View source for Financial Math FM/Annuities

<noinclude>{{nav}}</noinclude>
== Learning objectives ==
The Candidate will be able to calculate present value, current value, and accumulated value for sequences of non-contingent payments.
== Learning outcomes ==
The Candidate will be able to:
* Define and recognize the definitions of the following terms: annuity-immediate, annuity due, perpetuity, payable m-thly or payable continuously, level payment annuity, arithmetic increasing/decreasing annuity, geometric increasing/decreasing annuity, term of annuity.
* For each of the following types of annuity/cash flows, given sufficient information of immediate or due, present value, future value, current value, interest rate, payment amount, and term of annuity, calculate any remaining item.
:* Level annuity, finite term.
:* Level perpetuity.
:* Non-level annuities/cash flows.
::* Arithmetic progression, finite term and perpetuity.
::* Geometric progression, finite term and perpetuity.
::* Other non-level annuities/cash flows.

==Geometric series formulas==
Recall the following formulas, which are useful for deriving the formulas for different types of annuities.
* <math>a+ar+ar^2+\cdots\overset{\text{ def }}=\sum_{k=0}^{\infty}ar^k=\frac{a}{1-r},\quad |r|<1 </math>;
* <math>\underbrace{a+ar+ar^2+\cdots+ar^n}_{n+1\text{ terms }}\overset{\text{ def }}=\sum_{k=0}^n ar^k=a\left(\frac{1-r^{n+1}}{1-r}\right),\quad r\ne 1</math>.

==Level annuities==
{{colored definition|
(Annuity)
An {{colored em|annuity}} is a series of payments made at {{colored em|equal}} time intervals.
}}
{{colored remark|
* The length of each time interval is arbitrary, but it is usually one year here.
* An {{colored em|annuity}} is level if all payments are equal in amount, non-level otherwise.
:* We will discuss mainly level annuities in this section, and some special types of non-level annuities will be discussed later.
* We will only discuss annuities with non-contingent (or certain) payments, but there exist annuities with contingent (or uncertain) payments.
}}
{{colored example|
* A fund provides payments at {{colored em|June 1st}} for each year. 
:* Then, it is an annuity, since the time intervals for payments are all one year <ref>For simplicity, assume there are always 365 days in on year, i.e. February 29th does not exist.</ref>.
:* (Rough) time diagram for illustration:
<pre>
     ↓         ↓         ↓  
-----*---------*---------*----------
          1         2         3
|---------|---------|---------|
  1st yr    2nd yr     3rd yr
</pre>
* A fund provides payments at the {{colored em|end}} of <math>n</math>th year if <math>n</math> is odd. 
:* Then, it is an annuity, since the time intervals for payments are all two years.
:* Time diagram:
<pre>
     ↓             ↓  
-----*-------------*----------
     1      2      3     
|----|      |------|
1st yr       3rd yr
</pre>
* A fund provides payments at the {{colored em|beginning}} of <math>k</math>th year if <math>k</math> is prime. 
:* Then, it is {{colored em|not}} an annuity, since the time intervals are not equal. E.g., payments are made at the beginning of 2nd, 3rd and 5th years, and time intervals vary here.
:* Time diagram:
<pre>
     ↓     ↓           ↓   
-----*-----*-----*-----*-----*-----
     1     2     3     4     5
     |-----|-----|     |-----|
     2nd yr 3rd yr      5th yr
</pre>
{{colored exercise|
<quiz display=simple>
{Select all annuity (or annuities).
|type="[]"}
+ A series of payments of <math>k</math> at the end of <math>k</math>th year, in which <math>k</math> is a positive integer.
|| Although payments are not equal in amount, the series is still (non-level) annuity.
+ A series of payments of 1 at the beginning of each year.
+ A series of payments payable monthly.
+ A series of payments payable daily.
- A series of payments at the end of <math>m</math>th year if <math>m</math> is a multiple of 2 or 3.
|| The payments are made at the end of 2nd, 3rd, 4th, 6th, ... year. We can see that time intervals vary.
</quiz>
}}
}}
===Annuity-immediate===
{{colored definition|
(Annuity-immediate)
An {{colored em|annuity-immediate}} is an annuity under which payments are made at the {{colored em|end}} of each period for <math>n</math> periods (<math>n</math> is a positive integer).
