SlideShare a Scribd company logo

MATH1206 by Zomak Assignments

Download as PPTX, PDF
Z
zomakassignments

Chapter 12 focuses on ordinary general annuities, covering key calculations such as future value, present value, payment amounts, number of periods, and interest rates. It differentiates between 'ordinary' and 'general' annuities based on payment timing and interest conversion periods, providing examples of payment structures and rate conversions. The chapter also includes formulas and practical examples for computing annuity values and payments using various scenarios, including mortgages and educational savings.

Education
1 of 50
Download to read offline
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
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Chapter 12 Ordinary General Annuities
Objectives 1. Compute the future value (or accumulated value) for ordinary general annuities. 2. Compute the present value (or discounted value) for ordinary general annuities. 3. Compute the payment for ordinary general annuities. 4. Compute the number of periods for ordinary general annuities. 5. Compute the interest rate for ordinary general annuities. 6. Compute future value and present value for constant- growth annuities.
Ordinary General Annuities (1 of 4) • When the length of the interest conversion period is different from the length of the payment interval, these annuities are called general annuities. • In Canada, home mortgages are usually compounded semi-annually and payments are often made on a monthly, semi-monthly, or weekly basis. • With the exception of variable rate mortgages, all mortgages are compounded semi-annually, by law.
Ordinary General Annuities (2 of 4) Definition • “Ordinary” means that the payments are made or received at the end of each payment interval. • “General” means that the payment interval and interest conversion intervals are different. Example • Payments of $100 made at the end of every month for five years earns interest at 5% compounded semi- annually. • Note that the payments are at the end of the month (ordinary) and are made monthly. • The interest conversion period is semi-annually which is every six months. • The length of the payment period is different from the length of the conversion period.
Ordinary General Annuities (3 of 4) • It’s important to understand the relationship between the payment interval and the number of interest conversion periods per payment interval. • The number of interest conversion periods per payment interval, designated by the letter c, can be determined from the following ratio: 𝒄 = 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑰𝑵𝑻𝑬𝑹𝑬𝑺𝑻 𝑪𝑶𝑵𝑽𝑬𝑹𝑺𝑰𝑶𝑵 𝑷𝑬𝑹𝑰𝑶𝑫𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑷𝑨𝒀𝑴𝑬𝑵𝑻𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹
Ordinary General Annuities (4 of 4) Table 12.1 Some Possible Combinations of Payment Intervals and Interest Conversion Periods (1 of 2) Numerator: Interest Conversion Period Denominator: Payment Interval Number of Interest Conversion Periods per Payment Interval monthly semi-annually 𝑐 = 12 2 = 6 monthly quarterly 𝑐 = 12 4 = 3 quarterly annually 𝑐 = 4 1 = 4 annually semi-annually 𝑐 = 1 2 = 0.5
Table 12.1 Some Possible Combinations of Payment Intervals and Interest Conversion Periods (2 of 2) Numerator: Interest Conversion Period Denominator: Payment Interval Number of Interest Conversion Periods per Payment Interval annually quarterly 𝑐 = 1 4 = 0.25 semi-annually monthly 𝑐 = 2 12 = 1 6 quarterly monthly 𝑐 = 4 12 = 1 3 annually monthly 𝑐 = 1 12
Equivalent Rate of Interest Per Payment Period “p” • To change the given rate of interest to the equivalent interest per payment period “p” of the problem, use the following formula: 𝑝 = 1 + 𝑖 𝑐 − 1 𝒄 = 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑰𝑵𝑻𝑬𝑹𝑬𝑺𝑻 𝑪𝑶𝑵𝑽𝑬𝑹𝑺𝑰𝑶𝑵 𝑷𝑬𝑹𝑰𝑶𝑫𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑷𝑨𝒀𝑴𝑬𝑵𝑻𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹
Ordinary General Annuities • Computing p, allows us to treat the ordinary general annuity problem similar to an ordinary simple annuity problem. • FVg = the future value (or accumulated value) of an ordinary general annuity • PMT = the size of the periodic payment • n = the number of periodic payments • c = the number of interest conversion periods per payment interval • i = the interest rate per interest conversion period • p = the equivalent rate of interest per payment period
Future Value of an Ordinary General Annuity (1 of 3) 𝐹𝑉 𝑔 = 𝑃𝑀𝑇 1 + 𝑝 𝑛 − 1 𝑝 where 𝑝 = 1 + 𝑖 𝑐 − 1 𝒄 = 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑰𝑵𝑻𝑬𝑹𝑬𝑺𝑻 𝑪𝑶𝑵𝑽𝑬𝑹𝑺𝑰𝑶𝑵 𝑷𝑬𝑹𝑰𝑶𝑫𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑷𝑨𝒀𝑴𝑬𝑵𝑻𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹
Future Value of an Ordinary General Annuity (2 of 3) • To attend school, Sam deposits $1500 at the end of every six months for four and one-half years. What is the accumulated value of the deposits if interest is 6% compounded quarterly? • 6%/4 = 1.5% = 0.015 every quarter (3 months) so there are four quarters per year • Payments are every 6 months or two times per year 𝒄 = 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑰𝑵𝑻𝑬𝑹𝑬𝑺𝑻 𝑪𝑶𝑵𝑽𝑬𝑹𝑺𝑰𝑶𝑵 𝑷𝑬𝑹𝑰𝑶𝑫𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑷𝑨𝒀𝑴𝑬𝑵𝑻𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹 = 𝟒 𝟐 = 𝟐 𝑝 = 1 + 0.015 2 − 1 = 0.030225 • 3.0225% every six months
Interest Conversion Using the BA II PLUS • Given interest rate is 6% compounded quarterly. • Payments are semi-annually. • Convert 6% compounded quarterly to the equivalent semi-annual rate. KEY DISPLAY KEY DISPLAY 2nd Blank ↓ NOM = 6 INCONV 0 CPT NOM = 6.045 (nominal semi-annual rate equivalent to 6% compounded quarterly). 2nd Blank Blank Blank CLR WORK NOM = 0 Blank Blank 6 ENTER NOM = 6 CLR 6.045 ↑ C/Y = 2 (default) ÷ Blank 4 ENTER C/Y = 4 2 Blank ↑ EFF = 0 (default) = 3.0225 (periodic rate – every 6 months) CPT EFF = 6.136355 (effective annual rate) See previous slide for manual computation. Blank ↓ C/Y = 4 Blank Blank 2 ENTER C/Y = 2 Blank Blank
Future Value of an Ordinary General Annuity (3 of 3) • Number of payments n = 4.5 × 2 = 9 𝐹𝑉 𝑔 = 1500 1 + 0.030225 9 − 1 0.030225 𝐹𝑉 𝑔 = 1500 1.030225 9 − 1 0.030225 𝐹𝑉 𝑔 = 1500 10.168425 = $15,252.64
