Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-1 C ompound Chapter 9

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-2 Calculate the… Learning Objectives After completing this chapter, you will be able to: … effective interest rate when given a nominal interest rate … interest rate and term in compound interest applications … equivalent interest rate at another compounding frequency when given a nominal interest rate LO-1 LO-2 LO-3

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-3 Some key Keys!

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-4 The nominal interest rate ( I nterest / Y ear) 1.Number of compoundings (for lump payments) 2.Number of payments (for annuities) Represents the Periodic Annuity P ay m en t (used in chapter 10) Tells the calculator to compute (CPT) Present Value or initial(first) lump sum Find the following KEYS: Future Value or terminal(last) lump sum

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-5 However, we can now input the number of compoundings per year into the financial calculator. This can be performed by using the symbol Find the following KEYS: …it is rare for interest to be compounded only once per year! Previously, it was noted that To access this symbol use: …and you will see

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-6 The 12 is a default setting This display is referred to as “the worksheet”. … represents the number of P ayments per y ear … represents the number of C ompoundings per y ear To access use: Note: You can override these values by entering new ones! …more Appears automatically Appears automatically

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-7 Find the following KEYS: The Power function Key. Used to calculate the value of exponents. Used to access symbols located “above” another key, i.e. its acts like the SHIFT key on a computer keyboard. Use a calculator and algebraic sequencing Changes the sign of the data value of the number being displayed.

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-8 Some calculators have the y x symbol above the calculator key. (1.04) 8 is… The key stroke sequence to evaluate an EXPONENT that is… Positive Find the following KEYS: Use a calculator and algebraic sequencing Negative (1.04) - 8 is…1.048

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-9 This calculator can store up to 10 values. Find the following KEYS: Use a calculator and algebraic sequencing Used to Sto re or save displayed values. Used to Recall the saved values. Let’s Practise Therefore, the calculator must be informed as to where the values are to be stored.

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-10 Use a calculator and algebraic sequencing Using the key e.g. you want to store the value ’45’. The key stroke sequence ‘to store’ is: 45..choose from …’clear’ display The key stroke sequence ‘to recall’ is: …where you stored the value!

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-11 What amount must you invest now at 6% compounded daily to accumulate to $5000 after 1 year? n = 1 x 365 = 365 i =.06/365 FV = $5000 Solve… PV = FV(1 + i) - n Formula Extract necessary data... LO-1

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© = $ Solve… PV = FV(1 + i) - n Formula 4, n = 1 x 365 = 365 i =.06/365 FV = $

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© , = $ n = 1 x 365 = 365 i =.06/365 FV = $

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-14 Suppose you have $5000 and you want it to grow to $9000 in 7 years so you could buy a car. What rate of interest do you need to accomplish your goal? A ssume interest is compounded annually. Solving for i this becomes… i = (FV/PV) 1/n - 1 PV = $5000 n = 7 x 1 = 7 FV = $9000 FV = PV(1 + i) n Formula Extract necessary data... See next slide for steps in developing this formula

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-15 Formula i = (FV/PV) 1/n - 1 Developing the Formula FV = PV(1 + i) n 2. Divide both sides by PV 1. Base formula FV = PV(1 + i) n PV FV PV 3. To remove n as exponent raise both sides to the power 1/n = (1 + i) n 1/n 4. Complete reducingFV PV 1/n = 1 + i i = FV PV 1/n - 1 i = (FV/PV) 1/n - 1 FV PV = (1 + i) n Back to Solving the problem

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© % Solve… Suppose you have $5000 and you want it to grow to $9000 in 7 years so you could buy a car. What rate of interest do you need to accomplish your goal? A ssume interest is compounded annually. Formula i = (FV/PV) 1/n - 1 PV = $5000 n = 1 x 7 = 7 FV = $9000

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© % I/Y = Suppose you have $5000 and you want it to grow to $9000 in 7 years so you could buy a car. What rate of interest do you need to accomplish your goal? A ssume interest is compounded annually. PV = $5000 n = 1 x 7 = 7 FV = $9000

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© % per month 1.9% per quarter % daily 1.3 * 12 = 1.9 * 4 = * 365 = Find the Nominal Rate if the Periodic Rate is: The formula gives the Periodic Rate (i) but in practice we usually refer to the Nominal(annual)Rate 15.6% per year 7.6% per year = 18.4% per year

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-19 Suppose you have $5000 and you want it to grow to $9000 in 7 years so you could buy a car. What rate of interest do you need to accomplish your goal? PV = $5000 FV = $9000 n = 7 x 12 = 84 Solve… A ssume interest is compounded monthly. Formula i = (FV/PV) 1/n - 1 Extract necessary data...

