Compound Interest Interest Interest Chapter 8 McGraw-Hill Ryerson©

Learning Objectives Calculate the…
After completing this chapter, you will be able to: Calculate the… LO-1 …Maturity Value(MV), Future Value (FV), and Present Value(PV) in compound interest applications, by both the algebraic method and the pre-programmed financial calculator method …Maturity Value of compound interest for Guaranteed Investment Certificates (GICs) …Price of "strip" bonds

Learning Objectives And be able to… Calculate the…
LO-2 … Redemption Value of a compound interest bearing Canada Savings Bond …Payment on any date that is equivalent to one or more payments on other dates …Economic Value of a payment stream And be able to… …Adapt the concepts and equations of compound interest to cases of compound growth

Compound Interest 8 LO-1 Interest Compound Interest Interest

The formula on which we base our calculation is…
Compound Interest 8 To better understand how Compound Interest is calculated, let’s review how we calculate Simple Interest! The formula on which we base our calculation is… Formula I = Prt Review Here we have an amount, the Principal, which is multiplied by the Interest Rate and the Time over which the Interest is earned! As we will now see, Compound Interest uses the Sum of P & I as a base on which to calculate new Interest!

Is More Compound Interest
- Future Value Is …the interest on the principal plus the interest of prior periods e.g. Principal + prior period interest = $ $ $100.00 Interest for the next period is calculated on $ This method will continue over the life of the loan or investment. (See later example) More

...is the value of a loan or investment TODAY!
Compound Interest - Future Value Future Value …is the compounded amount and is the FINAL amount of the loan or investment at the end of the last period! Contrast this with… Present Value ...is the value of a loan or investment TODAY! More

Compounding Compound Interest
- Future Value Compounding …the calculation of interest over the life of the loan or investment Let’s assume that the interest rate is 10% pa. Example: Principal + prior period interest = $ Interest is now calculated on $ Principal(Compounded) * = $110.00 New P $ to start next period Graphically…

Compound Interest - Future Value Interest Interest Interest Interest
100 110 Interest Interest 121 Interest 133.1 Amount $1000 1 2 3 4 1331 100 110 121 1210 100 110 1100 100 1000 Compounding Period Compounding Period Compounding Period Compounding Period Time(Years)

Compound Interest - Future Value What happens if the interest rate changes during the life of an investment? Example…

Compound Interest - Future Value Q You hold an investment for a period of 4 years. Rates of return for each year are 4%, 8%, % and 9% respectively If you invested $1000 at the beginning of the term, how much will you have at the end of the last year?

Q Compound Interest Year 1 Year 2 Year 3 Year 4 $1000 $1040 $1123.20
- Future Value You hold an investment for a period of 4 years. Rates of return for each year are 4%, 8%, -10% and 9% respectively. If you invested $1000 at the beginning of the term, how much will you have at the end of the last year? Q Year 1 Year 2 Year 3 Year 4 $1000 $1040 $ $ $1000 * ( ) $1040 * ( ) $ * ( ) $ * (1 +.09) = $1040 = $ = $ = $ …Alternative

Q Note: Compound Interest 1000(1.04)(1.08)(.90)(1.09) = $1101.86
- Future Value You hold an investment for a period of 4 years Rates of return for each year are 4%, 8%, -10% and % respectively. If you invested $1000 at the beginning of the term, how much will you have at the end of the last year? Q 1000(1.04)(1.08)(.90)(1.09) = $ Solving Alternative 1 -10% Solve for all 4 years at once! It is rare for interest to be compounded only once per year! Note:

Compounding Frequencies and Periods
Table 8 - 1 Compounding Frequencies and Periods Frequency No. per Year Period Annually year Semiannually months Quarterly months Monthly month Daily day

n Symbols Formula Development of a Total Number of
Nominal or Annual Rate ( j ) Number of compoundings per year m Symbols Periodic Rate per period ( i ) n Total Number of Periods Determining values for n and i

# of Compounding Frequencies p.a. (m)
Formulae To Determine n * Time(Years) # of Compounding Frequencies p.a.(m) To Determine i Annual Interest Rate(j) # of Compounding Frequencies p.a. (m)

