Time Value of Money MATH 100 Survey of Mathematical Ideas J. Robert Buchanan Department of Mathematics Fall 2014
Terminology Money borrowed or saved increases in value over time. principal: the amount borrowed or deposited. interest: the increase in value of the principal. interest rate: percentage increase in the principal (usually per year).
Terminology Money borrowed or saved increases in value over time. principal: the amount borrowed or deposited. interest: the increase in value of the principal. interest rate: percentage increase in the principal (usually per year). Interest calculated on principal only is called simple interest. Interest calculated on principal plus any previously earned interest is called compound interest.
Simple Interest Definition If P is the principal, r is the annual interest rate, and t is the time (in years), then the simple interest I is calculated as I = P r t.
Simple Interest Definition If P is the principal, r is the annual interest rate, and t is the time (in years), then the simple interest I is calculated as Definition I = P r t. The principal plus interest (for either a loan or a deposit) is called the future value (or future amount or amount due or maturity value). The future value is denoted A and is calculated as A = P + I = P + P r t = P(1 + r t). The principal P will be referred to as the present value.
Example Suppose $800 is deposited for 3 years at 3.5% annual simple interest. The interest earned is I = P r t = (800)(0.035)(3) = $84. The future value of the principal is A = P(1 + r t) = 800(1 + (0.035)(3)) = $884.
Example Suppose $800 is deposited for 3 years at 3.5% annual simple interest. The interest earned is I = P r t = (800)(0.035)(3) = $84. The future value of the principal is A = P(1 + r t) = 800(1 + (0.035)(3)) = $884. Use your i>clicker2 to enter the future value after 7 years.
Compound Interest After a certain period, accounts earning compound interest are credited with interest (added to the principal) and this sum of the principal plus interest earns interest for the next period.
Compound Interest After a certain period, accounts earning compound interest are credited with interest (added to the principal) and this sum of the principal plus interest earns interest for the next period. Definition If P dollars are deposited at an annual interest rate of r, compounded m times per year, and the money is left on deposit for time t, then the future value is A = P ( 1 + r m ) m t. The principal amount P is called the present value of A.
Compound Interest After a certain period, accounts earning compound interest are credited with interest (added to the principal) and this sum of the principal plus interest earns interest for the next period. Definition If P dollars are deposited at an annual interest rate of r, compounded m times per year, and the money is left on deposit for time t, then the future value is A = P ( 1 + r m ) m t. The principal amount P is called the present value of A. If the future value A is known, the following formula calculates the present value. ( P = A 1 + r ) m t. m
Example Revisited Suppose $800 is deposited for 3 years at 3.5% annual interest compounded quarterly (four times per year). The future value will be ( A = 800 1 + 0.035 ) (4)(3) = 800(1.110203) = $888.16. 4
Example Revisited Suppose $800 is deposited for 3 years at 3.5% annual interest compounded quarterly (four times per year). The future value will be ( A = 800 1 + 0.035 ) (4)(3) = 800(1.110203) = $888.16. 4 Compounding Frequency m Annually 1 Semiannually 2 Quarterly 4 Monthly 12 Weekly 52 Daily 365
Comparison of Simple and Compound Interest Suppose $1000 is deposited and earns 5% annual interest. The future values when the interest is simple versus compounded monthly are listed in the table below. ( t (years) P(1 + r t) P 1 + r ) 12t 12 0 1000.00 1000.00 1 1050.00 1051.16 2 1100.00 1104.94 3 1150.00 1161.47 4 1200.00 1220.90 5 1250.00 1283.36 6 1300.00 1349.02 7 1350.00 1418.04 8 1400.00 1490.59 9 1450.00 1566.85
Examples Use your i>clicker2 to enter the following results. Round all results to the nearest dollar (no cents). Find the future value of $14, 000 compounded quarterly at 3.79% for 4 years.
Examples Use your i>clicker2 to enter the following results. Round all results to the nearest dollar (no cents). Find the future value of $14, 000 compounded quarterly at 3.79% for 4 years. Find the present value of $11, 000 compounded semiannually at 6% for 10 years.
Doubling Time The length of time necessary for a present value to double is called the doubling time. ( 2P = P 1 + r ) n m ( 2 = 1 + r ) n ( m ln 2 = n ln 1 + r ) m ln 2 n = ln ( 1 + r ) m
Doubling Time Example Find the number of compounding periods necessary for an amount deposited to double in value if it is earning 3.25% interest compounded daily. n = ln 2 ln ( 1 + r ) = m ln 2 ln ( 1 + 0.0325 365 ) = 0.693147 = 7784.9 8.904 10 5
Doubling Time Example Find the number of compounding periods necessary for an amount deposited to double in value if it is earning 3.25% interest compounded daily. n = ln 2 ln ( 1 + r ) = m ln 2 ln ( 1 + 0.0325 365 ) = 0.693147 = 7784.9 8.904 10 5 It will take approximately 7785 days (21.3 years) for the principal amount to double.
Example An investor deposits $1,000 in an account earning 5% interest compounded monthly. Use your i>clicker2 to enter the number of months equal to the doubling time. Round your answer to the nearest whole month.
Effective Annual Yield Mortgage lenders and credit card issuers often state two interest rates associated with their loans. nominal rate: the annual interest rate (often with compounding) effective annual rate: the equivalent simple interest rate which will produce the same future value (this is the APR in the fine print).
Effective Annual Yield Mortgage lenders and credit card issuers often state two interest rates associated with their loans. nominal rate: the annual interest rate (often with compounding) effective annual rate: the equivalent simple interest rate which will produce the same future value (this is the APR in the fine print). Definition A nominal interest rate of r, compounded m times per year is equivalent to an effective annual rate of Y = ( 1 + r m ) m 1.