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© per month Solve… % nominal rate.0843 Suppose you have $5000 and you want it to grow to $9000 in 7 years so you could buy a car. What rate of interest do you need to accomplish your goal? A ssume interest is compounded monthly. Formula i = (FV/PV) 1/n - 1 PV = $5000 n = 7 x 12 = 84 FV = $9000

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© % I/Y = Suppose you have $5000 and you want it to grow to $9000 in 7 years so you could buy a car. What rate of interest do you need to accomplish your goal? A ssume interest is compounded monthly. 12

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-22 Suppose you have $19,500 and you want it to grow to $23,000 so you could buy a car. You know that you can get 3% interest compounded daily. How long will it take to reach your goal? Solving for n this becomes… n = ln(FV/PV) / ln(1+ i) PV = $19500 i =.03/ 365 =.0001 FV = $23000 FV = PV(1 + i) n Formula Extract necessary data... See next slide for steps in developing this formula

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-23 Back to Solving the problem n = ln(FV/PV) / ln(1+ i) Formula FV = PV(1 + i) n 1. Base formula Developing the Formula 2. To isolate n, divide both sides by PV FV = PV(1 + i) n PV FV PV = (1 + i) n 3. Take the natural logarithm (LN or ln) of both sides FV PV = n * ln(1 + i) ln 4. Complete reducing FV PV ln (1 + i) n = n = ln(FV/PV) / ln(1+ i)

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© or 2009days Solve… Formula n = ln(FV/PV) / ln(1+ i) Suppose you have $19,500 and you want it to grow to $23,000 so you could buy a car. You know that you can get 3% interest compounded daily. How long will it take to reach your goal? PV = $19500 i =.03/ 365 =.0001 FV = $23000 …

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© Suppose you have $19,500 and you want it to grow to $23,000 so you could buy a car. You know that you can get 3% compounded daily. How long will it take to reach your goal? N = days Conversion of Days

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-26 Conversion of Days (years) Months 5 Years 6 months Years 6 Months

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-27 How long will it take at 4.2% compounded monthly for $800 to grow to $1100? Formula n = ln(FV/PV) / ln(1+ i) PV = $800 FV = $1100 i =.042/ 12 =.0033 Solve… Extract necessary data...

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© months Solve… 800 Formula n = ln(FV/PV) / ln(1+ i) years 7.1 months PV = $800 FV = $1100 i =.042/ 12 =.0033 How long will it take at 4.2% compounded monthly for $800 to grow to $1100?

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© N = How long will it take at 4.2% compounded monthly for $800 to grow to $1100? 91.1months 7 years 7.1 months

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-30 How long will it take for money to triple when invested at 7.2% compounded semi-annually ? Formula n = ln(FV/PV) / ln(1+ i) PV = $1 FV = $3 i =.072/ 2 =.036 Solve… Note: When no FV or PV given use the easiest number! Note: When no FV or PV given use the easiest number! $1$1 $1$1 Extract necessary data... FV =

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© sa periods Solve… years 3.2 months How long will it take for money to triple when invested at 7.2% compounded semi- annually ? Formula n = ln(FV/PV) / ln(1+ i) PV = $1 FV = $3 i =.072/ 2 =.036

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© N = How long will it take for money to triple when invested at 7.2% compounded semi- annually ? 31.1 sa periods 15 years 3.2 months

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-33 How long will it take for money to double when invested at 10.0% compounded annually ? Note: This results in a “rule of thumb” that you can use for comparisons or ballparking your other solutions. Note: This results in a “rule of thumb” that you can use for comparisons or ballparking your other solutions. Formula n = ln(FV/PV) / ln(1+ i) PV Extract necessary data... i =.010/ 2 =.005FV = $2 Solve… Use $1 = $1

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© N = How long will it take for money to double when invested at 10.0% compounded annually ? 7.3 Years

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-35 Rule of 72 …Rule of thumb to quickly estimate the time needed to double your money! Doubling time (in years) = 72 % annual rate of return … investment earning 9% compounded annually will double in approx… 72/9 = 8 years … investment earning 12% compounded annually will double in approx… 72/12 = 6 years … investment earning 4% compounded annually will double in approx… 72/4 = 18 years