# of Compounding Frequencies per year (m)
Determining values for n Q If you compounded $100 for 3 years at 6% annually, semiannually, or quarterly, what are the values for n and i ? # of Compounding Frequencies per year (m) Time(Years) * Formula n No. Annually Semiannually Quarterly 3 * 1 2 4 = 3 = 6 = 12

# of Compounding Frequencies per year(m)
Determining values for i Q If you compounded $100 for 3 years at 6% annually, semiannually, or quarterly, what are the values for n and i ? Annual Interest Rate (j) # of Compounding Frequencies per year(m) Formula Rate - i No. 6% / Annually Semiannually Quarterly 1 2 4 = 6% = 3% = 1.5%

PV= Present Value(Principal)
Formula Development of a for Future Value FV = PV(1 + i)n Where… PV= Present Value(Principal) i = rate per period n = number of periods

Q Step Step Compound Interest FV = PV(1 + i)n Formula PV = n = i =
- Future Value FV = PV(1 + i)n Formula Q Steve Smith deposited $1,000 in a savings account for 4 years at a rate of 8% compounded semiannually What is Steve’s interest and compounded amount? Step Extract necessary data... PV = n = i = $1000 4 X 2 = 8 .08/2 = .04 Solve… Step

Step Compound Interest FV = PV(1 + i)n Formula = $1,368.57
- Future Value FV = PV(1 + i)n Formula Solve… Step Using PV = $1000 n = 8 i= .04 FV = $1000( )8 = $1000( ) = $1,368.57 Principal $1,000.00 + Interest Compounded $1,368.57

Use a calculator and algebraic sequencing
There are two methods that can be used to calculate compound interest: 1. Use a calculator and algebraic sequencing BOTH ways will be shown! CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst 2. Use the TI BAII Plus financial calculator!

Advanced Business Analyst
Use a calculator and algebraic sequencing Solve… $1000( )8 ON/OFF CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst = .04 + 1 = $1,368.57 y x 8 = X 1000 Financial =

Use a calculator and algebraic sequencing CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst Find the following KEYS: The Power function Key Used to calculate the value of exponents. y x 2nd Used to access symbols located “above” another key, i.e. its acts like the SHIFT key on a computer keyboard. +/- Changes the sign of the data value of the number being displayed.

Find the following KEYS:
Use a calculator and algebraic sequencing Find the following KEYS: CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst Some calculators have the yx symbol above the calculator key. The key stroke sequence to evaluate an EXPONENT that is… (1.04)8 is… 1.04 8 Positive 2nd y x = Negative (1.04)-8 is… 1.04 8 2nd y x =

Use a calculator and algebraic sequencing CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst Find the following KEYS: Used to Store or save displayed values. STO Used to Recall the saved values. RCL This calculator can store up to 10 values. Therefore, the calculator must be informed as to where the values are to be stored. Let’s Practise

Use a calculator and algebraic sequencing Using the key STO CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst e.g. you want to store the value ’45’. The key stroke sequence ‘to store’ is: 45 STO ..choose from 0 - 9 …’clear’ display CE/C The key stroke sequence ‘to recall’ is: RCL …where you stored the value!

Using your Financial Calculator CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst Some key Keys!

Number of compoundings (for lump payments) Number of payments (for annuities) CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst I/Y The nominal interest rate (Interest/Year) PV Present Value or initial(first) lump sum PMT Represents the Periodic Annuity Payment (used in chapter 10) FV Future Value or terminal(last) lump sum CPT Tells the calculator to compute (CPT)

Previously, it was noted that CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst …it is rare for interest to be compounded only once per year! However, we can now input the number of compoundings per year into the financial calculator This can be performed by using the symbol P/Y To access this symbol use: 2nd I/Y …and you will see

The 12 is a default setting
P/Y= 12 This display is referred to as “the worksheet”. … represents the number of Payments per Year P/Y … represents the number of Compoundings per Year C/Y To access use: C/Y C/Y= 12 Appears automatically Note: You can override these values by entering new ones! …more

If the calculation does not involve more than one payment
Note: If the calculation does not involve more than one payment must be given the same value as P/Y C/Y Illustration

Illustration C/Y 2nd QUIT CPT
… represents the number of Compoundings per Year C/Y In Compound Interest, P/Y must be given the same value as C/Y. Setting a new value for P/Y will automatically change the entry for C/Y to the same value as the default, i.e. P/Y We must key in this sequence to close any worksheet you have opened. …to scroll 2nd CPT QUIT

Advanced Business Analyst 2.
There are two methods that can be used to calculate compound interest: CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst Using the TI BAII Plus financial calculator! 2.