Example A credit union offers two types of savings account. Find the effective annual yield of each account if compounding takes place monthly. Rate Super Saver Account 2.75% Premier Savings Account 3.25% Yield
Example A credit union offers two types of savings account. Find the effective annual yield of each account if compounding takes place monthly. Y = Rate Yield Super Saver Account 2.75% 2.78% Premier Savings Account 3.25% ( 1 + r ) m ( 1 = 1 + 0.0275 ) 12 1 = 0.027849 m 12
Example A credit union offers two types of savings account. Find the effective annual yield of each account if compounding takes place monthly. Y = Rate Yield Super Saver Account 2.75% 2.78% Premier Savings Account 3.25% 3.30% ( 1 + r ) m ( 1 = 1 + 0.0325 ) 12 1 = 0.032989 m 12
Example What is the effective annual yield of an account earning interest at 5% compounded quarterly? Round your answer to the nearest hundredth of a percent and submit your answer using your i>clicker2.
Inflation Recall: we have stated that when money is borrowed or lent it gains value over time through interest. We generally use money to buy things. When compared to the amount of things money can buy, money tends to lose value over time due to the process of inflation. The Bureau of Labor Statistics publishes the consumer price index (CPI) which attempts to measure the percentage change in the prices of commonly purchased items such as food, housing, fuel, clothing, etc.
Consumer Price Index Average % Change Year CPI-U CPI-U 1982 96.5 6.2 1983 99.6 3.2 1984 103.9 4.3 1985 107.6 3.6 1986 109.6 1.9 1987 113.6 3.6 1988 118.3 4.1 1989 124.0 4.8 1990 130.7 5.4 1991 136.2 4.2 1992 140.3 3.0 1993 144.5 3.0 1994 148.2 2.6 1995 152.4 2.8 1996 156.9 3.0 1997 160.5 2.3 Average % Change Year CPI-U CPI-U 1998 163.0 1.6 1999 166.6 2.2 2000 172.2 3.4 2001 177.1 2.8 2002 179.9 1.6 2003 184.0 2.3 2004 188.9 2.7 2005 195.3 3.4 2006 201.6 3.2 2007 207.3 2.8 2008 215.3 3.8 2009 214.5 0.4 2010 218.1 1.6 2011 224.9 3.2 2012 229.6 2.1 2013 233.0 1.5
Inflation Proportion Definition For a given consumer product or service subject to average inflation, prices in two different years are related through the equation: Price in year A CPI in year A = Price in year B CPI in year B.
Inflation Proportion Definition For a given consumer product or service subject to average inflation, prices in two different years are related through the equation: Price in year A CPI in year A = Price in year B CPI in year B. Suppose college tuition in 2001 was $7, 500. What would college tuition have been in 2011?
Inflation Proportion Definition For a given consumer product or service subject to average inflation, prices in two different years are related through the equation: Price in year A CPI in year A = Price in year B CPI in year B. Suppose college tuition in 2001 was $7, 500. What would college tuition have been in 2011? Price in 2011 Price in 2001 Price in 2011 7500 CPI in 2011 = CPI in 2001 = 224.9 177.1 Price in 2011 = 7500 224.9 = $9, 524 177.1
Example Suppose the cost of an X-ray of your arm in 1995 was $179. What would the same X-ray have cost in 2007? Round your answer to the nearest dollar and enter your result using your i>clicker2.
Continuous Compounding Definition If an initial deposit of P dollars earns continuously compounded interest at an annual rate r for t years, then the future value is given by A = P e r t. The principal amount P is the present value of A.
Continuous Compounding Definition If an initial deposit of P dollars earns continuously compounded interest at an annual rate r for t years, then the future value is given by A = P e r t. The principal amount P is the present value of A. If the future value A is known the present value can be calculated as P = Ae rt.
Continuous Compounding Definition If an initial deposit of P dollars earns continuously compounded interest at an annual rate r for t years, then the future value is given by A = P e r t. The principal amount P is the present value of A. If the future value A is known the present value can be calculated as P = Ae rt. Example Assume the average price of a gallon of gasoline is $4.00. Find the price of a gallon of gasoline in 2018 assuming inflation is 3% (compounded continuously).
Continuous Compounding Definition If an initial deposit of P dollars earns continuously compounded interest at an annual rate r for t years, then the future value is given by A = P e r t. The principal amount P is the present value of A. If the future value A is known the present value can be calculated as P = Ae rt. Example Assume the average price of a gallon of gasoline is $4.00. Find the price of a gallon of gasoline in 2018 assuming inflation is 3% (compounded continuously).
Example Assume the average price of a gallon of gasoline is $4.00. Find the price of a gallon of gasoline in 2018 assuming inflation is 10% (compounded continuously). Use your i>clicker2 to enter the answer. Round your answer to the nearest cent.
Years to Double Since the inflation rate changes from year to year, we treat it as an approximation. Often we are interested in the amount of time it takes prices to double. 2P = Pe r t 2 = e r t ln 2 = r t t = ln 2 100 ln 2 = r 100r 70 years to double annual inflation rate
Years to Double Since the inflation rate changes from year to year, we treat it as an approximation. Often we are interested in the amount of time it takes prices to double. 2P = Pe r t 2 = e r t ln 2 = r t t = ln 2 100 ln 2 = r 100r 70 years to double annual inflation rate If inflation is 7% it takes approximately 10 years for prices to double.
Example Find the years to double using the rule of 70 assuming inflation is 3%, Round your result to the nearest tenth of a year and enter your answer using your i>clicker2.
Example Find the years to double using the rule of 70 assuming inflation is 3%, 5%. Round your result to the nearest tenth of a year and enter your answer using your i>clicker2.