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Feb 10, 2023 •306 likes •383 views

Section 5.3 d i E Interest Compounded Continuously a l l u d Dr. Abdulla Eid b A College of Science . r D MATHS 103: Mathematics for Business I Dr. Abdulla Eid (University of Bahrain) Continuous Interest 1 / 6 Recall: (Section

  1. Section 5.3 d i E Interest Compounded Continuously a l l u d Dr. Abdulla Eid b A College of Science . r D MATHS 103: Mathematics for Business I Dr. Abdulla Eid (University of Bahrain) Continuous Interest 1 / 6

  2. Recall: (Section 4.1) The compound interest formula is given by 1 + r � � nm A = P , m where m is the number of times an interest is paid. Suppose the interest is paid continuously ( frequently ) (i.e., m is a large number ( m → ∞ )). Then d how much should we earn after n years? i E Recall: � � x 1 + 1 a e = lim l l x x → ∞ u d So we have b � 1 + 1 � nm 1 + r � � mn A A = = P m m . r r Set x = m r , so we have m = xr and we get D � x � nr � � nxr �� 1 + 1 1 + 1 A = P = = P x x Take x → ∞ , we get A = Pe nr Dr. Abdulla Eid (University of Bahrain) Continuous Interest 2 / 6

  3. Note: Think of the interest compounded continuously as an interest which will be paid in every second. Effective rate: r e = e r − 1 d Present Value: i E P = Ae − nr a l l u d Example b If 1000 BD is deposited in a saving account that earns interest at an A annual rate of 5.5% compounded continuously. What is the value of the . r account at the end of 3 years. D Solution: P = 1000, A = ? , n = 3, and r = 5.5% = 0.055. Thus A = Pe nr = 1000 e 3 ( 0.055 ) = 1179.39 BD Dr. Abdulla Eid (University of Bahrain) Continuous Interest 3 / 6

  4. Exercise Find the compounded amount and the compounded interest if 400 BD is invested for 5 years in an account with interest compounded continuously 1 5 1 4 % 2 10%. d i E a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Continuous Interest 4 / 6

  5. Exercise Find the effective rate that corresponds to the given annual rate compounded continuously 1 3% 2 7%. d i 3 2% E a 4 10%. l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Continuous Interest 5 / 6

  6. Example What is the annual rate compounded continuously corresponding to effective rate of 5%? d Solution: i E a l r e = e r − 1 l u d 0.05 = e r − 1 b A 1.05 = e r . ln 1.05 = r r D 0.0488 = r 4.88% = r Dr. Abdulla Eid (University of Bahrain) Continuous Interest 6 / 6

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