Using the TI BAII Plus financial calculator Steve Smith deposited $1,000 in a savings account for years at a rate of 8% compounded semiannually What is Steve’s interest and compounded amount? Q CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst Step 1 Step 2 Set the frequency of interest compounding Input values into the financial keys Financial Using

Advanced Business Analyst I/Y ON/OFF +/- 2nd PV = P/Y N ENTER 2nd PMT
Using the TI BAII Plus financial calculator Set the frequency of interest compounding Step 1 Input values into the financial keys Step 2 8.0 CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst I/Y FV= 1000 ON/OFF Steve Smith deposited $1,000 in a savings account for years at a rate of 8% compounded semiannually What is Steve’s interest and compounded amount? +/- 2nd PV 4 * 2 = P/Y 2 N ENTER 2nd PMT $1,368.57 QUIT CPT FV Financial

Advanced Business Analyst Keep in Mind
CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst Keep in Mind …there is no need to keep inputting each time! The calculator remembers this step! You only need to input the values that have changed!

CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst Keep in Mind Cash Flows ..a term that refers to payments that can be either … Inflows Outflows … payments received e.g. receipts … payments made e.g. cheques Treated as: Positives + Negatives -

What is the effect on the Future Value of different Compounding Periods of Interest?
Q

Q Compound Interest FVA = 100(1.06)3 $119.10 FVS = 100(1.03)6 $119.41
- Future Value Q If you compounded $100 for 3 years at 6% annually, semiannually, or quarterly, what are the final amounts that you would have at the end of the three (3) years ? Annual FVA = 100(1.06)3 $119.10 FVS = 100(1.03)6 $119.41 Semi- Semi = 6%/2 Quarterly FVQ = 100(1.015)12 $119.56 Quarterly = 6%/4

The Components of the Future Value of $100
Compound Interest - Future Value The Components of the Future Value of $100 Interest on Interest 250 FV=PV(1+i) n 200 Future Value S or FV Interest on Original Principal 150 S=P(1+rt) 100 Original Principal 50 Time(Years) 1 2 3 4 5 6 7 8 9 10 11

Comparisons

Simple Vs Compound Interest
Al Jones deposited $1,000 in a savings account for 5 years at 10% p.a.. Annual Simple Interest Rate of 10% Annual Compound Rate of 10% What is Al’s Simple Interest and Maturity Value? What is Al’s Interest and Compounded Value?

Simple Vs Compound Interest
Al Jones deposited $1,000 in a savings account for 5 years at 10% n = 5 * 1 = 5 i = .10 Simple Compound Formulae I = Prt FV = PV(1 + i)n I = FV – PV = $ $1000 I = $1,000 * .10 * 5 = $500 Compare = $610.51 FV = $1000(1.1)5 = $1,000 *1.6105 = $1,610.51 FV = $1,000 + $500 = $1,500 Compare

Future Values of $100 at Various Rates of Interest Compounded Annually
8 Future Values of $100 at Various Rates of Interest Compounded Annually 1800 6% 10% 8% 12% 1600 1400 1200 Future Value FV 1000 800 600 400 100 200 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Years to Maturity, n

$1,000 + 6% Nominal Rates of Interest Compared $1,060.00 $1,060.90
Beginning Balance Nominal Rate Compounding Period Ending Balance Annual Semiannual Quarterly Daily $1,060.00 $1,060.90 $1,000 + 6% $1,061.36 $1,061.83

12% Compounded monthly 12% Compounded Annually
Interest 8 Future Values of $100 at the same Nominal Rate but Different Compounding Frequencies 500 1000 1500 2000 2500 5 10 15 20 25 Future Value FV 12% Compounded monthly 12% Compounded Annually 100 Time (years)

Compounding Daily Interest
8 Compounding Daily Interest Calculate the Future Value of $2, compounded daily for 4 years at 4.5%. FV = PV(1 + i)n Formula i = n = 4 * 365 = 1460 .045 /365 = FV = $2000( /365)1460 = $2,000 * = $2,394.41

