Lecture Compound Interest Formula by alicejenny

VIEWS: 7 PAGES: 12

									                 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

								
To top