Calculator Solution (1 of 2) Manually KEY DISPLAY KEY DISPLAY CE/C Blank ) 1.0168425 1500 Blank = 15,252.63701 × Blank Blank Blank ( Blank Blank Blank 1.030225 Blank Note: You could also compute the terms inside the square brackets first and then multiply by 1500 – this eliminates the need for brackets on the calculator. Blank yx Blank Blank Blank 9 Blank Blank Blank −1 1.03073416 Blank Blank ÷ 0.03073406 Blank Blank 0.030225 Blank 𝐹𝑉 𝑔 = 1500 1.030225 9 − 1 0.030225 Blank
Calculator Solution (2 of 2) TVM Financial functions KEY DISPLAY KEY DISPLAY 2nd Blank 6* Blank CLR TVM 0 I/Y* I/Y = 6* 2nd Blank 0 Blank P/Y Blank PV PV = 0 2 ENTER P/Y = 2 −1500 Blank ↓ Blank PMT PMT = −1500 4 ENTER C/Y = 4 CPT Blank 2nd Blank FV FV = 15,252.64 QUIT 0 * Enter the nominal rate given as C/Y (set to 4 for quarterly). There is no need for a conversion. Blank 9 Blank Blank Blank N N = 9 Blank Blank
Ordinary General Annuities— Finding the Present Value PV (1 of 3) 𝑃𝑉 𝑔 = 𝑃𝑀𝑇 1 − 1 + 𝑝 −𝑛 𝑝 𝑝 = 1 + 𝑖 𝑐 − 1
Ordinary General Annuities— Finding the Present Value PV (2 of 3) • A 30-year mortgage on a condominium requires payments of $915.60 at the end of each month. If interest is 5.5% compounded semi-annually, what was the mortgage principal? • i = 5.5  2 = 2.75% = 0.0275 every 6 month period (twice per year). • Payments are monthly which means twelve payments per year. 𝒄 = 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑰𝑵𝑻𝑬𝑹𝑬𝑺𝑻 𝑪𝑶𝑵𝑽𝑬𝑹𝑺𝑰𝑶𝑵 𝑷𝑬𝑹𝑰𝑶𝑫𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑷𝑨𝒀𝑴𝑬𝑵𝑻𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹 = 𝟐 𝟏𝟐 = 𝟏 𝟔 𝑝 = 1 + 0.0275 1 6 − 1 = 1.0275 1 6 − 1 = 0.00453168 • 0.453168% per month (5.438018% compounded monthly)
Interest Conversion Using the BA II PLUS • Given interest rate is 5.5% compounded semi-annually. • Payments are monthly. • Convert 5.5% compounded semi-annually to the equivalent monthly rate. KEY DISPLAY KEY DISPLAY 2nd Blank ↓ NOM = 5.5 INCONV 0 CPT NOM = 5.438018062 (nominal monthly rate equivalent to 5.5% compounded semi- annually). 2nd Blank Blank Blank CLR WORK NOM = 0 Blank Blank 5.5 ENTER NOM = 5.5 CLR 5.438018062 ↑ C/Y = 2 (default) ÷ Blank 2 ENTER C/Y = 2 12 Blank ↑ EFF = 0 (default) = 0.453168172 (periodic rate – every month) CPT EFF = 5.575625 (effective annual rate) See previous slide for manual computation. Blank ↓ C/Y = 2 Blank Blank 12 ENTER C/Y = 12 Blank Blank
Ordinary General Annuities— Finding the Present Value PV (3 of 3) • Number of payments n = 12 × 30 = 360 𝑃𝑉 𝑔 = 915.60 1 − 1 + 0.00453168 −360 0.00453168 𝑃𝑉 𝑔 = 915.60 1 − 1.00453168172 −360 0.00453168172 𝑃𝑉 𝑔 = 915.60 177.334432 = $162,367.41
Calculator Solution (1 of 2) Manually KEY DISPLAY KEY DISPLAY CE/C Blank = 177.334432 1.00453168172* Blank × Blank yx Blank 915.60 Blank 360 Blank = 162,367.41* + | − −360 Blank = 0.196377 Note: You could also compute the terms inside the square brackets first and then multiply by 1500 – this eliminates need for brackets on the calculator. Blank + | − −0.196377 Blank Blank + Blank Blank Blank 1 Blank Blank Blank ÷ 0.8036231 Blank Blank 0.00453168172* Blank 𝑃𝑉 𝑔 = 915.60 1 − 1.00453168 −360 0.00453168 Blank
Calculator Solution (2 of 2) TVM Financial functions KEY DISPLAY KEY DISPLAY 2nd Blank 5.5* Blank CLR TVM 0 I/Y* I/Y = 5.5* 2nd Blank 0 Blank P/Y Blank FV FV = 0 12 ENTER P/Y = 12 −915.60 Blank ↓ Blank PMT PMT = −915.60 2 ENTER C/Y = 2 CPT Blank 2nd Blank PV PV = 162,367.41 QUIT 0 * Enter the nominal rate given as C/Y has been set to 2 for semi- annually. There is no need for a conversion. * Using the Time value of money (TVM) keys the solution is computed with 9 decimal places for the rate. Blank 360 Blank Blank Blank N N = 360 Blank Blank
Ordinary General Annuities— Finding the Periodic Payment PMT • When the future value of a general annuity is known 𝑃𝑀𝑇 = 𝐹𝑉 𝑔𝑝 1 + 𝑝 𝑛 − 1
Ordinary General Annuities—Finding the Periodic Payment PMT when FV Known • What payment made at the end of each year for 18 years will amount to $16,000 at 4.2% compounded monthly? 𝑃𝑀𝑇 = 𝐹𝑉 𝑔𝑝 1 + 𝑝 𝑛 − 1 • p = (1 + 0.0035)12/1 −1 = 0.042818 • n = 18 𝑃𝑀𝑇 = 16000 × 0.042818 1.042818 18 − 1 = $607.92
Calculator Solution (1 of 2) Manually KEY DISPLAY KEY DISPLAY CE/C Blank 1 Blank 16000 Blank ) 1.126930957 × Blank = 607.92 0.042818007 Blank Blank Blank = 685.088112 Blank Blank ÷ Blank 𝑃𝑀𝑇 = 16000 × 0.042818007 1.042818007 18 − 1 = $607.92 Blank ( Blank Blank Blank 1.042818007 Blank Blank Blank yx Blank Blank Blank 18 Blank Blank Blank − Blank Blank Blank
Calculator Solution (2 of 2) TVM Financial functions KEY DISPLAY KEY DISPLAY 2nd Blank I/Y* I/Y = 4.2* CLR TVM 0 0 Blank 2nd Blank PV PV = 0 P/Y Blank 16000 Blank 1 ENTER P/Y = 1 FV FV = 16000 ↓ Blank CPT Blank 12 ENTER C/Y = 12 PMT PV = 607.92 2nd Blank Blank Blank QUIT 0 * Enter the nominal rate given because C/Y has been set to 12 for semi-annually. There is no need for a conversion. * Using the time value of money (TVM) keys, the solution is computed with 9 decimal places for the rate. Blank 18 Blank Blank Blank N N = 18 Blank Blank 4.2* Blank Blank Blank
Ordinary General Annuities—Finding the Periodic Payment PMT (3 of 4) When the PV of an annuity is known
Ordinary General Annuities—Finding the Periodic Payment PMT (4 of 4) • When the present value of a general annuity is known; 𝑃𝑀𝑇 = 𝑃𝑉 𝑔𝑝 1 − 1 + 𝑝 −𝑛
Ordinary General Annuities—Finding the Periodic Payment PMT when FV Known • What payment is required at the end of each month for five years to repay a loan of $6000 at 7% compounded semi- annually? 𝑃𝑀𝑇 = 𝑃𝑉 𝑔𝑝 1 − 1 + 𝑝 −𝑛 • p = (1 + 0.035)2/12 −1 = 0.005750039 • n = 12 × 5 = 60 𝑃𝑀𝑇 = 6000 × 0.005750039 1 − 1.005750039 −60 = $118.52
Calculator Solution (1 of 2) Manually KEY DISPLAY KEY DISPLAY CE/C Blank yx Blank 6000 Blank −60 Blank × Blank ) 0.708918835 0.005750039 Blank ) 0.291081165 = 34.500234 = 118.52 ÷ Blank 𝑃𝑀𝑇 = 6000 × 0.005750039 1 − 1.005750039 −60 = $118.52 Blank ( Blank Blank Blank 1 Blank Blank Blank − Blank Blank Blank ( Blank Blank Blank 1.005750039 Blank Blank Blank
Calculator Solution (2 of 2) TVM Financial functions KEY DISPLAY KEY DISPLAY 2nd Blank I/Y* I/Y = 7* CLR TVM 0 6000 Blank 2nd Blank PV PV = 6000 P/Y Blank 0 Blank 12 ENTER P/Y = 12 FV FV = 0 ↓ Blank CPT Blank 2 ENTER C/Y = 2 PMT PV = 118.52 2nd Blank Blank Blank QUIT 0 * Enter the nominal rate as given because C/Y has been set to 2 for semi-annually. There is no need for a conversion. * Using the time value of money (TVM) keys, the solution is computed with 9 decimal places for the rate. Blank 60 Blank Blank Blank N N = 60 Blank Blank 7* Blank Blank Blank
