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Mathematics of Finances 1. Simple Interest- is calculated by the equation I=Prt where; I = dollar amount of interest earned P= principal amount in dollars r = annual interest rate in decimals t = time in years EXAMPLE: If $8000 is invested for 2 years at an annual interest rate of 9%, how much interest will be received at the end of the 2-year period? Solution: The interest is I= Prt = 8000 (0.09)(2) = $1440. 2. Future value of an investment or a loan is calculated by: S= P + I where; or S = P + Prt S = the future value I = interest P = principal r = annual interest rate in decimals t = time in years EXAMPLE: If $6000 is invested for 4 years at an annual interest rate of 8%, how much interest will be received at the end of the 4-year period and what will be the future value of the investment? Solution: The simple interest is I =Prt = 6000 (0.08 )(4) = $1920 The future value is S=P+I= 6000 + 1920 = $7920. 3. Annual Compounding- The interest for each period is added to the principal before interest is calculated for the next year. S = P ∗ (1 + r ) n where; S = future value P= principal in dollars r = interest rate in decimal n= number of years EXAMPLE: If $3000 is invested for 4 years at 9% compounded annually, what is the future value of investment? Solution: The future value is the 4 S=$3000 ∗ (1+0.09) =$3000 ∗ (1.4115816) =$4234.7448 =$4234.74 NOTE: Future Value is a dollar amount. Therefore, it must be rounded to the nearest cent. The Math Center ■ Valle Verde ■ Tutorial Support Services ■ EPCC 1 4. Periodic Compounding- is used when the interest on an investment or a loan is compounded more than once a year. Future Value: r⎞ ⎛ S = P ∗ (1 + i ) = P ∗ ⎜1 + ⎟ ⎝ m⎠ ( m∗t ) n S = future value P = principal or present value i = interest rate per compounding period, n- number of compounding periods r , where r is the nominal interest rate m and m is the number of times compounded per year. This is expressed as a decimal. - n is found by the equation n = m ∗ t where m is the number of times compounded per year and t is the number of years invested. -This method can be used when compounding semi- annually (2 times per year), quarterly (4 times per year), monthly (12 times per year) or daily (365 times per year). - interest(i) is found by the equation i = EXAMPLE: What amount must be invested now in order to have $12,000 after 3 years if money is worth 6% compounded semiannually? Solution: Find the preset value P. The future value is 0.06 S=$12,000. Use i= =0.03 and n=3(2)=6. 2 n S=P ∗ (1+i) $12,000=P ∗ (1+0.03) 6 = P ∗ (1.03) 6 = P ∗ (1.1940523) 12,000 = $10,049.81, to the nearest cent. P= 1.1940523 5. Continuous Compounding- is used to determine the interest that results from compounding continuously. Future Value: S = P ∗ e (r ∗t ) P = principal r = interest rate, t = number of years Example: Find the future value if $1000 is invested for 20 years at 8%, compounded continuously. Solution: The future value is (0.08 )∗( 20 ) S=$1000*e =$1000 ∗ e 1.6 =$1000*(4.95303) =$4953.03 The Math Center ■ Valle Verde ■ Tutorial Support Services ■ EPCC 2