}}
{{colored remark|
* The {{colored em|present value}} of <math>n</math>-period annuity-immediate with payments of 1 is denoted by <math>a_{\overline n|}</math> (<math>\overline n|</math> is read "angle-n"). 
:* If the (effective) interest rate per period is <math>i</math>, then it can also be denoted by <math>a_{\overline n|i }</math>. This holds for other similar notations.
* The {{colored em|accumulated value}} (or future value) at time <math>n</math> of <math>n</math>-period annuity-immediate with payments of 1 (i.e. at the end of <math>n</math>th period) is dentoed by <math>s_{\overline n|}</math>. 
* Annuity-immediate is "immediate" in the sense that the payments start at the {{colored em|end}} of {{colored em|first}} year, {{colored em|without deferring}} to later year, instead of "start at the beginning of each year, without delay".
}}
Time diagram:
<pre>
     ↓     ↓           ↓
*----*-----*-----------*------
0    1     2    ...    n      t
</pre>
{{colored proposition|
(Formula of present value of annuity-immediate)
<math>a_{\overline n|i}=\frac{1-v^n}{i}</math>.
}}
{{colored proof|
<pre>
PV
v^n                  1
...
v^2         1    
v     1    
*-----*-----*---...--*
0     1     2   ...  n
</pre>
From the time diagram, we have
<math>a_{\overline n|i}=\underbrace{v+v^2+\dotsb+v^n}_{n\text{ terms}}=\frac{v(1-v^n)}{1-v}=\frac{(1-v^n)/1+i}{i/1+i}=\frac{1-v^n}{i}</math>. 
}}
{{colored remark|
* We can calculate the value of <math>a_{\overline n|i}</math> using BA II Plus.
:* If the (annual effective) interest rate is <math>k\%</math>, then press '''k I/Y'''. 
:* Since the amount of each payment is 1 by definition, press '''1 PMT''' (in general, if the amount is <math>m</math>, then press '''m PMT''') (positive (negative) value should be inputted for cash inflow (outflow) by convention, we input payment as cash inflow from the annuity's owner's perspective, since the owner {{colored em|receives}} payments).
:* If the annuity lasts for <math>n</math> years, then press '''n N'''.
:* Finally, press '''CPT PV''' to compute the present value, i.e. the value of <math>a_{\overline n|i}</math>. Negative value is obtained, since it tells how much cash outflow is needed in present value to exchange for the cash inflows inputted, which is the same as the present value of cash inflows by definition.
::* Alternatively, we can input negative value to '''PMT''' (treating payments as cash outflows for the cash inflow at present) and then '''CPT PV''' will yield positive value (present value is the cash inflow).
::* We can input numbers in arbitrary order before computing the present value .
:* We can also press '''CPT FV''' to compute the future value at time <math>n</math> (negative value is obtained, since it tells cash outflow needed in future value).
:* Similarly, given information about other things, we input numbers accordingly and compute the desired thing.
:* Press '''2ND CE{{!}}C 2ND FV''' for clearing calculator memory (memory for inputted values).
* For accumulated (or future) value at time <math>n</math> of payments from this annuity, we have <math>s_{\overline n|i}=a_{\overline n|i}(1+i)^n</math> from the relationship <math>FV=PVa(n),\quad a(t)=(1+i)^t</math> (the present value of annuity accumulates with accumulation function <math>a(t)=(1+i)^t</math>).
:* Explicitly, it follows from this proposition that <math>s_{\overline n|}=\frac{(1+i)^n-1}{i}</math>. 
* This formula provides a mnemonic: for annuity-{{colored em|i}}mmediate, the present value <math>a_{\overline n|}=\frac{1-v^n}{{\color{darkgreen}i}}</math> ("<math>i</math>" in the denominator matches with "i" in immediate). 
}}
{{colored example|
* An annuity pays 1000 at the end of each year for 10 years. The effective interest rate is 5%. 
* Then, the present value of payments is <math>1000a_{\overline{10}|0.05}\approx 7721.73</math> (by pressing '''5 I/Y 1000 PMT 10 N CPT PV''', which yields negative value, representing cash outflow). 
* The present value is approximately 8110.90 if the interest rate is 4% instead (by pressing '''4 I/Y CPT PV''' without clearing memory).
* The effective interest rate, such that the present value of payments is 4500, is approximately 17.96% (by pressing '''4500 +{{!}}- PV CPT I/Y''' without clearing memory, and the output is the number before "%", and we input negative value into '''PV''' since it is cash outflow).