Ordinary General Annuities— Finding the Term n • When the future value of a general annuity is known; 𝑛 = 𝑙𝑛 𝐹𝑉 𝑔𝑝 𝑃𝑀𝑇 + 1 𝑙𝑛 1 + 𝑝
Ordinary General Annuities—Finding the Term n when the FV is Known • For how long must contributions of $2000 be made at the end of each year to accumulate to $100,000 at 6% compounded quarterly? 𝑛 = 𝑙𝑛 𝐹𝑉 𝑔𝑝 𝑃𝑀𝑇 + 1 𝑙𝑛 1 + 𝑝 • p = (1 + 0.015)4/1 − 1 = 0.061363551effective • Can also use ICONV function. 𝑛 = 𝑙𝑛 100000 × 0.061363551 2000 + 1 𝑙𝑛 1.061363551 = 23.561549 • 23.56 years
Calculator Solution (1 of 2) Manually KEY DISPLAY KEY DISPLAY CE/C Blank ÷ Blank 100000 Blank 1.061363551 Blank × Blank LN 0.05955445 0.061363551 Blank = 23.561549* ÷ Blank * years Blank 2000 Blank 𝑛 = 𝑙𝑛 100000 × 0.061363551 2000 + 1 𝑙𝑛 1.061363551 Blank + 3.06817755 Blank Blank 1 Blank Blank Blank = 4.06817755 Blank Blank LN 1.403195123 Blank Blank
Calculator Solution (2 of 2) TVM Financial functions KEY DISPLAY KEY DISPLAY 2nd Blank PV PV = 0 CLR TVM 0 −2000 Blank 2nd Blank PMT PMT = −2000 P/Y Blank 100000 Blank 1 ENTER P/Y = 1 FV FV = 100000 ↓ Blank CPT Blank 4 ENTER C/Y = 4 N N = 23.561549* 2nd Blank Blank Blank QUIT 0 *years Blank 6 Blank Blank Blank I/Y I/Y = 6 Blank Blank 0 Blank Blank Blank
Ordinary General Annuities— Finding the Term n • When the present value of a general annuity is known; 𝑛 = 𝑙𝑛 1 − 𝑃𝑉 𝑔𝑝 𝑃𝑀𝑇 −𝑙𝑛 1 + 𝑝
Finding the Periodic Rate of Interest i (1 of 2) • Using preprogrammed financial calculators • Determining i without a financial calculator is extremely time consuming. • When the future value or present value, the periodic payment (PMT), and the term n of a general annuity are known, retrieve the value of I/Y by pressing CPT I/Y.
Finding the Periodic Rate of Interest i (2 of 2) • Victoria saved $416 every six months for eight years. What nominal rate of interest compounded annually is earned if the savings account amounts to $7,720 in eight years? KEY DISPLAY KEY DISPLAY 2nd Blank PV PV = 0 CLR TVM 0 −416 Blank 2nd Blank PMT PMT = −416 P/Y Blank 7720 Blank 2 ENTER P/Y = 2 FV FV = 7720 ↓ Blank CPT Blank 1 ENTER C/Y = 1 I/Y I/Y = 3.9247* 2nd Blank Blank Blank QUIT 0 *compounded annually (effective) Blank 16 Blank Blank Blank N N = 16 Blank Blank 0 Blank Blank Blank
Constant-Growth Annuities (1 of 12) • Constant-growth annuities differ from fixed- payment-size annuities in that the periodic payments change (usually grow) at a constant rate. • The assumption of constant growth is often used in sales forecasting and long-term financial planning. • It is consistent with the indexing of pensions or annuities.
Constant-Growth Annuities (2 of 12) • In general, if the first payment is represented by PMT and the constant rate of growth by k, the constant growth factor for the annuity payments is (1 + k). • The size of the successive payments is as follows: • 1st payment = PMT • 2nd payment = PMT(1 + k) • 3rd payment = PMT(1 + k)2 • 4th payment = PMT(1 + k)3 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ • 10th payment = PMT(1 + k)9 • SIZE OF THE nth PAYMENT = PMT(1 + k)n – 1
Constant-Growth Annuities (3 of 12) • PMT is the size of the first payment. • k is the periodic compounding rate. • n is the number of payments.
Constant-Growth Annuities (4 of 12) • Sum of the Periodic Constant-Growth Payments is equal to 𝑃𝑀𝑇 1 + 𝑘 𝑛 − 1 𝑘
Constant-Growth Annuities (5 of 12) • Future value of an ordinary simple constant-growth annuity: 𝐹𝑉 = 𝑃𝑀𝑇 1 + 𝑖 𝑛 − 1 + 𝑘 𝑛 𝑖 − 𝑘 • PMT = the size of the first annuity payment • i = the interest rate per conversion period • k = the constant growth rate of the annuity payments • n = the number of conversion periods
Constant-Growth Annuities (6 of 12) • Seven deposits increasing at a constant rate of 1.5% are made at the end of each of six successive years. The size of the first deposit is $5,000 and the fund earns interest at 6% compounded annually. 1. What is the size of the last deposit? 2. How much was deposited in total? 3. What is the accumulated value of the deposits? 4. What is the interest earned by the deposits?
Constant-Growth Annuities (7 of 12) • What is the size of the last deposit? 𝑠𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑡ℎ𝑑𝑒𝑝𝑜𝑠𝑖𝑡 = 𝑃𝑀𝑇 1 + 𝑘 𝑛−1 𝑠𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 7𝑡ℎ𝑑𝑒𝑝𝑜𝑠𝑖𝑡 = 5,000 1 + 0.015 7−1 𝑠𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 7𝑡ℎ 𝑑𝑒𝑝𝑜𝑠𝑖𝑡 = 5,000 1.015 6 𝑠𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 7𝑡ℎ𝑑𝑒𝑝𝑜𝑠𝑖𝑡 = 5,000 1.015 6 𝑠𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 7𝑡ℎ𝑑𝑒𝑝𝑜𝑠𝑖𝑡 = $𝟓, 𝟒𝟔𝟕. 𝟐𝟐
Constant-Growth Annuities (8 of 12) • How much was deposited in total? 𝑃𝑀𝑇 1+𝑘 𝑛−1 𝑘 k = the constant growth rate of the annuity payments 5,000 1 + 0.015 7 − 1 0.015 = $𝟑𝟔, 𝟔𝟏𝟒. 𝟗𝟕
Constant-Growth Annuities (9 of 12) • What is the accumulated value of the deposits? 𝐹𝑉 = 𝑃𝑀𝑇 1 + 𝑖 𝑛 − 1 + 𝑘 𝑛 𝑖 − 𝑘 𝐹𝑉 = 5,000 1 + 0.06 7 − 1 + 0.015 7 0.06 − 0.015 𝐹𝑉 = 5,000 1.06 7 − 1.015 7 0.045 = $𝟒𝟑, 𝟕𝟓𝟑. 𝟗𝟑
Constant-Growth Annuities (10 of 12) • What is the interest earned by the deposits? 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 43,753.93 − 36,614.97 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = $𝟕, 𝟏𝟑𝟖. 𝟗𝟔
Constant-Growth Annuities (11 of 12) • Present value of an ordinary simple constant- growth annuity; 𝑃𝑉 = 𝑃𝑀𝑇 1 − 1 + 𝑘 𝑛 1 + 𝑖 −𝑛 𝑖 − 𝑘 • PMT = the size of the first annuity payment • i = the interest rate per conversion period • k = the constant growth rate of the annuity payments • n = the number of conversion periods
Constant-Growth Annuities (12 of 12) • When the constant-growth rate of the annuity payments (k) and the periodic interest rate (i) are the same, use 𝐹𝑉 = 𝑛 𝑃𝑀𝑇 1 + 𝑖 𝑛−1 𝑃𝑉 = 𝑛 𝑃𝑀𝑇 1 + 𝑖 −1
Summary In an ordinary general annuity, the payment interval and interest conversion interval do not coincide (General). It is necessary to calculate an equivalent rate of interest p so that the payment interval and interest conversion interval do coincide. Once an equivalent rate of interest is computed, the FV, PV, PMT, and n can be calculated in the same manner as ordinary simple annuities. Constant-growth (ordinary simple) annuities have a first payment PMT and then subsequent payments increase by a constant amount k.

