 0.06/365)365 = 1.0618, so bank A give 6.183% true interest. Bank B: ( 1 + 0.0602/4)4 =1.06157, so 6.157% true interest. Bank C: er = e0.0598 =1.0616, so 6.16% true interest. Best deal is bank A.