Solve FV = $2000( /365)1460 ON/OFF CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst .045 . 365 + 1 = y x = $2,394.41 1460 = X 2000 Financial =

Set the frequency of interest compounding Step 1 Input values into the financial keys Step 2 CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst 4.5 FV= I/Y 2000 ON/OFF Calculate the Future Value of $2, compounded daily for 4 years at 4.5%. PV 2nd 4 * 365 = P/Y N 365 ENTER PMT 2nd CPT QUIT $2,394.41 FV

What amount will you have 41/2 years after the initial investment?
Future Value Q You invested $6000 at 4.5% compounded quarterly. After 2 years, the rate changed to 5.2% compounded monthly. What amount will you have 41/2 years after the initial investment? Required: Prepare a ‘time-line’ as part of the solution

What amount will you have 41/2 years after the initial investment?
Future Value You invested $6000 at 4.5% compounded quarterly. After 2 years, the rate changed to 5.2% compounded monthly. What amount will you have 41/2 years after the initial investment? Q 2 years 4.5 years $6000 FV1 = PV2 FV2 i = .045/4 n = (2*4) = 8 i = .052/12 n = 2.5*12 = 30 FV1 = 6000(1+.045/4)8 = 6000(1.0936) = FV2 = (1+.052/12)30 = (1.1385) = $ Financial

Advanced Business Analyst +/- PV ON/OFF = 2nd N P/Y I/Y ENTER CPT 2nd
Using the TI BAII Plus financial calculator FV1 = PV2 Input values into the financial keys Step 2 Set the frequency of interest compounding Step 1 6000 CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst +/- 6,571.75 You invested $6000 at 4.5% compounded quarterly After 2 years, the rate changed to 5.2% compounded monthly What amount will you have 41/2 years after the initial investment? PV ON/OFF 4 * 2 = 2nd N 4.5 P/Y 4 I/Y ENTER CPT $6,561.75 2nd FV STO QUIT FV2 Financial

Advanced Business Analyst RCL
Using the TI BAII Plus financial calculator FV2 Input values into the financial keys Step 2 Set the frequency of interest compounding Step 1 CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst RCL You invested $6000 at 4.5% compounded quarterly After 2 years, the rate changed to 5.2% compounded monthly. What amount will you have 41/2 years after the initial investment? ON/OFF +/- 2nd PV 2.5*12 = P/Y 12 N 5.2 ENTER I/Y 2nd CPT $7,470.61 QUIT FV

Future Value Q Required: Prepare a ‘time-line’ as part of the solution
You borrowed $5000 at 7% compounded monthly. On the first and second anniversaries of the loan, you made payments of $ What is the balance outstanding immediately following the second payment? Required: Prepare a ‘time-line’ as part of the solution

Future Value Q Financial
You borrowed $5000 at 7% compounded monthly On the first and second anniversaries of the loan, you made payments of $2500. What is the balance outstanding immediately following the second payment? 1 year 2 years FV1 - $2500 = PV2 $5000 FV2 i = .07/12 n = 12 i = .07/12 n = 12 FV1 = 5000(1+.07/12)12 = 5000( ) = PV2 = – = FV2 = (1+.07/12)12 = ( ) = $ = $ – Financial New Balance = $568.30

Using the TI BAII Plus financial calculator FV1 – 2500 = PV2 Step 1 7.0 CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst ON/OFF I/Y 5000 12 You borrowed $5000 at 7% compounded monthly On the 1st. and 2nd anniversaries of the loan, you made payments of $2500. What is the balance outstanding immediately after the 2nd payment? FV= PV N 2nd CPT P/Y FV 12 + ENTER = 2500 2nd $2,861.45 QUIT FV2 Financial

Using the TI BAII Plus financial calculator FV2 Step 2 CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst PV You borrowed $5000 at 7% compounded monthly On the 1st. and 2nd anniversaries of the loan, you made payments of $2500. What is the balance outstanding immediately after the 2nd payment? - 2,861.45 568.30 CPT FV - 2500 = $568.30

P resent V alue

Formula for Present Value
PV = FV(1 + i)-n Formula i + CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst Keys 1 = This is the only change to the usual sequence! y x N +/- = X FV $PV =

