Answer: mag aral kapo <3 Step-by-step explanation: hehe thanks AnnuitiesAn annuity is a type of investment in which regular payments are made over the course of multiple periods. Learning Objectives Classify the different types of annuity Key TakeawaysKey Points
Key Terms
An annuity is a type of multi- period investment where there is a certain principal deposited and then regular payments made over the course of the investment. The payments are all a fixed size. For example, a car loan may be an annuity: In order to get the car,
you are given a loan to buy the car. In return you make an initial payment (down payment), and then payments each month of a fixed amount. There is still an interest rate implicitly charged in the loan. The sum of all the payments will be greater than the loan amount, just as with a regular loan, but the payment schedule is spread out over time.
Future Value of AnnuityThe future value of an annuity is the sum of the future values of all of the payments in the annuity. Learning Objectives Calculate the future value of different types of annuities Key TakeawaysKey Points
Key Terms
The future value of an annuity is the sum of the future values of all of the payments in the annuity. It is possible to take the FV of all cash flows and add them together, but this
isn't really pragmatic if there are more than a couple of payments. A=m[(1 +r/n)nt−1]r/n\displaystyle{\text{A}=\frac{\text{m}[(1+\text{r}/\text{n})^{\text{nt}}-1]}{\text{r}/\text{n}}}
where m is the payment amount, r is the interest rate, n is the number of periods per year, and t is the length of time in years. A=m[( 1+r/n)nt+1−1]r/n−m\displaystyle{\text{A}=\frac{\text{m}[(1+\text{r}/\text{n})^{\text{nt}+1}-1]}{\text{r}/\text{n}}-\text{m}} Provided you know m, r, n, and t, therefore, you can find the future value (FV) of an annuity. Present Value of AnnuityThe PV of an annuity can be found by calculating the PV of each individual payment and then summing them up. Learning Objectives Calculate the present value of annuities Key TakeawaysKey Points
Key Terms
The Present Value (PV) of an annuity can be found by calculating the PV of each individual payment and then summing them up. As in the case of finding the Future Value (FV) of an annuity, it is important to note when each payment occurs. Annuities-due have payments at the beginning of each period, and ordinary annuities have them at the end. P0=Pn(1+i)n =P1−(1+i)−ni⋅(1+i)\displaystyle{ \text{P}_0 = \frac{\text{P}_\text{n}}{(1+\text{i})^\text{n}} = \text{P} \frac{1-(1+\text{i})^{-\text{n}}}{\text{i}} \cdot (1+\text{i}) } where P\text{P} is the size of the payment (sometimes A \text{A} or pmt\text{pmt} ), i\text{i} is the interest rate, and n\text{n} is the number of periods. P0=Pn(1 +i)n=P⋅∑k=1n1(1+i)n+k−1=P⋅1 −[1(1+i)n]i\displaystyle{ { \text{P} }_{ 0 }=\frac { { \text{P} }_{ \text{n} } }{ { (1+\text{i}) }^{ \text{n} } } =\text{P}\cdot \sum _{ \text{k}=1 }^{ \text{n} }{ \frac { 1 }{ { (1+\text{i}) }^{ \text{n}+\text{k}-1 } } } =\text{P}\cdot \frac { 1-\left[ \frac { 1 }{ { (1+\text{i} })^{ \text{n} } } \right] }{ \text{i} }} where, again, P\text{P} , i\text{i} , and n\text{n} are the size of the payment, the interest rate, and the number of periods, respectively. ExamplesExample 1 Suppose you have won a lottery that pays $1,000 per month for the next 20 years. But, you prefer to have
the entire amount now. If the interest rate is 8%, how much will you accept? x(1+.0812)240\displaystyle{ {\text{x} \left( \frac{1+.08}{12} \right)} ^{240} } Since Mr. Credit is receiving a sequence of payments, or an annuity, of $1,000 per month, its future value is given by the annuity formula: 1000[(1+0.0812)240−1]0.0812 \displaystyle{ \frac{1000 \left[ \left( \frac{1+0.08}{12} \right) ^{240}-1 \right]} {\frac{0.08}{12}} } The only way Mr. Cash will agree to the amount he receives is if these two future values are equal. So we set them equal and solve for the unknown: x(1+.0812)240=1000[(1+0.0812)240−1] 0.0812\displaystyle{ {\text{x} \left( \frac{1+.08}{12} \right)} ^{240} = \frac{1000 \left[ \left( \frac{1+0.08}{12} \right) ^{240}-1 \right]} {\frac{0.08}{12}}} x⋅(4.9268)=1,000⋅(589.02041)\text{x}\cdot (4.9268) = \\ 1,000 \cdot (589.02041) x⋅4.9268=589,020.41\text{x} \cdot 4.9268 = \\ 589,020.41 x=119,554.36\text{x} = \\ 119,554.36 The reader should also note that if Mr. Cash takes his lump sum of $119,554.36 and invests it at 8% compounded monthly, he will have $589,020.41 in 20 years. Example 2 Find the monthly payment for a car costing $15,000 if the loan is amortized over five years at an interest rate of 9%. $15,000⋅(1+.0912)60\displaystyle{ \$ 15,000\cdot { \left( \frac{1+.09}{12} \right) }^{60}} Mr. Credit wishes to make a sequence of payments, or an annuity, of x dollars per month, and its future value is given by the annuity formula: x[(1+0.0912)60−1]0.0912\displaystyle{ \frac { \text{x}{ \left[ { \left( \frac{1+0.09}{12} \right) }^{ 60 }-1 \right] } }{ \frac{0.09}{12} }} We set the two future amounts equal and solve for the unknown: 15,000⋅(1+.0912 )60=x[(1+0.0912)60−1]0.0912\displaystyle{15,000\cdot { \left( \frac{1+.09}{12} \right) }^{60} = \frac { \text{x}{ \left[ { \left( \frac{1+0.09}{12} \right) }^{ 60 }-1 \right] } }{ \frac{0.09}{12} }} 15,000⋅1.5657=x⋅75.4241\ 15,000 \cdot 1.5657 = \text{x} \cdot 75.4241 311.38=x\ 311.38 = \text{x} Calculating AnnuitiesUnderstanding the relationship between each variable and the broader concept of the time value of money enables simple valuation calculations of annuities. Learning Objectives Calculate the present or future value of various annuities based on the information given Key TakeawaysKey Points
Key Terms
Annuities Defined To understand how to calculate an annuity, it's useful to understand the variables that impact the calculation. An annuity is essentially a loan, a multi-period investment that is paid back over a fixed (or perpetual, in the case of a perpetuity ) period of time. The amount paid
back over time is relative to the amount of time it takes to pay it back, the interest rate being applied, and the principal (when creating the annuity, this is the present value). VariablesThis gives us six simple variables to use in our calculations:
Calculating AnnuitiesWith all of the inputs above at hand, it's fairly simply to value various types of annuities. Generally investors, lenders, and borrowers are interested in the present and future value of annuities. Present ValueThe present value of an annuity can be calculated as follows: PV(A)=Ai⋅[1−1(1 +i)n]{\displaystyle \text{PV}(\text{A})\,=\,{\frac {\text{A}}{\text{i}}}\cdot \left[{1-{\frac {1}{\left(1+\text{i}\right)^{\text{n}}}}}\right]} For a growth annuity (where the payment amount changes at a predetermined rate over the life of the annuity), the present value can be calculated as follows: PV=A(i−g)[1−(1 +g1+i)n]{\displaystyle \text{PV}\,=\,{\text{A} \over (\text{i}-\text{g})}\left[1-\left({1+\text{g} \over 1+\text{i}}\right)^{\text{n}}\right]} Future ValueThe future value of an annuity can be determined using this equation: FV(A)=A⋅(1+i)n−1 i{\displaystyle \text{FV}(\text{A})\,=\,\text{A}\cdot {\frac {\left(1+\text{i}\right)^{\text{n}}-1}{\text{i}}}} In a situation where payments grow over time, the future value can be determined using this equation: FV(A)=A⋅(1+i)n−(1+g)ni−g {\displaystyle \text{FV}(\text{A})\,=\,\text{A}\cdot {\frac {\left(1+\text{i}\right)^{\text{n}}-\left(1+\text{g}\right)^{\text{n}}}{\text{i}-\text{g}}}} Various Formula ArrangementsIt is also possible to use existing information to solve for missing information. Which is to say, if you know interest and time, you can solve for the following (given the following): Annuities Equations: This table is a useful way to view the calculation of annuities variables from a number of directions. Understanding how to manipulate the formula will underline the relationship between the variables, and provide some conceptual clarity as to what annuities are. Licenses and AttributionsCC licensed content, Shared previously
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What is the sum of future values of all payments to be made during the entire term of the annuity?The future value of any annuity equals the sum of all the future values for all of the annuity payments when they are moved to the end of the last payment interval. For example, assume you will make $1,000 contributions at the end of every year for the next three years to an investment earning 10% compounded annually.
What is the future value of an annuity?The future value of an annuity is the value of a group of recurring payments at a certain date in the future, assuming a particular rate of return, or discount rate. The higher the discount rate, the greater the annuity's future value.
What is the formula for the future value of general annuity due?How Is the Formula for Future Annuity Due Derived? In the first alternative, FV = PV (1 + r) n, i.e., you can multiply (1 + r) n by the current value of annuity due. The formula for current value of annuity due is (1 + r) * P {1 - (1 + r) - n} / r.
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The sum of future values of all the payments to be made during the entire term of annuity.
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