More Related Content

PDF
Time value Ch6.pdf
abdullahmamun649356
 
PDF
Texas Instrument Financial Calculator Guide
Siti Syahirah
 
PPTX
Time Value of Money- managerial finance
Tanjin Tamanna urmi
 
PPTX
CHAPTER 2 - Presentation for Teachers.pptx
Jimma Technology Institute,Jimma University
 
PPT
Nominal and Effective interest Rate fore
ateif2
 
PPTX
F.M - Chapter 5.pptx
MostafaGamal216952
 
PPT
Chapter 6 microeconomics
Noor Asma Zakaria
 
PPT
Topic 5 Valuation of Future Cashflows.ppt
ssuser71f60d
 
Time value Ch6.pdf
abdullahmamun649356
 
Texas Instrument Financial Calculator Guide
Siti Syahirah
 
Time Value of Money- managerial finance
Tanjin Tamanna urmi
 
CHAPTER 2 - Presentation for Teachers.pptx
Jimma Technology Institute,Jimma University
 
Nominal and Effective interest Rate fore
ateif2
 
F.M - Chapter 5.pptx
MostafaGamal216952
 
Chapter 6 microeconomics
Noor Asma Zakaria
 
Topic 5 Valuation of Future Cashflows.ppt
ssuser71f60d
 

Similar to MATH1206 by Zomak Assignments (20)

PPTX
Understanding the time value of money (annuity)
DIANN MOORMAN
 
PPT
Time Value of Money
Mustafa El Awady
 
PDF
Rates
David Keck
 
PPT
Fm11 ch 02 time value of money
Nhu Tuyet Tran
 
PDF
Rates of return pdf
David Keck
 
PPT
Fm11 ch 02 show
Adi Susilo
 
PPT
Time value of money
Usman Nazir
 
PPTX
10_General_Annuity.pptx General Mathematics 11
jaysongulla1
 
PPT
Leland_Tarquin_Engineering_Economy_Chapter_4_Nominal_Effective_Interest_Rates...
ssuser1ba731
 
PPT
Ch06
MufeezUrRehman
 
PDF
Lecture 2
Farzad Javidanrad
 
PPT
Week1_Lecture2.ppt
Phanny ITH
 
PPT
Chapter 4 nominal & effective interest rates
Bich Lien Pham
 
PPT
Chapter 4 nominal & effective interest rates
Bich Lien Pham
 
PDF
Ch4.pdf
Usamanajam5
 
PPTX
General Annuity PPT -General Mathematics
IndigoGabrielAnderso
 
PPT
Discounted measures of project wealth.ppt
DafaAllaMD
 
PPT
Time Value Of Money -Finance
Zoha Qureshi
 
DOCX
0132994879.DS_Store__MACOSX0132994879._.DS_Store01329.docx
mercysuttle
 
PDF
unit three.pdf
SagarChaulagai
 
Understanding the time value of money (annuity)
DIANN MOORMAN
 
Time Value of Money
Mustafa El Awady
 
Rates
David Keck
 
Fm11 ch 02 time value of money
Nhu Tuyet Tran
 
Rates of return pdf
David Keck
 
Fm11 ch 02 show
Adi Susilo
 
Time value of money
Usman Nazir
 
10_General_Annuity.pptx General Mathematics 11
jaysongulla1
 
Leland_Tarquin_Engineering_Economy_Chapter_4_Nominal_Effective_Interest_Rates...
ssuser1ba731
 
Ch06
MufeezUrRehman
 
Lecture 2
Farzad Javidanrad
 
Week1_Lecture2.ppt
Phanny ITH
 
Chapter 4 nominal & effective interest rates
Bich Lien Pham
 
Chapter 4 nominal & effective interest rates
Bich Lien Pham
 
Ch4.pdf
Usamanajam5
 
General Annuity PPT -General Mathematics
IndigoGabrielAnderso
 
Discounted measures of project wealth.ppt
DafaAllaMD
 
Time Value Of Money -Finance
Zoha Qureshi
 
0132994879.DS_Store__MACOSX0132994879._.DS_Store01329.docx
mercysuttle
 
unit three.pdf
SagarChaulagai
 
Ad

More from zomakassignments (14)

PPTX
DMGS_SSK_18663 R2.pptx by Zomak Assignments
zomakassignments
 
PPTX
Ryan Jarvis - London Teacher Who Secretly Recorded by Zomak Assignments
zomakassignments
 
PPTX
Strategies to Maximize Shareholders Wealth by Zomak Assignments
zomakassignments
 
PPTX
Marketing a New Boutique Resort.pptx by Zomak Assignments
zomakassignments
 
PPTX
BSBSTR502.pptx by Zomak Assignments
zomakassignments
 
PPTX
New Forest Natural Park by Zomak Assignments
zomakassignments
 
PPTX
Diversity Implementation Plan by Zomak Assignments
zomakassignments
 
PPTX
BSBWHS411.pptx by Zomak Assignments
zomakassignments
 
PPTX
Assessment 3 BUS5002 by Zomak Assignments
zomakassignments
 
PPTX
Assessment 5 - Presentation Research Proposal by Zomak Assignments
zomakassignments
 
PPTX
Manage Diversity by Zomak Assignments
zomakassignments
 
PPTX
Overcoming Problems of Intangibility and Perceived Risks.pptx
zomakassignments
 
PPTX
BSBCMM511.pptx
zomakassignments
 
PPTX
risk management process.pptx
zomakassignments
 
DMGS_SSK_18663 R2.pptx by Zomak Assignments
zomakassignments
 
Ryan Jarvis - London Teacher Who Secretly Recorded by Zomak Assignments
zomakassignments
 
