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Lecture 5: Compound Interest Formula MA170: Introduction to Mathematics of Finance January 18, 2011 Mathematics of Finance (MA170) Lecture 5: Compound Interest Formula January 18, 2011 1 / 12 Outline 1 Equivalent Compound Interest Rates 2 Discounted Value Text: Chapter 2, Sect. 2.2, 2.3 Mathematics of Finance (MA170) Lecture 5: Compound Interest Formula January 18, 2011 2 / 12 Equivalent Compound Interest Rates Annual Eﬀective Rate Deﬁnition For a given nominal rate jm compounded m time per year, we deﬁne the corresponding annual eﬀective rate to be that rate j which if compounded annually will produce the same amount of interest per year Formula • At the rate i = jm /m, $1 will accumulate at the end of one year to $(1 + i)m • At the rate j, $1 will accumulate at the end of one year to $(1 + j) • Therefore, 1 + j = (1 + i)m ⇒ j = (1 + i)m − 1 = (1 + jm /m)m − 1 Mathematics of Finance (MA170) Lecture 5: Compound Interest Formula January 18, 2011 3 / 12 Equivalent Compound Interest Rates Annual Eﬀective Rate: Example #1 (2.2.1a,b) Determine the annual eﬀective rate (two decimals) equivalent to the following rates: a) j2 = 7%; b) j4 = 16% Mathematics of Finance (MA170) Lecture 5: Compound Interest Formula January 18, 2011 4 / 12 Equivalent Compound Interest Rates Annual Eﬀective Rate: Example #1 (2.2.7) Application The eﬀective interest rate is useful for comparison of investments that have interest rates with diﬀerent compounding periods A trust company oﬀers guaranteed investment certiﬁcates paying j2 = 8.9% and j1 = 9%. Which option yields the higher annual eﬀective rate of interest? Mathematics of Finance (MA170) Lecture 5: Compound Interest Formula January 18, 2011 5 / 12 Equivalent Compound Interest Rates Equivalent Rates • Recall that the rates that have the same eﬀect on money over the same period of time are called equivalent rates • Two nominal compound interest rates are equivalent if they yield the same accumulated values at the end of one year, and hence at the end of any number of years • Let the nominal rate jm be compounded m times per year. Let the nominal rate jn be compounded n times per year. These rates are equivalent if m n jm jn 1+ = 1+ m n Mathematics of Finance (MA170) Lecture 5: Compound Interest Formula January 18, 2011 6 / 12 Equivalent Compound Interest Rates Annual Eﬀective Rate: Example #3 (2.2.2c,d) Determine the nominal rate (two decimals) equivalent to the given annual eﬀective rate: c) j = 10% determine j12 ; d) j = 17% determine j365 Mathematics of Finance (MA170) Lecture 5: Compound Interest Formula January 18, 2011 7 / 12 Equivalent Compound Interest Rates Annual Eﬀective Rate: Example #4 (2.2.4a,c) Determine the nominal rate equivalent to the given nominal rate? a) j2 = 8% determine j4 ; c) j12 = 18% determine j4 Mathematics of Finance (MA170) Lecture 5: Compound Interest Formula January 18, 2011 8 / 12 Discounted Value Discounted Value at Compound Interest • The accumulated value of P after n period at compound interest rate i will be S = P(1 + i)n • Being given the accumulated value S, ﬁnd the principal P: S P= = S(1 + i)−n (1 + i)n • P is called the present value of S or the discounted value of S. • $P invested at time 0 grows to $S after n periods. • The factor 1 (1+it)n is called the discount factor. Mathematics of Finance (MA170) Lecture 5: Compound Interest Formula January 18, 2011 9 / 12 Discounted Value Discounted Value: Example #5 (2.3.6) What amount of money invested today will grow to $1000 at the end of 5 years if j4 = 8%? Mathematics of Finance (MA170) Lecture 5: Compound Interest Formula January 18, 2011 10 / 12 Discounted Value Discounted Value: Example #5 (2.3.12) A note dated October 1, 2007, calls for the payment of $800 in 7 years. On October 1, 2008, it is sold at a price that will yield the investor j4 = 12%. How much is paid for the note? Mathematics of Finance (MA170) Lecture 5: Compound Interest Formula January 18, 2011 11 / 12 Discounted Value Recommended Exercises Section 2.2 Part A: 1(d,e), 2(a,b), 3(b,d), 4, 8a Section 2.3 Part A: 7, 9, 11, 14 Mathematics of Finance (MA170) Lecture 5: Compound Interest Formula January 18, 2011 12 / 12