Calculating Present Value
Q You expect to need $1,500 in 3 years Your bank offers 4% interest compounded semiannually How much money must you put in the bank today (PV) to reach your goal in 3 years? Required: Prepare the solution…(a) algebraically, and (b) by financial calculator

PV = FV(1 + i)-n Formula You expect to need $1,500 in 3 years Your bank offers 4% interest compounded semiannually. How much money must you put in the bank today (PV) to reach your goal in 3 years? Q n = 3 * 2 = 6 i = .04/2 = .02 1,331.96 (a) PV = $1500(1+.02)-6 1.02 = $1500 * .8880 = $1,331.96 y x 6 = +/- X 1500 Financial =

You expect to need $1,500 in 3 years Your bank offers 4% interest compounded semiannually. How much money must you put in the bank today (PV) to reach your goal in 3 years? Q 3 * 2 = CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst 1500 N FV Financial (b) PV= -1,331.96 4 I/Y PMT 2nd CPT P/Y 2 PV ENTER $ QUIT 2nd

PV = FV(1 + i)-n Formula Q What amount must you invest now at 5% compounded daily to accumulate to $6000 after 1 year? j = 5% 365 .05/365 1*365 = 365 $6000 PV = $6000(1+.05/365)-365 1.001 0.0001 0.9512 5,707.40 = $6000 * .9512 = $5,707.40 m = .05 . +/- i = 365 + n = X 1 6000 FV = y x 365

Q What amount must you invest now at 5% compounded daily to accumulate to $6000 after 1 year? 1 * 365 = CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst 6000 N PV= - 5,707.40 FV 5 I/Y PMT 2nd Financial CPT P/Y 365 PV ENTER 2nd QUIT $

Q Required: Equivalent Payments
Two payments of $2200 each must be made 1 and 4 years from now. If money can earn 5% compounded monthly, what single payment 3 years from now would be equivalent to the two scheduled payments? Required: Step 1 Draw a Time-line Step 2 Find the FV of the payment that is moved from Year 1 to Year 3 Step 3 Find the PV of the payment that is moved from Year 4 to Year 3 Prepare the solution…(a) algebraically, and (b) by financial calculator

Find the FV of the payment that is moved from Year 1 to Year 3
Equivalent Payments Two payments of $2200 each must be made 1 and 4 years from now. If money can earn 5% compounded monthly, what single payment 3 years from now would be equivalent to the two scheduled payments? Q Draw a Time-line Step 1 year years years years PV2 FV2 FV1 PV1 $ $2200 i = .05/12 n = 2*12 = 24 Step 2 Find the FV of the payment that is moved from Year 1 to Year 3 (a) FV1 = 2200(1+.05/12)24 = 2200(1.1049) = Now STO Financial

Find the FV of the payment that is moved from Year 1 to Year 3
(b) Financial $ $2200 year years years years i = .05/12 n = 2*12 = 24 FV1 PV1 PV2 FV2 2*12 2200 N Step 2 Find the FV of the payment that is moved from Year 1 to Year 3 +/- 5 PV I/Y 2nd PMT P/Y 12 CPT ENTER Now STO FV QUIT 2nd

Find the PV of the payment that is moved from Year 4 to Year 3
Equivalent Payments $ $2200 year years years years FV1 PV1 PV2 Find the PV of the payment that is moved from Year 4 to Year 3 Step 3 i = .05/12 n =1*12=12 (a) PV2 = 2200(1+.06/12)-12 = 2200(0.9513) = Finally, this PV amount can be added to that put into memory… + = RCL $ Financial

Financial n =1*12=12 $ $2200 year years years years PV2 FV1 PV1 CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst 2,092.92 4,523.79 Finally, this PV amount can be added to that put into memory… 1*12 = N + = Some of the values have not changed so there is no need to enter them again! RCL $ 2200 +/- FV CPT PV

Q What regular payment will an investor receive from a $10,000, 3 year, monthly payment GIC earning a nominal rate of 4.8% compounded monthly? Interest rate per payment interval is: i = j/m = .048/12 = …the monthly payment will be: PV * I = $10000 * = $40.00 Making a choice!…

Which rate should an investor choose for an investment of $1,000?
Making a choice!… Suppose a bank quotes nominal annual interest rates of % compounded annually, % compounded semi-annually, and % compounded monthly on five-year GICs. Which rate should an investor choose for an investment of $1,000?