* Suppose the annuity lasts for <math>n</math> years instead (and other conditions are the same). Then, the least value of <math>n</math> such that the present value of payments is 4500, is 6 (by pressing '''5 I/Y CPT N''' and get 5.22 without clearing memory, which implies the least value is 6 since <math>n</math> is integer).
* Suppose the annuity pays <math>k</math> at the end of each year, such that the present value of payments is 4500. Then, <math>k\approx 582.77</math> (by pressing '''10 N CPT PMT''' without clearing memory)
{{colored exercise|
<quiz display=simple>
{(Calculator exercise) Let <math>x_1,x_2,\dotsc,x_6</math> be the present value of payments if the effective interest rate is 1%, 2%, ..., 6% respectively. Calculate <math>x_1+x_2+\dotsb+x_6</math> ('''Hint''': you may utilize the memory in BA II Plus calculator, i.e. use the '''STO''' and '''RCL''' keys (check calculator manual for details)) 
|type="()"}
+ 50176.81
|| We may utilize the memory in the BA II Plus calculator to ease the calculation.
|| Press '''1000 PMT 10 N 1 I/Y CPT PV STO 1 2 I/Y CPT PV STO 2 3 I/Y CPT PV STO 3 4 I/Y CPT PV STO 4 5 I/Y CPT PV STO 5 6 I/Y CPT PV STO 6'''.
|| After that, pressing '''RCL 1 + RCL 2 + RCL 3 + RCL 4 + RCL 5 + RCL 6 +''' yields the answer.
- 42816.72
|| This is <math>x_1+\dotsb+x_5</math>. 
- 4054.08
|| This is <math>1000a_{\overline {1+2+3+4+5+6}|0.05}</math>. 
</quiz>
}}
}}
{{colored example|
Calculate <math>i</math> such that <math>a_{\overline {10}|i}=a_{\overline 5|0.03}</math>. 

'''Solution''':
* Using BA II Plus and pressing '''5 N 3 I/Y 1 PMT CPT PV''', <math>a_{\overline 5|0.03}\approx 4.58</math>. 
* After that, pressing '''10 N CPT I/Y''' without clearing memory (so that the number stored in '''PV''' is still the previous answer) yields <math>i\approx 17.47\%</math>.
{{colored exercise|
<quiz display=simple>
{Calculate the minimum value of <math>n</math> such that <math>a_{\overline n|0.05}\ge a_{\overline {10}|0.1}</math>.
|type="()"}
- 6
- 7
+ 8
|| Without using BA II Plus, <math>a_{\overline n|0.05}\ge a_{\overline{10}|0.1}\Rightarrow \frac{1-1.05^{-n}}{0.05}\ge\frac{1-1.1^{-10}}{0.1}\Rightarrow 1-1.05^{-n}\ge 0.307\Rightarrow 1.05^{-n}\le 0.692\Rightarrow n\ge -\frac{\ln 0.692}{\ln 1.05}\approx 7.55</math> 
|| So, the minimum value of <math>n</math> is 8.
|| Alternatively, using BA II Plus and pressing '''10 N 10 I/Y 1 PMT CPT PV 5 I/Y CPT N''' yields <math>n\approx 7.52</math>. If we can notice that <math>a_{\overline n|}</math> is an increasing function of <math>n</math> (since a greater <math>n</math> means more payments, so the present value cannot decrease with greater <math>n</math>), then we know that the minimum value of <math>n</math> is 8 (with <math>n=8</math>, "<math>\ge</math>" holds). 
- 9
- 10
</quiz>
}}
}}
{{colored example|
Show that <math>a_{\overline n|i}</math> is a decreasing function of <math>i</math> for each nonnegative <math>i</math>.
{{colored proof|
* It is a decreasing function of <math>i</math> if and only if <math>\frac{d}{di}a_{\overline n|i}\le 0</math>, for each nonnegative <math>i</math>. 