Strategies to Maximize Shareholders Wealth by Zomak Assignments
zomakassignments
 
Marketing a New Boutique Resort.pptx by Zomak Assignments
zomakassignments
 
BSBSTR502.pptx by Zomak Assignments
zomakassignments
 
New Forest Natural Park by Zomak Assignments
zomakassignments
 
Diversity Implementation Plan by Zomak Assignments
zomakassignments
 
BSBWHS411.pptx by Zomak Assignments
zomakassignments
 
Assessment 3 BUS5002 by Zomak Assignments
zomakassignments
 
Assessment 5 - Presentation Research Proposal by Zomak Assignments
zomakassignments
 
Manage Diversity by Zomak Assignments
zomakassignments
 
Overcoming Problems of Intangibility and Perceived Risks.pptx
zomakassignments
 
BSBCMM511.pptx
zomakassignments
 
risk management process.pptx
zomakassignments
 
Ad

Recently uploaded (20)

PDF
RTP_AR_KS1_Tutor's Guide_English [FOR REPRODUCTION].pdf
RobertMentino1
 
PPTX
UV-Visible spectroscopy..pptx UV-Visible Spectroscopy – Electronic Transition...
Samruddhi Khonde
 
PDF
Classroom Observation Tools for Teachers
Nicaramirez
 
PDF
Paper A Mock Exam 9_ Attempt review.pdf.
SameeraNaveed2
 
PDF
Black Hat USA 2025 - Micro ICS Summit - ICS/OT Threat Landscape
Chris Sistrunk
 
PPTX
Final Presentation General Medicine 03-08-2024.pptx
ZaheerAhmad228692
 
PPTX
History, Philosophy and sociology of education (1).pptx
tmdth1
 
PDF
medical_surgical_nursing_10th_edition_ignatavicius_TEST_BANK_pdf.pdf
victor kamau
 
PPTX
Digestion and Absorption of Carbohydrates, Proteina and Fats
rekhapositivity
 
PDF
Indian roads congress 037 - 2012 Flexible pavement
offersjackpot
 
PPTX
202450812 BayCHI UCSC-SV 20250812 v17.pptx
home
 
PPTX
Tissue processing ( HISTOPATHOLOGICAL TECHNIQUE
mathiasmelwyn7
 
PDF
LNK 2025 (2).pdf MWEHEHEHEHEHEHEHEHEHEHE
NielDestora
 
PDF
Weekly quiz Compilation Jan -July 25.pdf
Nitin Suresh
 
PPTX
A powerpoint presentation on the Revised K-10 Science Shaping Paper
JericEstaco2
 
PPTX
CHAPTER IV. MAN AND BIOSPHERE AND ITS TOTALITY.pptx
greciamaycalambro
 
PPTX
Radiologic_Anatomy_of_the_Brachial_plexus [final].pptx
atiaruhi95
 
PDF
A systematic review of self-coping strategies used by university students to ...
CleeveMoralde1
 
PDF
RMMM.pdf make it easy to upload and study
sajedul2
 
PDF
GENETICS IN BIOLOGY IN SECONDARY LEVEL FORM 3
ElishamichaelSamwel
 
RTP_AR_KS1_Tutor's Guide_English [FOR REPRODUCTION].pdf
RobertMentino1
 
UV-Visible spectroscopy..pptx UV-Visible Spectroscopy – Electronic Transition...
Samruddhi Khonde
 
Classroom Observation Tools for Teachers
Nicaramirez
 
Paper A Mock Exam 9_ Attempt review.pdf.
SameeraNaveed2
 
Black Hat USA 2025 - Micro ICS Summit - ICS/OT Threat Landscape
Chris Sistrunk
 
Final Presentation General Medicine 03-08-2024.pptx
ZaheerAhmad228692
 
History, Philosophy and sociology of education (1).pptx
tmdth1
 
medical_surgical_nursing_10th_edition_ignatavicius_TEST_BANK_pdf.pdf
victor kamau
 
Digestion and Absorption of Carbohydrates, Proteina and Fats
rekhapositivity
 
Indian roads congress 037 - 2012 Flexible pavement
offersjackpot
 
202450812 BayCHI UCSC-SV 20250812 v17.pptx
home
 
Tissue processing ( HISTOPATHOLOGICAL TECHNIQUE
mathiasmelwyn7
 
LNK 2025 (2).pdf MWEHEHEHEHEHEHEHEHEHEHE
NielDestora
 
Weekly quiz Compilation Jan -July 25.pdf
Nitin Suresh
 
A powerpoint presentation on the Revised K-10 Science Shaping Paper
JericEstaco2
 
CHAPTER IV. MAN AND BIOSPHERE AND ITS TOTALITY.pptx
greciamaycalambro
 
Radiologic_Anatomy_of_the_Brachial_plexus [final].pptx
atiaruhi95
 
A systematic review of self-coping strategies used by university students to ...
CleeveMoralde1
 
