Annuities GEN-MATH-MODULE-PDF

**SAINT TONIS COLLEGE, INC.** (Formerly: Kalinga Christian Learning Center) **United Church of Christ in the Philippines** Purok 4 Bulanao Centro, Tabuk City, Kalinga Philippines 3800 Tel. No. (074) 627-5930, Email Address: Doc. 09 s. 2024 I. **Introduction** II. **Topic** III. **Objectiv...

**SAINT TONIS COLLEGE, INC.** (Formerly: Kalinga Christian Learning Center) **United Church of Christ in the Philippines** Purok 4 Bulanao Centro, Tabuk City, Kalinga Philippines 3800 Tel. No. (074) 627-5930, Email Address: Doc. 09 s. 2024 I. **Introduction** II. **Topic** III. **Objectives** a. illustrate simple and general annuities; b. distinguish between simple and general annuities; and c. find the future value and present value of simple annuity. IV. **Discussion** 1. **Annuities Classified by Length of Payment Intervals and Interest Compounding Period** a. **Simple Annuity** -- the payment interval coincides with the interest compounding periods or the interest is computed on payment date. For example, when the payment interval is six months, the interest is compounded semi-annually. In case the payment of an annuity is made at the beginning of each quarter, the interest is also computed and converted at the beginning of each quarter. b. **General Annuity** -- the payment interval does not coincide with the interest conversion periods. For example, when the payment interval is three months, and the interest is converted semi-annually. 2. **Annuities Classified by Term** c. **Annuity Certain** -- the term of annuity certain begins and ends on a definite date. For example, a loan of Php 100,000 may be settled within three years, let's say from March 21, 2012 to March 21, 2015. d. **Perpetuity** - the term of perpetuity begins on a definite date but never ends. The principal remains intact and earning interest. The length of the term is endless. e. **Contingent Annuity** -- the term of this annuity begins on a definite date; however, the ending date is not yet fixed in advance. The ending date depends on some certain conditions that will happen in the future. For example, life insurance premiums are paid as long as the person insured is alive, we are not certain when a person will die. 3. **Annuities Classified by Dates of Payment** f. **Ordinary Annuity** -- periodic payments are made at the end of each payment interval. g. **Annuity Due** -- periodic payments are made at the beginning of each payment interval. h. **Deferred Annuity** - periodic payment is not given in the beginning or end of the period but instead on the later time. ##### Ordinary Annuity ##### Sum or Amount of an Ordinary Annuity (F ord) Formula +-----------------------------------+-----------------------------------+ | Simple Annuity | General Annuity | +===================================+===================================+ | \ | \ | | [\$\$F = R\\left\\lbrack | [\$\$F = R\\left\\lbrack | | \\frac{\\left( 1 + i | \\frac{\\left( 1 + i | | \\right)\^{n} - 1}{i} | \\right)\^{n} - 1}{i} | | \\right\\rbrack\$\$]{.math | \\right\\rbrack\$\$]{.math | |.display}\ |.display}\ | | | | | Where: [\$i = | Where: | | \\frac{r}{m}\$]{.math.inline} | | | and [*n* = *mt*]{.math.inline} | \ | | | [*i*]{.math.display}\ | | F- future value | | | | \ | | A- equal payment | [\$\$= {(1 + | | | \\frac{r}{m\_{2}})}\^{\\frac{m\_{ | | *i-* interest rate per interest | 2}}{m\_{1}}} | | period | - 1\$\$]{.math.display}\ | | | | | r-nominal rate | [*m*~1~]{.math.inline}-is the | | | payment interval. (Payment | | m- number of compounding periods | interval refers to the frequency | | in a year | or period between payments, | | | specifying when payments are | | t- the term of investment or loan | made). | | | | | | [*m*~2~]{.math.inline}- number | | | of compounding periods in a year. | | | | | | n=[(*m*~1~)(*t*)]{.math.inline} | +-----------------------------------+-----------------------------------+ Example number 1. Suppose Mrs. Remoto would like to save 3,000 pesos every month in a fund that gives 9% compounded monthly. How much is the amount or the future value of her savings after 6 months. What do you think about this class, is it simple annuity or general