* Since <math>a_{\overline n|i}=v+v^2+\dotsb+v^n</math> (it is more convenient to use this form here, rather than <math>a_{\overline n|i}=\frac{1-v^n}{i}</math>), 
<math display=block>
\begin{align}
\frac{d}{di}a_{\overline n|i}&=\frac{d}{di}((1+i)^{-1}+(1+i)^{-2}+\dotsb+(1+i)^{-n})\\
&=-\underbrace{(1+i)^{-2}}_{\ge 0}-2\underbrace{(1+i)^{-3}}_{\ge 0}-\dotsb-n\underbrace{(1+i)^{-n-1}}_{\ge 0}\\
&=-\big(\underbrace{(1+i)^{-2}+2(1+i)^{-3}+\dotsb+n(1+i)^{n-1}}_{\ge 0}\big)\\
&\le 0.
\end{align}
</math> 
}}
}}
{{colored exercise|
<quiz display=simple>
{Select all correct expression(s) of <math>\frac{a_{\overline {2n}|i}}{a_{\overline n|i}}</math> in terms of <math>i</math> and <math>n</math> (<math>i\ne 0</math>).
|type="[]"}
+ <math>1+(1+i)^{-n}</math> 
|| <math>\frac{a_{\overline {2n}|i}}{a_{\overline n|i}}=\frac{(1-v^{2n})/i}{(1-v^n)/i}=\frac{(1-v^n)(1+v^n)}{1-v^n}=1+v^n=1+(1+i)^{-n}</math>.
- <math>1-(1+i)^{-n}</math> 
- <math>(1+i)^{-n}-1</math> 
- <math>(1+i)^{-n}</math> 
- <math>1-(1+i)^{n}</math> 
</quiz>
}}
===Annuity-due===
{{colored definition|
(Annuity-due)
An {{colored em|annuity-due}} is an annuity under which payments are made at the {{colored em|beginning}} of each period for <math>n</math> periods (<math>n</math> is a positive integer).
}}
{{colored remark|
* The present value of <math>n</math>-period annuity-due with payments of 1 is denoted by <math>\ddot a_{\overline n|}</math>. 
* The future value at time <math>n</math> (i.e. at the {{colored em|end}} of <math>n</math>th year) of <math>n</math>-period annuity-due with payments of 1 is denoted by <math>\ddot s_{\overline n|}</math>. 
* Annuity-due is "due" in the sense that payment is due as soon as the annuity starts, so payments are made at the beginning of a year.
}}
Time diagram:
<pre>
↓    ↓     ↓           ↓
*----*-----*-----------*------
0    1     2    ...   n-1      t
</pre>
{{colored proposition|
(Relationship between present values of annuity-immediate and annuity-due)
<math>\ddot a_{\overline n|i}=a_{\overline n|i}(1+i)</math>. 
}}
{{colored proof|
* Consider the time diagram for annuity-immediate with payments of 1:
<pre>
     1     1     1     1
*----*-----*-----*-----*------
0    1     2    ...    n   ... t
</pre>
* The present value of this annuity is <math>a_{\overline n|i}</math>. 
* So, the value of this annuity at <math>t=1</math> is <math>a_{\overline n|i}(1+i)</math>. 
* Then, if we regard <math>t=1</math> as present (i.e. <math>t=0</math>) by changing time labels, the time diagram becomes:
<pre>
     1     1     1     1
*----*-----*-----*-----*------
-1   0     1    ...   n-1  ... t
</pre>
* We can observe that this is the time diagram for annuity-due with payments of 1, and the value at <math>t=0</math> (i.e. present value) is <math>\ddot a_{\overline n|i}</math>. 
* It follows that <math>\ddot a_{\overline n|i}=a_{\overline n|i}(1+i)</math>, since these two expressions tell the value at the same time point (with different labels only), with the same series of payment. 
}}
{{colored remark|
* It follows from this proposition that <math>\ddot a_{\overline n|}=(1+i)a_{\overline n|}=(1+i)\cdot\frac{1-v^n}{i}=\frac{1-v^n}{d}</math> (<math>d</math> is equivalent to <math>i</math>). 
:* This formula provides a mnemonic: for annuity-{{colored em|d}}ue, the present value <math>\ddot a_{\overline n|i}=\frac{1-v^n}{{\color{darkgreen}d}}</math> ("<math>d</math>" in the denominator matches with "d" in due).
* Because of this relationship, we can calculate the value of <math>\ddot a_{\overline n|i}</math> using BA II Plus, by calculating the value of <math>a_{\overline n|i}</math> first, and then divide it by <math>1+i</math> to get <math>\ddot a_{\overline n|i}</math>.