RMMM.pdf make it easy to upload and study
sajedul2
 
GENETICS IN BIOLOGY IN SECONDARY LEVEL FORM 3
ElishamichaelSamwel
 

MATH1206 by Zomak Assignments

  • 2. Objectives 1. Compute the future value (or accumulated value) for ordinary general annuities. 2. Compute the present value (or discounted value) for ordinary general annuities. 3. Compute the payment for ordinary general annuities. 4. Compute the number of periods for ordinary general annuities. 5. Compute the interest rate for ordinary general annuities. 6. Compute future value and present value for constant- growth annuities.
  • 3. Ordinary General Annuities (1 of 4) • When the length of the interest conversion period is different from the length of the payment interval, these annuities are called general annuities. • In Canada, home mortgages are usually compounded semi-annually and payments are often made on a monthly, semi-monthly, or weekly basis. • With the exception of variable rate mortgages, all mortgages are compounded semi-annually, by law.
  • 4. Ordinary General Annuities (2 of 4) Definition • “Ordinary” means that the payments are made or received at the end of each payment interval. • “General” means that the payment interval and interest conversion intervals are different. Example • Payments of $100 made at the end of every month for five years earns interest at 5% compounded semi- annually. • Note that the payments are at the end of the month (ordinary) and are made monthly. • The interest conversion period is semi-annually which is every six months. • The length of the payment period is different from the length of the conversion period.
  • 5. Ordinary General Annuities (3 of 4) • It’s important to understand the relationship between the payment interval and the number of interest conversion periods per payment interval. • The number of interest conversion periods per payment interval, designated by the letter c, can be determined from the following ratio: 𝒄 = 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑰𝑵𝑻𝑬𝑹𝑬𝑺𝑻 𝑪𝑶𝑵𝑽𝑬𝑹𝑺𝑰𝑶𝑵 𝑷𝑬𝑹𝑰𝑶𝑫𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑷𝑨𝒀𝑴𝑬𝑵𝑻𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹
  • 6. Ordinary General Annuities (4 of 4) Table 12.1 Some Possible Combinations of Payment Intervals and Interest Conversion Periods (1 of 2) Numerator: Interest Conversion Period Denominator: Payment Interval Number of Interest Conversion Periods per Payment Interval monthly semi-annually 𝑐 = 12 2 = 6 monthly quarterly 𝑐 = 12 4 = 3 quarterly annually 𝑐 = 4 1 = 4 annually semi-annually 𝑐 = 1 2 = 0.5
  • 7. Table 12.1 Some Possible Combinations of Payment Intervals and Interest Conversion Periods (2 of 2) Numerator: Interest Conversion Period Denominator: Payment Interval Number of Interest Conversion Periods per Payment Interval annually quarterly 𝑐 = 1 4 = 0.25 semi-annually monthly 𝑐 = 2 12 = 1 6 quarterly monthly 𝑐 = 4 12 = 1 3 annually monthly 𝑐 = 1 12
  • 8. Equivalent Rate of Interest Per Payment Period “p” • To change the given rate of interest to the equivalent interest per payment period “p” of the problem, use the following formula: 𝑝 = 1 + 𝑖 𝑐 − 1 𝒄 = 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑰𝑵𝑻𝑬𝑹𝑬𝑺𝑻 𝑪𝑶𝑵𝑽𝑬𝑹𝑺𝑰𝑶𝑵 𝑷𝑬𝑹𝑰𝑶𝑫𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑷𝑨𝒀𝑴𝑬𝑵𝑻𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹
  • 9. Ordinary General Annuities • Computing p, allows us to treat the ordinary general annuity problem similar to an ordinary simple annuity problem. • FVg = the future value (or accumulated value) of an ordinary general annuity • PMT = the size of the periodic payment • n = the number of periodic payments • c = the number of interest conversion periods per payment interval • i = the interest rate per interest conversion period • p = the equivalent rate of interest per payment period
  • 10. Future Value of an Ordinary General Annuity (1 of 3) 𝐹𝑉 𝑔 = 𝑃𝑀𝑇 1 + 𝑝 𝑛 − 1 𝑝 where 𝑝 = 1 + 𝑖 𝑐 − 1 𝒄 = 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑰𝑵𝑻𝑬𝑹𝑬𝑺𝑻 𝑪𝑶𝑵𝑽𝑬𝑹𝑺𝑰𝑶𝑵 𝑷𝑬𝑹𝑰𝑶𝑫𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑷𝑨𝒀𝑴𝑬𝑵𝑻𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹
  • 11. Future Value of an Ordinary General Annuity (2 of 3) • To attend school, Sam deposits $1500 at the end of every six months for four and one-half years. What is the accumulated value of the deposits if interest is 6% compounded quarterly? • 6%/4 = 1.5% = 0.015 every quarter (3 months) so there are four quarters per year • Payments are every 6 months or two times per year 𝒄 = 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑰𝑵𝑻𝑬𝑹𝑬𝑺𝑻 𝑪𝑶𝑵𝑽𝑬𝑹𝑺𝑰𝑶𝑵 𝑷𝑬𝑹𝑰𝑶𝑫𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑷𝑨𝒀𝑴𝑬𝑵𝑻𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹 = 𝟒 𝟐 = 𝟐 𝑝 = 1 + 0.015 2 − 1 = 0.030225 • 3.0225% every six months
  • 12. Interest Conversion Using the BA II PLUS • Given interest rate is 6% compounded quarterly. • Payments are semi-annually. • Convert 6% compounded quarterly to the equivalent semi-annual rate. KEY DISPLAY KEY DISPLAY 2nd Blank ↓ NOM = 6 INCONV 0 CPT NOM = 6.045 (nominal semi-annual rate equivalent to 6% compounded quarterly). 2nd Blank Blank Blank CLR WORK NOM = 0 Blank Blank 6 ENTER NOM = 6 CLR 6.045 ↑ C/Y = 2 (default) ÷ Blank 4 ENTER C/Y = 4 2 Blank ↑ EFF = 0 (default) = 3.0225 (periodic rate – every 6 months) CPT EFF = 6.136355 (effective annual rate) See previous slide for manual computation. Blank ↓ C/Y = 4 Blank Blank 2 ENTER C/Y = 2 Blank Blank
  • 13. Future Value of an Ordinary General Annuity (3 of 3) • Number of payments n = 4.5 × 2 = 9 𝐹𝑉 𝑔 = 1500 1 + 0.030225 9 − 1 0.030225 𝐹𝑉 𝑔 = 1500 1.030225 9 − 1 0.030225 𝐹𝑉 𝑔 = 1500 10.168425 = $15,252.64
  • 14. Calculator Solution (1 of 2) Manually KEY DISPLAY KEY DISPLAY CE/C Blank ) 1.0168425 1500 Blank = 15,252.63701 × Blank Blank Blank ( Blank Blank Blank 1.030225 Blank Note: You could also compute the terms inside the square brackets first and then multiply by 1500 – this eliminates the need for brackets on the calculator. Blank yx Blank Blank Blank 9 Blank Blank Blank −1 1.03073416 Blank Blank ÷ 0.03073406 Blank Blank 0.030225 Blank 𝐹𝑉 𝑔 = 1500 1.030225 9 − 1 0.030225 Blank
