Math 418.1                                               Lecture 3  

                          Present Value, Discount Rate, and Nominal Rates  

 

 

 

Compound Interest

 

$1                1+ i                 

|___________|___________|____________|____________|_____________________

0                    1                     2                      3                       4        .  .  .  .               time

 

 

Present Value

                  v                     $1

|___________|___________|____________|____________|_____________________

0                    1                     2                      3                       4        .  .  .  .               time

 

 

 

Simple Interest

 

$1                1+ i                1+2i

|___________|___________|____________|____________|_____________________

0                    1                     2                      3                       4        .  .  .  .               time

 

 

Present Value

                          $1

|___________|___________|____________|____________|_____________________

0                    1                     2                      3                       4        .  .  .  .               time

 

 

Examples

Find the amount that must be invested at a rate of simple interest rate of 8% per annum in order to accumulate $100 at the end of three years.

 

 

 

Find the amount that must be invested at a rate of compound interest rate of 8% per annum in order to accumulate $100 at the end of three years.

 

 

 

 

Discount Rate

 

$1-d              $1

|___________|___________|____________|____________|_____________________

0                    1                     2                      3                       4        .  .  .  .               time

 

Interest – paid at the end of the period on the balance at the beginning of the period

Discount – paid at the beginning of the period on the balance at the end of the period

 

Effective rate of discount during the nth period

 

 

 

 

Compound Discount

 

         1-d                  $1

|___________|___________|____________|____________|_____________________

0                    1                     2                      3                       4        .  .  .  .               time

 

 

Simple Discount

 

1-2d             1-d                  $1

|___________|___________|____________|____________|_____________________

0                    1                     2                      3                       4        .  .  .  .               time

 

 

 

Examples

Find the amount that must be invested at a rate of simple discount rate of 8% per annum in order to accumulate $100 at the end of three years.

 

 

 

 

 

Find the amount that must be invested at a rate of compound discount rate of 8% per annum in order to accumulate $100 at the end of three years.

 

 

 

 

Remark:

Two rates of interest or discount are said to be equivalent if a given amount of principal invested for the same length of time  at each of the rates produces the same accumulated value.

 

 

 

 

 

 

 

Nominal Interest Rate

 

$1                                          

|___________|___________|____________|____________|_____________________

0                  1/4                   1/2                   3/4                     1        .  .  .  .               time

                 

 

 

 

Nominal Discount Rate

 

                              $1

      |___________|___________|____________|____________|_______________

     0                  1/4                   1/2                   3/4                      1        .  .  .  .              time

                 

 

Examples

 

Find the accumulated value of $100 invested for 5 years at 6% per annum convertible

(a) quarterly

 

 

 

(b) semiannually

 

 

 

(c) monthly

 

 

 

 

Find the present value of $100 to be paid at the end of 4 years at 6% per annum payable in advance and convertible

(a) semiannually

 

 

 

 

(b) quarterly

 

 

 

 

 

Find the nominal interest rate convertible quarterly which is equivalent to a nominal discount rate of 7% per annum convertible monthly.

 

 

 

 

 

 

 

Nov. 2001

#24  David can receive one of the following two payment streams:

(i)                  100 at time 0, 200 at time n, and 300 at time 2n

(ii)                600 at time 10

At an annual effective interest rate of i, the present values of the two streams are equal. Given , determine i.

 

 

 

 

 

Ans. 3.5%

 

May 2001

#7  Seth, Janice, and Lori each borrows 5000 for five years at a nominal interest rate of 12% compounded semi-annually. Seth has interest accumulated over the five years and pays all the interest and principal in a lump sum at the end of five years. Janice pays the interest at the end of every six-month period as it accrues and the principal at the end of five years. Lori repays her loan with 10 level payments at the end of every six-month period. Calculate the total amount of interest paid on all three loans.

 

 

 

 

 

Ans.8747.64

 

 

May 2001

#12  Bruce and Robbie each open up new bank accounts at time 0. Bruce deposits 100 into his bank account, and Robbie deposits 50 into his. Each account earns an annual effective discount rate of d. The amount of interest earned in Bruce’s account during the 11th year is equal to X. The amount of interest earned in Robbie’s account during the 17th year is also equal to X. Calculate X.

 

 

 

 

 

Ans.38.88

 

 

May 2001

#49  Tawny makes a deposit into a bank account which credits interest at a nominal interest rate of 10% per annum, convertible semiannually. At the same time, Fabio deposits 1000 into a different bank account, which is credited with simple interest. At the end of 5 years, the forces of interest on the two accounts are equal, and Fabio’s account has accumulated to Z. Determine Z.

 

 

 

 

 

Ans.1953