annuity? R=3,000 r=9% or 0.09 t=6 months or 0.5-year m= 12 n=mt 12(0.5) =6 [\$i = \\frac{r}{m} = \\frac{0.09}{12}\$]{.math.inline}=0.0075 Example number 2: In order to save for her high school graduation. Marie decided to save 200 pesos at the end of each month. If the bank pays 0.25% compounded monthly, how much will her money be at the end of 6 years? Given: R=200 t= 6 years r=0.0025 n=mt =12(6) = 72 [\$i = \\frac{r}{m} = \\frac{0.0025}{12}\$]{.math.inline} Solution: \ [\$\$F = R(\\frac{\\left( 1 + i \\right)\^{n} - 1}{i})\$\$]{.math.display}\ **General Annuity** is an annuity where the length of the payment interval is [not] the same as the length of the interest compounding period. A General Ordinary Annuity is a general annuity in which the periodic payment is made at the end of the payment interval. 1. [Monthly] installment payment of a car, lot, or house with an interest rate that is [compounded annually]. 2. Paying a debt [semi-annually] when the interest is [compounded monthly]. Example number 3: What is the future value of an annuity of 2,000.00 payable annually for 9 years if the money is worth 5% compounded quarterly? given are: R=2,000.00 r=5% or 0.05 t=9 [*m*~1~]{.math.inline}=1 [*m*~2~]{.math.inline}=4 n=[*m*~1~]{.math.inline}t=1(9) =9 \ [\$\$i = {(1 + \\frac{r}{m\_{2}})}\^{\\frac{m\_{2}}{m\_{1}}} - 1\$\$]{.math.display}\ \ [\$\$i = {(1 + \\frac{0.05}{4})}\^{\\frac{4}{1}} - 1\$\$]{.math.display}\ \ [*i* = 0.05094533691]{.math.display}\ Sir, using the formula for general annuity: \ [\$\$F = R\\left\\lbrack \\frac{\\left( 1 + i \\right)\^{n} - 1}{i} \\right\\rbrack\$\$]{.math.display}\ \ [\$\$F = 2000(\\frac{\\left( 1 + 0.05094533691 \\right)\^{9} - 1}{0.05094533691})\$\$]{.math.display}\ [*F* = 22, 139.17]{.math.inline}pesos Example number 4: Alex invests 2,000 pesos quarterly for 6 years in a saving account earning 6% compounded monthly. Calculate for the future value. the given are: R= 2000 r=0.06 t=6 [*m*~1~]{.math.inline}=4 [*m*~2~]{.math.inline}=12 n=[*m*~1~]{.math.inline}t =4(6) =24 SOLUTIONS: \ [\$\$i = {(1 + \\frac{r}{m\_{2}})}\^{\\frac{m\_{2}}{m\_{1}}} - 1\$\$]{.math.display}\ \ [\$\$i = {(1 + \\frac{0.06}{12})}\^{\\frac{12}{4}} - 1\$\$]{.math.display}\ \ [*i* = 0.015075125]{.math.display}\ \ [\$\$F = R\\left\\lbrack \\frac{\\left( 1 + i \\right)\^{n} - 1}{i} \\right\\rbrack\$\$]{.math.display}\ \ [\$\$F = 2,000(\\frac{\\left( 1 + 0.015075125 \\right)\^{24} - 1}{0.015075125})\$\$]{.math.display}\ [*F* = 53, 318.83]{.math.inline} pesos A **cash flow** is a term that refers to payments received (cash inflows) or payments or deposits made (cash outflows). Cash inflows can be represented by positive numbers and cash outflows can be represented by negative numbers. ##### Deferred Annuity +-----------------------------------+-----------------------------------+ | Present value of a deferred | Future value of a deferred | | annuity | annuity | +===================================+===================================+ | \ | \ | | [\$\$PV = P\\lbrack\\frac{1 - | [\$\$FV = P\\lbrack\\frac{\\left( | | \\left( 1 + i \\right)\^{- | 1 + i \\right)\^{n} - | | \\left( n + d \\right)}}{i} - | 1}{i}\\rbrack\$\$]{.math | | \\frac{1 - \\left( 1 + i |.display}\ | | \\right)\^{- | | | d}}{i}\\rbrack\$\$]{.math | Where: | |.display}\ | | | | FV=future value | | Where: | | | | P=Regular payment | | PV= present Value | | | | i=rate per conversion period | | P=Regular payment | [\$\\begin{pmatrix} i = | | | \\frac{r}{K},\\ where\\ r\\ is\\ | | i-rate per conversion period | the\\ annual\\ rate\\ and\\ K | | [\$\\begin{pmatrix} i = | \\\\ \\text{\\ is\\ the\\ | | \\frac{r}{K},\\ where\\ r\\ is\\ | number\\ of\\ conversion\\ | | the\\ annual\\ rate\\ and\\ K | periods} \\\\ | | \\\\ \\text{\\ is\\ the\\ | \\end{pmatrix}\$]{.math.inline} | | number\\ of\\ conversion\\ | | | periods} \\\\ | n=number of paying | | \\end{pmatrix}\$]{.math.inline} | [(*n*=*t*×*K*, *where* *is* *the* | | |  *number*  *of* *years*)]{.math | | n=number of paying |.inline} | | [(*n*=*t* ×*K*, *where* *is* *the | | | * *number*  *of* *years*)]{.math | | |.inline} | | | | | | d= number of deferred periods | | +-----------------------------------+-----------------------------------+ Example 1. Find the present value of 10 semi-annual payments of ₱2,000.00 each if the first payment is due at the end of 3 years and money is worth 8% compounded semi-annually. Given: P=₱2,000 t=5 r=8% [\$i = \\frac{8\\%}{2} = 0.04\$]{.math.inline} K=2 n=5(2) =10 SOLUTION: \ [\$\$PV = P\\lbrack\\frac{1 - \\left( 1 + i \\right)\^{- \\left( n + d \\right)}}{i} - \\frac{1 - \\left( 1 + i \\right)\^{- d}}{i}\\rbrack\$\$]{.math.display}\ \ [\$\$PV = 2\\ 000\\lbrack\\frac{1 - \\left( 1 + 0.04 \\right)\^{- \\left( 10 + 5 \\right)}}{0.04} - \\frac{1 - \\left( 1 + 0.04 \\right)\^{- 5}}{0.04}\\rbrack\$\$]{.math.display}\ PV=₱ 13,333.13 EXAMPLE 2. Given: P=₱1,500 t=8 years r=6% [\$i = \\frac{6\\%}{4} = 0.015\$]{.math.inline} K=2 n=8(4) =32 d=3(4) =12 SOLUTION: \ [\$\$PV = P\\lbrack\\frac{1 - \\left( 1 + i \\right)\^{- \\left( n + d \\right)}}{i} - \\frac{1 - \\left( 1 + i \\right)\^{- d}}{i}\\rbrack\$\$]{.math.display}\ \ [\$\$PV = 1,\\ 500\\lbrack\\frac{1 - \\left( 1 + 0.015 \\right)\^{- \\left( 32 + 12 \\right)}}{0.015} - \\frac{1 - \\left( 1 + 0.015 \\right)\^{- 12}}{0.015}\\rbrack\$\$]{.math.display}\ =₱31, 699.68 EXAMPLE 3. A deferred annuity is purchased that will pay ₱5, 000 per quarter for 10 years after being deferred for 5 years and with interest rate of 6% compounded quarterly. What is the present value of annuity? Given: P=₱5,000 t=10years r=6% [\$i = \\frac{6\\%}{4} = 0.015\$]{.math.inline} K=2 n=10(4) =40 d=5(4) =20 SOLUTION: \ [\$\$PV = P\\lbrack\\frac{1 - \\left( 1 + i \\right)\^{- \\left( n + d \\right)}}{i} - \\frac{1 - \\left( 1 + i \\right)\^{- d}}{i}\\rbrack\$\$]{.math.display}\ \ [\$\$PV = 5000\\lbrack\\frac{1 - \\left( 1 + 0.015 \\right)\^{- 60}}{0.015} - \\frac{1 - \\left( 1 + 0.015 \\right)\^{- 20}}{0.015}\\rbrack\$\$]{.math.display}\ = ₱111, 0558.15 V. Activity/Activities 1.) This refers to a sequence of payments made at equal (fixed) interval or periods of time. A. Annuity C. Future Value B. Term D. Present Value 2.) An annuity where the payment interval is the same as the interest period. A. General Annuity C. Ordinary Annuity B. Simple Annuity D. Annuity Due 3.) An annuity where the payment is not the same as the interest period. A. Simple Annuity C. Ordinary Annuity B. General Annuity D. Annuity Due 4.) A type of annuity in which the payments are made at the end of each payment interval. A. Simple Interest C. Ordinary Annuity B. General Annuity D. Annuity Due 5.) A type of annuity in which the payments are made at the beginning of each payment interval. A. Simple Interest C. Ordinary Annuity B. General Annuity D. Annuity Due 6.) It is the amount of each payment. A. Future Value C. Present Value of an Annuity B. Regular or Periodic Payment D. Annuity Due 7.) Time between the first payment interval and last payment interval. A. Term of an Annuity C. Annuity Due B. Periodic Payment D. Annuity Certain 8.) It is the sum of future values of all the payments to be made during the entire term of the annuity. A. Future Value C. Present Value of an Annuity B. Amount of an Annuity D. Term of an Annuity 9.) It is the sum of present values of all the payments to be made during the entire term of the annuity. A. Future Value C. Present Value of an Annuity B. Amount of an Annuity D. Term of an Annuity 10.) It is an annuity in which payments begin and end at definite times. A. Simple Annuity C. Annuity Certain B. General Annuity D. Annuity Due 11.) Determine the amount of the annuity of Php 1,500 every end of three months for two years and three months, money is worth 12% converted quarterly. 12.) Gian bought a piece of land with a down payment of Php 200,000 and Php 7,500 every end of three months for 5 years at 8% compounded quarterly. Determine the cash equivalent of the lot. 13.) Find the amount of an annuity of Php 400 every 3 months for 10 years if interest is 8% compounded annually. 14.) Find the amount of an annuity of Php 700 every 6 months for 12 years if interest is 6%, compounded monthly. 15.) What is the accumulated amount of an annuity of Php 2,000 every year for 15 years which has an interest of 8% compounded every three months? USE YELLOW PAD PAPER Prepared by: Rajad F. Liwag Joselyn G. Vargas Rishelle G. Taguibao