  • 15. Calculator Solution (2 of 2) TVM Financial functions KEY DISPLAY KEY DISPLAY 2nd Blank 6* Blank CLR TVM 0 I/Y* I/Y = 6* 2nd Blank 0 Blank P/Y Blank PV PV = 0 2 ENTER P/Y = 2 −1500 Blank ↓ Blank PMT PMT = −1500 4 ENTER C/Y = 4 CPT Blank 2nd Blank FV FV = 15,252.64 QUIT 0 * Enter the nominal rate given as C/Y (set to 4 for quarterly). There is no need for a conversion. Blank 9 Blank Blank Blank N N = 9 Blank Blank
  • 16. Ordinary General Annuities— Finding the Present Value PV (1 of 3) 𝑃𝑉 𝑔 = 𝑃𝑀𝑇 1 − 1 + 𝑝 −𝑛 𝑝 𝑝 = 1 + 𝑖 𝑐 − 1
  • 17. Ordinary General Annuities— Finding the Present Value PV (2 of 3) • A 30-year mortgage on a condominium requires payments of $915.60 at the end of each month. If interest is 5.5% compounded semi-annually, what was the mortgage principal? • i = 5.5  2 = 2.75% = 0.0275 every 6 month period (twice per year). • Payments are monthly which means twelve payments per year. 𝒄 = 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑰𝑵𝑻𝑬𝑹𝑬𝑺𝑻 𝑪𝑶𝑵𝑽𝑬𝑹𝑺𝑰𝑶𝑵 𝑷𝑬𝑹𝑰𝑶𝑫𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹 𝑻𝑯𝑬 𝑵𝑼𝑴𝑩𝑬𝑹 𝑶𝑭 𝑷𝑨𝒀𝑴𝑬𝑵𝑻𝑺 𝑷𝑬𝑹 𝒀𝑬𝑨𝑹 = 𝟐 𝟏𝟐 = 𝟏 𝟔 𝑝 = 1 + 0.0275 1 6 − 1 = 1.0275 1 6 − 1 = 0.00453168 • 0.453168% per month (5.438018% compounded monthly)
  • 18. Interest Conversion Using the BA II PLUS • Given interest rate is 5.5% compounded semi-annually. • Payments are monthly. • Convert 5.5% compounded semi-annually to the equivalent monthly rate. KEY DISPLAY KEY DISPLAY 2nd Blank ↓ NOM = 5.5 INCONV 0 CPT NOM = 5.438018062 (nominal monthly rate equivalent to 5.5% compounded semi- annually). 2nd Blank Blank Blank CLR WORK NOM = 0 Blank Blank 5.5 ENTER NOM = 5.5 CLR 5.438018062 ↑ C/Y = 2 (default) ÷ Blank 2 ENTER C/Y = 2 12 Blank ↑ EFF = 0 (default) = 0.453168172 (periodic rate – every month) CPT EFF = 5.575625 (effective annual rate) See previous slide for manual computation. Blank ↓ C/Y = 2 Blank Blank 12 ENTER C/Y = 12 Blank Blank
  • 19. Ordinary General Annuities— Finding the Present Value PV (3 of 3) • Number of payments n = 12 × 30 = 360 𝑃𝑉 𝑔 = 915.60 1 − 1 + 0.00453168 −360 0.00453168 𝑃𝑉 𝑔 = 915.60 1 − 1.00453168172 −360 0.00453168172 𝑃𝑉 𝑔 = 915.60 177.334432 = $162,367.41
  • 20. Calculator Solution (1 of 2) Manually KEY DISPLAY KEY DISPLAY CE/C Blank = 177.334432 1.00453168172* Blank × Blank yx Blank 915.60 Blank 360 Blank = 162,367.41* + | − −360 Blank = 0.196377 Note: You could also compute the terms inside the square brackets first and then multiply by 1500 – this eliminates need for brackets on the calculator. Blank + | − −0.196377 Blank Blank + Blank Blank Blank 1 Blank Blank Blank ÷ 0.8036231 Blank Blank 0.00453168172* Blank 𝑃𝑉 𝑔 = 915.60 1 − 1.00453168 −360 0.00453168 Blank
  • 21. Calculator Solution (2 of 2) TVM Financial functions KEY DISPLAY KEY DISPLAY 2nd Blank 5.5* Blank CLR TVM 0 I/Y* I/Y = 5.5* 2nd Blank 0 Blank P/Y Blank FV FV = 0 12 ENTER P/Y = 12 −915.60 Blank ↓ Blank PMT PMT = −915.60 2 ENTER C/Y = 2 CPT Blank 2nd Blank PV PV = 162,367.41 QUIT 0 * Enter the nominal rate given as C/Y has been set to 2 for semi- annually. There is no need for a conversion. * Using the Time value of money (TVM) keys the solution is computed with 9 decimal places for the rate. Blank 360 Blank Blank Blank N N = 360 Blank Blank
  • 22. Ordinary General Annuities— Finding the Periodic Payment PMT • When the future value of a general annuity is known 𝑃𝑀𝑇 = 𝐹𝑉 𝑔𝑝 1 + 𝑝 𝑛 − 1
  • 23. Ordinary General Annuities—Finding the Periodic Payment PMT when FV Known • What payment made at the end of each year for 18 years will amount to $16,000 at 4.2% compounded monthly? 𝑃𝑀𝑇 = 𝐹𝑉 𝑔𝑝 1 + 𝑝 𝑛 − 1 • p = (1 + 0.0035)12/1 −1 = 0.042818 • n = 18 𝑃𝑀𝑇 = 16000 × 0.042818 1.042818 18 − 1 = $607.92
  • 24. Calculator Solution (1 of 2) Manually KEY DISPLAY KEY DISPLAY CE/C Blank 1 Blank 16000 Blank ) 1.126930957 × Blank = 607.92 0.042818007 Blank Blank Blank = 685.088112 Blank Blank ÷ Blank 𝑃𝑀𝑇 = 16000 × 0.042818007 1.042818007 18 − 1 = $607.92 Blank ( Blank Blank Blank 1.042818007 Blank Blank Blank yx Blank Blank Blank 18 Blank Blank Blank − Blank Blank Blank
  • 25. Calculator Solution (2 of 2) TVM Financial functions KEY DISPLAY KEY DISPLAY 2nd Blank I/Y* I/Y = 4.2* CLR TVM 0 0 Blank 2nd Blank PV PV = 0 P/Y Blank 16000 Blank 1 ENTER P/Y = 1 FV FV = 16000 ↓ Blank CPT Blank 12 ENTER C/Y = 12 PMT PV = 607.92 2nd Blank Blank Blank QUIT 0 * Enter the nominal rate given because C/Y has been set to 12 for semi-annually. There is no need for a conversion. * Using the time value of money (TVM) keys, the solution is computed with 9 decimal places for the rate. Blank 18 Blank Blank Blank N N = 18 Blank Blank 4.2* Blank Blank Blank
  • 26. Ordinary General Annuities—Finding the Periodic Payment PMT (3 of 4) When the PV of an annuity is known
  • 27. Ordinary General Annuities—Finding the Periodic Payment PMT (4 of 4) • When the present value of a general annuity is known; 𝑃𝑀𝑇 = 𝑃𝑉 𝑔𝑝 1 − 1 + 𝑝 −𝑛
  • 28. Ordinary General Annuities—Finding the Periodic Payment PMT when FV Known • What payment is required at the end of each month for five years to repay a loan of $6000 at 7% compounded semi- annually? 𝑃𝑀𝑇 = 𝑃𝑉 𝑔𝑝 1 − 1 + 𝑝 −𝑛 • p = (1 + 0.035)2/12 −1 = 0.005750039 • n = 12 × 5 = 60 𝑃𝑀𝑇 = 6000 × 0.005750039 1 − 1.005750039 −60 = $118.52
  • 29. Calculator Solution (1 of 2) Manually KEY DISPLAY KEY DISPLAY CE/C Blank yx Blank 6000 Blank −60 Blank × Blank ) 0.708918835 0.005750039 Blank ) 0.291081165 = 34.500234 = 118.52 ÷ Blank 𝑃𝑀𝑇 = 6000 × 0.005750039 1 − 1.005750039 −60 = $118.52 Blank ( Blank Blank Blank 1 Blank Blank Blank − Blank Blank Blank ( Blank Blank Blank 1.005750039 Blank Blank Blank
  • 30. Calculator Solution (2 of 2) TVM Financial functions KEY DISPLAY KEY DISPLAY 2nd Blank I/Y* I/Y = 7* CLR TVM 0 6000 Blank 2nd Blank PV PV = 6000 P/Y Blank 0 Blank 12 ENTER P/Y = 12 FV FV = 0 ↓ Blank CPT Blank 2 ENTER C/Y = 2 PMT PV = 118.52 2nd Blank Blank Blank QUIT 0 * Enter the nominal rate as given because C/Y has been set to 2 for semi-annually. There is no need for a conversion. * Using the time value of money (TVM) keys, the solution is computed with 9 decimal places for the rate. Blank 60 Blank Blank Blank N N = 60 Blank Blank 7* Blank Blank Blank
  • 31. Ordinary General Annuities— Finding the Term n • When the future value of a general annuity is known; 𝑛 = 𝑙𝑛 𝐹𝑉 𝑔𝑝 𝑃𝑀𝑇 + 1 𝑙𝑛 1 + 𝑝