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-36 LO-2

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-37 … or Stated Rate is the rate on which the bank calculates interest … is the annually compounded rate that produces the same Future Value as the given nominal rate (j) Effective Rate = Interest for 1 year Principal

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-38 Ending Balance Ending Balance Compounding Period Compounding Period $1,000 Nominal Rates of Interest Compared $1, $1, $1, $1, Beginning Balance Beginning Balance Nominal Rate + 6% Annual Semiannual Quarterly Daily Effective Rate APY of Interest Effective Rate APY of Interest 6.00% 6.09% 6.14% 6.18% Invested for One Year

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-39 What is the effective rate of interest corresponding to 9% pa compounded daily? Formula f = (1 + i) m - 1 Extract necessary data... j =.09 m = 365 i =.09/ 365 =.0002 f = (1 +.09/365) 365 – 1 =.0942 = 9.42% Solve…

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-40 FV = = 9.42% Solve… What is the effective rate of interest corresponding to 9% pa compounded daily? Use $

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-41 Comparisons

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-42 Which is the most attractive of the following interest rates offered on 3 year GICs? A. 5.8% compounded semi-annually B. 5.68% compounded monthly C. 5.4% compounded daily Tip Faster if you use Comparing… Comparisons

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-43 Comparisons Approach Invest $100 at each of the rates for 1 year and compare A. 5.8% compounded semi-annually B. 5.68% compounded monthly C. 5.4% compounded daily

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-44 Comparisons Invest $100 for 1 year A. 5.8% compounded semi- annually FV = = 5.88%

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-45 Comparisons Invest $100 for 1 year FV = = 5.83% B. 5.68% compounded monthly

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-46 Comparisons Invest $100 for 1 year FV = = 5.55% C. 5.4% compounded daily

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-47 A. 5.8% compounded semi-annually 5.88% B. 5.68% compounded monthly 5.83% C. 5.4% compounded daily 5.55% Comparisons the 5.8% compounded semi-annually provides for the best rate of return on investment! Results

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-48 Interest rates that produce the same future value after one year Equivalent Interest Rates LO-3

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-49 Compounded quarterly Refers to compounding frequency … NOT ‘per quarter’! per quarter Refers to the compounding period! Example 4% compounded quarterly equals 1% for each quarter

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-50 Equivalent Interest Rates For a given interest rate of 8% compounded quarterly, what is the equivalent nominal rate of interest with monthly compounding? i 2 = (1+i 1 ) m 1 / m i 1 and m 1 represent the given values of i and m Formula i 2 and m 2 represent the equivalent values of i and m Explanation

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-51 Equivalent Interest Rates Explanation For a given interest rate of 8% i compounded quarterly m Represented by Value GivenEquivalent i1i1 m1m1 what is the equivalent nominal rate of interest with monthly compounding? i2i2 m2m2

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-52 Equivalent Interest Rates For a given interest rate of 8% pa, compounded quarterly, what is the equivalent nominal rate of interest with monthly compounding? i 2 = (1+i 1 ) m 1 / m Formula i 2 =(1+.08 ) 4 / /4 i 2 =.0066 nominal rate = j =.0066 * 12 = 7.95% Note…

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-53 is a fraction change it to a decimal number and STORE it before continuing. Equivalent Interest Rates Since the exponent 4 / 12 = i 2 = (1+.08 ) 4 / /4 Formula Solve…

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-54 i 2 = (1+.08 ) 4 / /4 Equivalent Interest Rates For a given interest rate of 8% pa, compounded quarterly, what is the equivalent nominal rate of interest with monthly compounding?

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-55 FV = For a given interest rate of 8% pa, compounded quarterly, what is the equivalent nominal rate of interest with monthly compounding? Calculate the FV of $100 invested for 1 year as given = 8.24% FV =

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-56 I/Y = For a given interest rate of 8% pa, compounded quarterly, what is the equivalent nominal rate of interest with monthly compounding? Compute the value of i 2 that also makes $100 grow to the same as FV in m 2 compoundings 7.95% compounded monthly = 8% compounded quarterly 7.95% compounded monthly = 8% compounded quarterly

Compound Interest Compound Interest 9 9 McGraw-Hill Ryerson© 9-57 This completes Chapter 9