= N = = I/Y N N +/- I/Y I/Y PV 2nd 2nd PMT P/Y P/Y 2nd ENTER ENTER P/Y
5*1 = j = 6.6% compounded annually j = 6.5% compounded semi-annually j = 6.4% compounded monthly N 6.6 5 * 2 5 * 12 = = I/Y N N Suppose a bank quotes nominal annual interest rates of % compounded annually, % compounded semi-annually, and % compounded monthly on five-year GICs Which rate should an investor choose for an investment of $1,000? 1000 6.5 6.4 +/- I/Y I/Y PV 2nd 2nd PMT P/Y P/Y 2 12 2nd ENTER ENTER P/Y 2nd QUIT 2nd QUIT 1 ENTER CPT CPT 2nd QUIT FV FV CPT Comparisons FV

Comparisons Results the 6.5% compounded semi-annually
j = 6.6% compounded annually j = 6.5% compounded semi-annually j = 6.4% compounded monthly the 6.5% compounded semi-annually provides for the best rate of return on investment!

Structure Structure of Interest Rates

…the interest rate does not change over the term of the GIC.
Structure of Interest Rates An investment in a GIC might have a… Fixed Rate …the interest rate does not change over the term of the GIC. Step-up Rate …the interest rate is increased every 6 months or every year according to a pre-determined schedule. Variable Rate ... is adjusted every year or every 6 months to reflect market rates… may be a minimum “floor” below which rates cannot drop

Payment of Interest Regular Interest version Compound Interest version
Interest is paid to the investor every year or every 6 months Interest is periodically converted to principal and paid at maturity

Canadian Savings Bonds

To view current rates of interest and redemption values
Canadian Savings Bonds - Can be purchased from financial institutions but funds go to federal government to help finance its debt - usual term is 10 or 12 years - variable interest rates - interest rate is changed on each anniversary, with minimum rates for subsequent 2 years To view current rates of interest and redemption values Go to

Canadian Savings Bonds
All CSBs issued up to 1988 (Series 1 to 43) have matured and are no longer earning interest. The rates of interest for Series 45 to 70 for subsequent years to maturity will be announced at future dates.

Valuation Principle

Concepts Valuation Principle
The fair market value of an investment is the sum of the Present Values of the expected cash flows. The discount rate used should be the prevailing market determined rate of return required on this type of investment.

S t r i p Bonds

S t r i p Bonds … owner will receive a single payment (called the face value of the bond) on the bond’s maturity date … the maturity date could be as much as 30 years in the future. No interest will be received in the interim! Example...

S t r i p Bonds Q Financial i = .0575/2 n = 18 * 2 = 36 FV = $10000
Suppose a $10,000 face value strip bond matures 18 years from now. The owner of this bond will receive a payment of $10, in 18 years. What is the appropriate price to pay for the bond today if the prevailing rate of return is 5.75%, compounded semi-annually? FV = $10000 i = .0575/2 n = 18 * 2 = 36 PV = 10000( /2)-36 = 10000(0.3605) = $ Financial

Q Suppose a $10,000 face value strip bond matures 18 years from now. The owner of this bond will receive a payment of $10,000 in 18 years.What is the appropriate price to pay for the bond today if the prevailing rate of return is 5.75%, compounded semi-annually? 18 * 2 = 10000 CPT 2nd N % INV LN STO RCL CE/C ENTER CF I/Y X ( 7 4 1 2 5 8 ) PV NPV IRR PMT x 1/ y 9 6 3 +/- ON/OFF FV . - + = QUIT SET DEL INS xP/Y P/Y AMORT BGN CLR TVM K HYP SIN COS TAN RAND X! e DATA STAT BOND ROUND DEPR BRKEVN DATE ICONV PROFIT CLR WORK MEM FORMAT RESET nPr nCr ANS Advanced Business Analyst j = 5.75% N FV 5.75 PV = -3,604.50 m = 2 I/Y n = 18*2 = 36 2nd PMT FV = $10000 Financial P/Y CPT 2 PV ENTER 2nd $ QUIT

This completes Chapter 8

Compound Interest Interest Interest Chapter 8 McGraw-Hill Ryerson©

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Compound Interest Interest Interest Chapter 8 McGraw-Hill Ryerson©

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