Math 166: Topics in Contemporary Mathematics II

Transcription

1 Math 166: Topics in Contemporary Mathematics II Xin Ma Texas A&M University October 28, 2017 Xin Ma (TAMU) Math 166 October 28, / 10

2 TVM Solver on the Calculator Unlike simple interest, it is much more complicated with compound to go backwards and find the amount of time that has passed or the interest rate. But it gets easier when we use our calculators. 1. Press APPS 2. Select Finance (Option 1) 3. Select TVM Solver (Option 1) N = number of compounding periods (i.e. N=mt) I% = interest rate (as a percent) PV = the present value (i.e. P) PMT = the payment amount (this is zero in this section) FV = future value (i.e. F ) P/Y = C/Y= the number of compounding periods in a year (i.e. m) PMT is when the period payments are made. In this class, we will always select END. 4. Enter the known information 5. Scroll to the line representing the unknown data. Press ALPHA and then ENTER to solve Xin Ma (TAMU) Math 166 October 28, / 10

3 TVM Solver on the Calculator Note: In the TVM Solver, the values for PV, PMT, and FV will sometimes be negative. This is done to represent the transfer or flow of money. We will usually look at these problems from the standpoint of the investor or borrower. An outflow of money is usually represented as a negative number. An inflow of money is represented as a positive number. Examples: Interest earned on your savings account is an inflow of money to you, but an outflow of money to the bank. Interest paid on a student loan is an outflow of money to you, but an inflow of money to the bank. You make a payment on your credit card. This is an outflow from the standpoint of your checking account and an inflow from the from the standpoint of your credit card account. Xin Ma (TAMU) Math 166 October 28, / 10

4 Examples of Compound Interest (Find F ) A savings account with an initial investment of $5000 earns annual interest of 6%. Find the amount in the account after 4 years if a. interest is compounded quarterly? N= 4 4 = 16 times I%=6 PV= 5000 PMT=0 FV= P/Y=C/Y= 4 b. interest is compounded monthly? N=12 4 = 48 times I%=6 PV=5000 PMT=0 FV= P/Y=C/Y=12 Xin Ma (TAMU) Math 166 October 28, / 10

5 Examples of Compound Interest (Find r) Peter invested $2000 in an account that compounded interest weekly (52 periods in a year) 2 years ago. Right now Peter earns $350 interest. What is the annual interest rate for this account? (Represent in a percent and round to two decimal places) N=52 2 = 104 I%=8.07 PV=2000 PMT=0 FV= =2350 P/Y=C/Y=52 Xin Ma (TAMU) Math 166 October 28, / 10

6 Examples of Compound Interest (Find P) How much should be placed into an account paying annual interest of 5% compounded quarterly so that after 4 years, the future value of the account will be $6000? N= 4 4 = 16 times I%=5 PV= PMT=0 FV=6000 P/Y=C/Y=4 Xin Ma (TAMU) Math 166 October 28, / 10

7 Examples of Compound Interest (Find t) Find the time in years (rounded to two decimal places) for an investment of $3000 compounding weekly at an interest rate of 8% per year to double. N=52t=451 whence t = 8.67 I%=8 PV=3000 PMT=0 FV=6000 P/Y=C/Y=52 Xin Ma (TAMU) Math 166 October 28, / 10

8 Effective Yield Definition: Suppose a sum of money is invested at an annual rate (or nominal rate) of r (expressed as a decimal) and is compounded m times a year. The effective yield r eff is r eff = (1 + r m )m 1 Find r eff by using calculator 1. Press APPS and select Finance (Option 1). 2. Scroll down and select Eff (Option C). 3. Enter r (as a percent), a comma, m, close parenthesis and hit ENTER. The effective rate will be given as a percent. Example: What is the effective annual yield on an account paying 8% interest per year, compounded monthly? (Round to four decimal places.) Xin Ma (TAMU) Math 166 October 28, / 10

9 Effective Yield Definition: Suppose a sum of money is invested at an annual rate (or nominal rate) of r (expressed as a decimal) and is compounded m times a year. The effective yield r eff is r eff = (1 + r m )m 1 Find r eff by using calculator 1. Press APPS and select Finance (Option 1). 2. Scroll down and select Eff (Option C). 3. Enter r (as a percent), a comma, m, close parenthesis and hit ENTER. The effective rate will be given as a percent. Example: What is the effective annual yield on an account paying 8% interest per year, compounded monthly? (Round to four decimal places.) r eff = (1 + 8% 12 )12 1 = 8.3% Xin Ma (TAMU) Math 166 October 28, / 10

10 Comparing Effective Yield Compare the following situations. Which would be the better choice for an investment? Which would be the better choice for a credit card? A. 8.1% compounded semiannually. B. 8% compounded daily Xin Ma (TAMU) Math 166 October 28, / 10

11 Comparing Effective Yield Compare the following situations. Which would be the better choice for an investment? Which would be the better choice for a credit card? A. 8.1% compounded semiannually. B. 8% compounded daily For A r eff = ( /2) 2 1 = 8.26% Xin Ma (TAMU) Math 166 October 28, / 10

12 Comparing Effective Yield Compare the following situations. Which would be the better choice for an investment? Which would be the better choice for a credit card? A. 8.1% compounded semiannually. B. 8% compounded daily For A r eff = ( /2) 2 1 = 8.26% For B r eff = ( /365) = 8.33% Xin Ma (TAMU) Math 166 October 28, / 10

13 Comparing Effective Yield Compare the following situations. Which would be the better choice for an investment? Which would be the better choice for a credit card? A. 8.1% compounded semiannually. B. 8% compounded daily For A r eff = ( /2) 2 1 = 8.26% For B r eff = ( /365) = 8.33% Then for an investment, choose the Bank B since it will pay more to you. For a credit card, choose Bank A since you will pay less to the bank. Xin Ma (TAMU) Math 166 October 28, / 10

14 Continuous Compounding of Interest Interest can also compound continuously. Suppose a principal P earns interest at an annual rate r (as a decimal) compounded continuously for t years. Then the future value F = Pe rt Example: If you place $5000 into a saving account that has 3% annual interest compounded continuously, how much money will you have after 20 years and how much interest have you earned? Xin Ma (TAMU) Math 166 October 28, / 10

15 Continuous Compounding of Interest Interest can also compound continuously. Suppose a principal P earns interest at an annual rate r (as a decimal) compounded continuously for t years. Then the future value F = Pe rt Example: If you place $5000 into a saving account that has 3% annual interest compounded continuously, how much money will you have after 20 years and how much interest have you earned? F = 5000e = Xin Ma (TAMU) Math 166 October 28, / 10