* Alternatively, we may press '''2ND PMT 2ND ENTER''' to change the calculation mode to "BGN" for BA II Plus (then at the top right corner, there will be a "BGN" sign), and then we can use the same keys to compute <math>a_{\overline n|i}</math> before to compute <math>\ddot a_{\overline n|i}</math> instead.
::* {{color|red|Warning}}: However, we should press '''2ND PMT 2ND SET''' to change back to the default calculation mode, "END", to compute the value of <math>a_{\overline n|i}</math>, otherwise the computed value will be {{color|red|wrong}} when we are computing <math>a_{\overline n|i}</math> even if we are using the same keys as before.
:* Thus, to avoid this, it may be better to use the first method to calculate <math>\ddot a_{\overline n|i}</math>. 
* Similarly, since the future value at time <math>n</math> is given by <math>FV=PVa(n),\quad a(t)=(1+i)^t</math>, we have <math>\ddot s_{\overline n|i}=\ddot a_{\overline n|i}(1+i)^n</math>. 
}}
{{colored example|
* A fund pays 500 at the end of each of first 5 years, and then pays 2000 at the beginning of each of next 5 years (after the end of 5th year).
* The annual effective interest rate is 10%.
* Then, the present value of payments is
<math display=block>
500a_{\overline 5|}+2000\ddot a_{\overline 5|}v^5
\approx 6174.99.
</math>
{{colored proof|
* Consider the time diagram:
<pre>
             2000
  500         500   2000        2000
---*----...----*------*----...----*-----*  
   1           5      6           9     10  t
</pre>
* For payments of 500, their present value is <math>500a_{\overline 5|}</math>.
* For payments of 2000, their value at <math>t=5</math> is <math>2000\ddot a_{\overline 5|}</math>. 
:* It follows that their present value (i.e. value at <math>t=0</math>) is <math>2000\ddot a_{\overline 5|}v^5</math>. 
* Then, using BA II Plus, pressing '''500 PMT 5 N 10 I/Y CPT PV''', and we compute that <math>500a_{\overline 5|}\approx 1895.39</math>. 
* Similarly, pressing '''2000 PMT CPT PV ÷ 1.1 {{=}} ÷ (1.1 y^x 5) {{=}}''' yields <math>2500a_{\overline 5|}v^5\approx 4279.60</math>. 
* Adding these two numbers up yields the desired result.
}}
{{colored exercise|
<quiz display=simple>
{To make the present value of payments not less than 7000, at least how many extra year(s) should the fund last? Assume the fund keeps paying 2000 at the beginning of each of the extra year(s).
|type="()"}
- 1
+ 2
|| Let <math>n</math> be the number of extra years.
|| <math>500a_{\overline 5|}+2000\ddot a_{\overline {5+n}|}v^5\ge 7000\Rightarrow a_{\overline {5+n}|}\ge 4.52.</math> 
|| Using BA II Plus, when <math>5+n\approx 6.31\Rightarrow n\approx 1.31</math>, the equality holds. So, the minimum number of extra years needed is 2.
- 3
- 6 
- 7 
|| We are calculating {{colored em|extra}} years.

{Calculate the annual effective interest rate such that the present value of payments is 5000.
|type="()"}
- 4.18%
|| We need to consider payments of 500 also.
+ 18.83%
|| Let <math>i</math> be the desired rate.
|| Then, <math>500a_{\overline 5|i}+2000\underbrace{\ddot a_{\overline 5|i}}_{=a_{\overline 5|i}v}v^5=5000\Rightarrow (500+2000v^6)a_{\overline 5|i}=5000</math>. 
|| Using BA II Plus, we get the answer.
- 21.89%
|| For the payments of 2000, the present value is not <math>2000a_{\overline 5|}v^5</math> 
- 25.18%
|| <math>\ddota_{\overline n|i}\ne (1+i)a_{\overline n|i}</math>. 
- 30.26%
</quiz>
}}
}}
===Annuities payable {{math|''m''}}thly===
Sometimes, annuities are not payable annually, and can be payable more or less frequently than annually.

To calculate the present value of these kind of annuities, we can simply change the measurement period, and calculate the interest rate during that period that is equivalent to the given interest (or discount) rate (or force of interest), 
and the new term of the annuity in the new measurement period.

Using new terms and new interest rates, we can calculate the present value of these kind of annuities by applying previously discussed method.

{{colored example|
* The annual interest rate is 12%.
