simple_and_compound_interest

simple_and_compound_interest - Business Calculus Vera...

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Unformatted text preview: Business Calculus Vera Klimkovsky MATHEMATICS OF FINANCE: SIMPLE AND COMPOUND INTEREST 1. SIMPLE INTEREST Simple interest is interest that is computed on the original principal only. Let P represent the principal amount, r represent an interest rate per year and t is time, in years, then we can define an interest earned (paid on principal) as I = Prt . The accumulated amount , A , is the sum of the principal and the interest paid on the principal, A = P + I = P + Prt = P (1 + rt ) Example: A bank pays simple interest at a rate of 8% per year for a deposit. If a customer deposits $1000 and makes no withdrawals for 3 years, what is the total amount on deposit at the end of 3 years? What is the interest earned in that period of time? Solution: P = $1000 (principal), r = 0.08 (interest rate), t = 3 (years) The total amount on deposit at the end of 3 years is given by A = P (1 + rt ) = 1000 (1 + 0.08 × 3) = $1, 240.00 The interest earned over the 3-year period is given by I = Prt = 1000 × 0.08 × 3 = $240.00 . Business Calculus Vera Klimkovsky 2. COMPOUND INTEREST Compound Interest is the interest computed on the principle amount to which interest earned to-date has been added. If amount of P dollars is invested over a term of t years, earning interest at the rate of r per year compounded annually, then the accumulated amount is A = P (1 + r ) t If interest at a nominal rate of r per year is compounded m times a year on a principal of P dollars, then the simple interest rate per conversion period is i= r Annual interest rate = m Periods per year Since there are n = mt (number of conversion periods in a year times the term, number of years), we can write the final general formula: A = P (1 + i ) n r⎞ ⎛ = P ⎜1 + ⎟ ⎝ m⎠ mt Example: Suppose $1000 is deposited in a bank for a term of 3 years, earning interest at the nominal rate of 8% per year compounded annually. Find the accumulated amount at the end of 3 years. Solution: The accumulated amount at the end of the third year is A = P(1 + r )3 = 1000 (1 + 0.08 ) ≈ $1259.71 3 Business Calculus Vera Klimkovsky Example Find the accumulated amount after 3 years if $1000 is invested at 8% per year compounded (a) annually (already done above) (b) semiannually (c) quarterly (d) monthly (e) daily. Solution: Here P = 1000, r = 0.08, t = 3 (a) Annually, m = 1 A = P (1 + i ) (b) mt n r⎞ 3 ⎛ = P ⎜1 + ⎟ = 1000 (1 + 0.08 ) = 1259.71 ⎝ m⎠ Semiannually, m = 2 mt 6 r⎞ ⎛ ⎛ 0.08 ⎞ A = P (1 + i ) = P ⎜1 + ⎟ = 1000 ⎜1 + ⎟ = 1265.32 2⎠ ⎝ m⎠ ⎝ (c) Quarterly, m = 4 n A = P (1 + i ) (d) mt n 36 r⎞ ⎛ ⎛ 0.08 ⎞ = P ⎜1 + ⎟ = 1000 ⎜1 + ⎟ = 1270.24 m⎠ 12 ⎠ ⎝ ⎝ Daily, m = 365 A = P (1 + i ) 12 r⎞ ⎛ ⎛ 0.08 ⎞ = P ⎜1 + ⎟ = 1000 ⎜1 + ⎟ = 1268.24 4⎠ ⎝ m⎠ ⎝ Monthly, m = 12 A = P (1 + i ) (e) mt n mt n 1095 r⎞ ⎛ ⎛ 0.08 ⎞ = P ⎜1 + ⎟ = 1000 ⎜1 + ⎟ 365 ⎠ ⎝ m⎠ ⎝ = 1271.22 Business Calculus Vera Klimkovsky 3. CONTINUOUS COMPOUNDING Continuous compounding occurs when the interest is computed and added to the principal at every instance, thus making the compounding period infinitely small and therefore, making m infinitely large. m 1⎞ ⎛ 1 + ⎟ as m becomes very If we examine the expression ⎜ ⎝ m⎠ large, we see that it approaches some fixed value. This value is known as constant e (Euler’s number) e ≈ 2.71828182845904523536... Business Calculus Vera Klimkovsky m mt 1⎞ r⎞ ⎛ ⎛ 1 + ⎟ and ⎜ 1 + ⎟ , Notice, that the two expressions, ⎜ ⎝ m⎠ ⎝ m⎠ are very similar but not exactly the same. Now, it can also be shown (by using mathematical manipulations) that mt r⎞ ⎛ A = P ⎜1 + ⎟ = Pe rt ⎝ m⎠ as m → ∞ (become infinitely large) Example Find the accumulated amount after 3 years if $1000 is invested at 8% per year compounded continuously. Solution: A = Pe rt = 1000 ⋅ e0.08×3 = $1, 271.25 ...
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This note was uploaded on 10/19/2011 for the course MATH 110 taught by Professor Staff during the Spring '11 term at S.F. State.

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