  • 32. Ordinary General Annuities—Finding the Term n when the FV is Known • For how long must contributions of $2000 be made at the end of each year to accumulate to $100,000 at 6% compounded quarterly? 𝑛 = 𝑙𝑛 𝐹𝑉 𝑔𝑝 𝑃𝑀𝑇 + 1 𝑙𝑛 1 + 𝑝 • p = (1 + 0.015)4/1 − 1 = 0.061363551effective • Can also use ICONV function. 𝑛 = 𝑙𝑛 100000 × 0.061363551 2000 + 1 𝑙𝑛 1.061363551 = 23.561549 • 23.56 years
  • 33. Calculator Solution (1 of 2) Manually KEY DISPLAY KEY DISPLAY CE/C Blank ÷ Blank 100000 Blank 1.061363551 Blank × Blank LN 0.05955445 0.061363551 Blank = 23.561549* ÷ Blank * years Blank 2000 Blank 𝑛 = 𝑙𝑛 100000 × 0.061363551 2000 + 1 𝑙𝑛 1.061363551 Blank + 3.06817755 Blank Blank 1 Blank Blank Blank = 4.06817755 Blank Blank LN 1.403195123 Blank Blank
  • 34. Calculator Solution (2 of 2) TVM Financial functions KEY DISPLAY KEY DISPLAY 2nd Blank PV PV = 0 CLR TVM 0 −2000 Blank 2nd Blank PMT PMT = −2000 P/Y Blank 100000 Blank 1 ENTER P/Y = 1 FV FV = 100000 ↓ Blank CPT Blank 4 ENTER C/Y = 4 N N = 23.561549* 2nd Blank Blank Blank QUIT 0 *years Blank 6 Blank Blank Blank I/Y I/Y = 6 Blank Blank 0 Blank Blank Blank
  • 35. Ordinary General Annuities— Finding the Term n • When the present value of a general annuity is known; 𝑛 = 𝑙𝑛 1 − 𝑃𝑉 𝑔𝑝 𝑃𝑀𝑇 −𝑙𝑛 1 + 𝑝
  • 36. Finding the Periodic Rate of Interest i (1 of 2) • Using preprogrammed financial calculators • Determining i without a financial calculator is extremely time consuming. • When the future value or present value, the periodic payment (PMT), and the term n of a general annuity are known, retrieve the value of I/Y by pressing CPT I/Y.
  • 37. Finding the Periodic Rate of Interest i (2 of 2) • Victoria saved $416 every six months for eight years. What nominal rate of interest compounded annually is earned if the savings account amounts to $7,720 in eight years? KEY DISPLAY KEY DISPLAY 2nd Blank PV PV = 0 CLR TVM 0 −416 Blank 2nd Blank PMT PMT = −416 P/Y Blank 7720 Blank 2 ENTER P/Y = 2 FV FV = 7720 ↓ Blank CPT Blank 1 ENTER C/Y = 1 I/Y I/Y = 3.9247* 2nd Blank Blank Blank QUIT 0 *compounded annually (effective) Blank 16 Blank Blank Blank N N = 16 Blank Blank 0 Blank Blank Blank
  • 38. Constant-Growth Annuities (1 of 12) • Constant-growth annuities differ from fixed- payment-size annuities in that the periodic payments change (usually grow) at a constant rate. • The assumption of constant growth is often used in sales forecasting and long-term financial planning. • It is consistent with the indexing of pensions or annuities.
  • 39. Constant-Growth Annuities (2 of 12) • In general, if the first payment is represented by PMT and the constant rate of growth by k, the constant growth factor for the annuity payments is (1 + k). • The size of the successive payments is as follows: • 1st payment = PMT • 2nd payment = PMT(1 + k) • 3rd payment = PMT(1 + k)2 • 4th payment = PMT(1 + k)3 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ • 10th payment = PMT(1 + k)9 • SIZE OF THE nth PAYMENT = PMT(1 + k)n – 1
  • 40. Constant-Growth Annuities (3 of 12) • PMT is the size of the first payment. • k is the periodic compounding rate. • n is the number of payments.
  • 41. Constant-Growth Annuities (4 of 12) • Sum of the Periodic Constant-Growth Payments is equal to 𝑃𝑀𝑇 1 + 𝑘 𝑛 − 1 𝑘
  • 42. Constant-Growth Annuities (5 of 12) • Future value of an ordinary simple constant-growth annuity: 𝐹𝑉 = 𝑃𝑀𝑇 1 + 𝑖 𝑛 − 1 + 𝑘 𝑛 𝑖 − 𝑘 • PMT = the size of the first annuity payment • i = the interest rate per conversion period • k = the constant growth rate of the annuity payments • n = the number of conversion periods
  • 43. Constant-Growth Annuities (6 of 12) • Seven deposits increasing at a constant rate of 1.5% are made at the end of each of six successive years. The size of the first deposit is $5,000 and the fund earns interest at 6% compounded annually. 1. What is the size of the last deposit? 2. How much was deposited in total? 3. What is the accumulated value of the deposits? 4. What is the interest earned by the deposits?
  • 44. Constant-Growth Annuities (7 of 12) • What is the size of the last deposit? 𝑠𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑡ℎ𝑑𝑒𝑝𝑜𝑠𝑖𝑡 = 𝑃𝑀𝑇 1 + 𝑘 𝑛−1 𝑠𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 7𝑡ℎ𝑑𝑒𝑝𝑜𝑠𝑖𝑡 = 5,000 1 + 0.015 7−1 𝑠𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 7𝑡ℎ 𝑑𝑒𝑝𝑜𝑠𝑖𝑡 = 5,000 1.015 6 𝑠𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 7𝑡ℎ𝑑𝑒𝑝𝑜𝑠𝑖𝑡 = 5,000 1.015 6 𝑠𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 7𝑡ℎ𝑑𝑒𝑝𝑜𝑠𝑖𝑡 = $𝟓, 𝟒𝟔𝟕. 𝟐𝟐
  • 45. Constant-Growth Annuities (8 of 12) • How much was deposited in total? 𝑃𝑀𝑇 1+𝑘 𝑛−1 𝑘 k = the constant growth rate of the annuity payments 5,000 1 + 0.015 7 − 1 0.015 = $𝟑𝟔, 𝟔𝟏𝟒. 𝟗𝟕
  • 46. Constant-Growth Annuities (9 of 12) • What is the accumulated value of the deposits? 𝐹𝑉 = 𝑃𝑀𝑇 1 + 𝑖 𝑛 − 1 + 𝑘 𝑛 𝑖 − 𝑘 𝐹𝑉 = 5,000 1 + 0.06 7 − 1 + 0.015 7 0.06 − 0.015 𝐹𝑉 = 5,000 1.06 7 − 1.015 7 0.045 = $𝟒𝟑, 𝟕𝟓𝟑. 𝟗𝟑
  • 47. Constant-Growth Annuities (10 of 12) • What is the interest earned by the deposits? 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = 43,753.93 − 36,614.97 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 = $𝟕, 𝟏𝟑𝟖. 𝟗𝟔
  • 48. Constant-Growth Annuities (11 of 12) • Present value of an ordinary simple constant- growth annuity; 𝑃𝑉 = 𝑃𝑀𝑇 1 − 1 + 𝑘 𝑛 1 + 𝑖 −𝑛 𝑖 − 𝑘 • PMT = the size of the first annuity payment • i = the interest rate per conversion period • k = the constant growth rate of the annuity payments • n = the number of conversion periods
  • 49. Constant-Growth Annuities (12 of 12) • When the constant-growth rate of the annuity payments (k) and the periodic interest rate (i) are the same, use 𝐹𝑉 = 𝑛 𝑃𝑀𝑇 1 + 𝑖 𝑛−1 𝑃𝑉 = 𝑛 𝑃𝑀𝑇 1 + 𝑖 −1
  • 50. Summary In an ordinary general annuity, the payment interval and interest conversion interval do not coincide (General). It is necessary to calculate an equivalent rate of interest p so that the payment interval and interest conversion interval do coincide. Once an equivalent rate of interest is computed, the FV, PV, PMT, and n can be calculated in the same manner as ordinary simple annuities. Constant-growth (ordinary simple) annuities have a first payment PMT and then subsequent payments increase by a constant amount k.

Editor's Notes

  • #2: If this PowerPoint presentation contains mathematical equations, you may need to check that your computer has the following installed: 1) MathType Plugin 2) Math Player (free versions available) 3) NVDA Reader (free versions available)