* Then, the equivalent monthly interest rate is <math>12(1.12^{1/12}-1)</math> .
* Since there are 120 months in 10 years, the present value of 10-year annuity-immediate with payments of 1 payable monthly is <math>a_{\overline {120}|12(1.12^{1/12}-1)}\approx 8.78</math>.
Time diagram:
<pre>
      1 1 1 1 1 1 1 1 1 1 1 1
----*-----------------------*------
    |-----------------------|
            12 %
    |-|   
12(1.12^{1/12}-1)
</pre>
{{colored exercise|
<quiz display=simple>
{Calculate the present value of the annuity with payments of 1 payable per two years instead.
|type="()"}
+ 2.67
|| The equivalent effective interest rate during 2 years is <math>1.12^2-1=0.2544</math>. 
|| There are 5 "two years" in 10 years.
|| So, the present value is <math>a_{\overline 5|0.2544}</math>. 
- 3.52
|| How many "two years" are there in 10 years?
- 3.60
|| What is the equivalent effective interest rate during 2 years?
- 3.93
|| How many "two years" are there in 10 years?
- 4.86
|| What is the equivalent effective interest rate during 2 years?
</quiz>
}}
}}
{{colored example|
* A fund provides payments with 4000 quarterly in arrears (i.e. at the end of each quarter) <ref>On the other hand, "quarterly in advance" means at the beginning of each quarter.</ref> for 36 months.
* The monthly discount rate is 4%.
* Calculate the present value of these payments.
'''Solution''':
* Let <math>i</math> be the equivalent quarterly interest rate. 
* Then, <math>(1+i)^4=(1-0.04)^{-12}\Rightarrow i\approx 0.13028</math>. 
* Also, there are 9 quarters in 36 months.
* Thus, the present value is 
<math display=block>4000a_{\overline 9|0.13028}\approx 20505.21.</math>
{{colored exercise|
<quiz display=simple>
{Suppose it is given that the monthly constant force of interest is 0.02 instead. Calculate the present value.
|type="()"}
- 11104.02
|| The equivalent quarterly interest rate is not <math>4(e^{0.02(4)}-1)</math>. 
+ 24649.58
|| Let <math>i'</math> be the equivalent quarterly interest rate.
|| Then, <math>1+i'=e^{0.02(4)}\Rightarrow i'\approx 0.08329</math>. 
|| So, the present value is <math>4000a_{\overline 9|i'}</math> 
- 32617.60
|| The given constant force of interest is not quarterly one.
- 35114.08
|| The given constant force of interest is not annual one.
- 45330.70
|| There are 9 quarters in 36 months.
</quiz>
}}
}}
===Annuities payable continuously===
* Recall from the motivation of force of interest that "payable continuously" is essentially "payable <math>\infty</math>thly (abuse of notation)"
* If an annuity pays 1 in each "infinitesimal" time interval, the present value of payments in a measurement period will be infinite, since there are infinitely many such time intervals during arbitrary measurement period.
* Because of this, it does not make sense to say an annuity has payments of xxx payable continuously. 
* Instead, we should use the notion of {{colored em|rate}} to describe the behaviour of continuous payment. 
{{colored example|
* A <math>n</math>-year annuity has payment continuously at a (uniform and constant) rate of 1 per year <ref>If we simply write "rate of <math>n</math>", then it implicitly shows that the rate is uniform and constant</ref>.
* Suppose the annual constant force of interest is <math>\delta</math> (and thus the annual interest rate is <math>e^\delta-1</math>). 
* The present value of this type of annuity is denoted by <math>\overline a_{\overline n|}</math>, and equals <math>\int_{0}^{n}e^{-\delta t}\,dt=\frac{1-e^{-n\delta}}{\delta}</math>. 
{{colored proof|
* The present value is <math>\underbrace{\int_{0}^{n}}_{\text{summing up}}\underbrace{a^{-1}(t)}_{\text{for PV}}\overbrace{(\underbrace{1}_{\text{rate}})\,\underbrace{dt}_{\text{`time'}}}^{\text{`amount'}}</math>.
* Since <math>a^{-1}(t)=e^{-\delta t}</math>, we get the desired integral. 
* Also, <math>\int_{0}^{n}e^{-\delta t}\,dt=\frac{-1}{\delta}\big[e^{-\delta t}\big]_0^n=\frac{-(e^{-n\delta }-1)}{\delta}=\frac{1-e^{-n\delta}}{\delta}</math>. 
* In the integral, we can interpret <math>t</math> is in years (so the lower limit is <math>t=0</math> (at the beginning of first year), and the upper limit is <math>t=n</math> (at the end of <math>n</math>th year).
}}
{{colored exercise|
<quiz display=simple>
{Select all expression(s) for the present value if the payment rate is <math>k</math> per year instead (<math>v=\frac{1}{1+i}</math>, in which <math>i</math> is the annual interest rate that is equivalent to the force of interest <math>\delta</math>).
|type="[]"}
- <math>k\int_{0}^{k}v^n\,dt</math> 
+ <math>k\int_{0}^{n}v^t\,dt</math> 
|| <math>v^t=e^{-\delta t}</math> 
+ <math>k\int_{0}^{n}e^{-\delta t}\,dt</math> 
|| We can replace the rate 1 in the integral in the example by <math>k</math> (i.e. multiply the integral by <math>k</math>). 
- <math>\int_{0}^{k}e^{-\delta t}\,dt</math> 
- <math>\int_{0}^{n}e^{-k\delta t}\,dt</math> 
</quiz>
}}
}}
{{colored remark|
* For {{colored em|variable}} force of interest <math>\delta_t</math> (<math>t</math> is in years), the present value (with payment rate <math>k</math>) is <math>\int_{0}^{n}k\exp\left(\int_{0}^{t}\delta_s\,ds\right)\,dt</math>, since <math>a^{-1}(t)=\exp\left(\int_{0}^{t}\delta_s\,ds\right)</math>.
* In general, the present value (with payment rate <math>k</math>) is <math>\int_{0}^{n}ka^{-1}(t)\,dt</math>. 
}}
{{colored example|
* Fund A provides payment continuously at a rate of 100 per year, at the force of interest <math>\delta_t=(1+t)^{-1}</math>. 
* Fund B provides level payments monthly in advance, at a rate of 1200 per year, at an annual constant force of interest 0.08.
* Both funds stop payments after the end of 10th year.
* After 10 years, calculate the difference between accumulated value of fund A and fund B.
'''Solution''':
* The accumulated value of fund A after 10 years is
<math display=block>\int_{0}^{10}100\exp\left(\int_{0}^{t}(1+s)^{-1}\,ds\right)\,dt
=\int_{0}^{10}100e^{\ln(1+t)}\,dt
=\int_{0}^{10}100(1+t)\,dt
=[50t^2+100t]_0^{10}
= 6000.
</math>
* For fund B, the monthly interest rate that is equivalent to the constant force of interest is <math>(e^{0.08})^{1/12}-1\approx 0.0067</math>. 
* Also, the monthly payment is <math>1200/12=100</math> (since all monthly payments are the same), and there are 120 monthly payments in 10 years.
* So, the accumulated value of fund B after 10 years is
<math display=block>100\ddot s_{\overline{120}|0.0067}\approx 18200.17.</math>
* It follows that the difference is approximately <math>18200.17-6000=12200.17</math> .
{{colored exercise|
<quiz display=simple>
{Calculate the difference between the present value of all payments of fund A and all payments of fund B.
|type="()"}
- 2232.56
+ 7938.07
|| The present value for fund A is <math>\int_{0}^{10}\frac{100}{1+t}\,dt=100\ln 11\approx 239.78953</math>. 
|| The present value for fund B is <math>100\ddot a_{\overline{120}|0.0067}\approx 8177.861</math>. 
- 7992.77
|| The present value for fund B is not <math>100a_{\overline{120}|0.0067}</math>. 
- 8047.84
|| <math>\ddot a_{\overline{120}|0.0067}\ne 1.0067 a_{\overline{120}|0.0067}</math> 
- 12119.11
</quiz>
}}
}}

===Perpetuities===
{{colored definition|(Perpetuity)
A {{colored em|perpetuity}} is an {{colored em|annuity}} whose payments continue {{colored em|forever}}. 
}}
{{colored remark|
* "continue forever" means the term of the annuity is infinite, or mathematically, tends to be infinity.
* A {{colored em|perpetuity-immediate}} is an {{colored em|annuity-immediate}} whose payments continue {{colored em|forever}}.
* A {{colored em|perpetuity-due}} is an {{colored em|annuity-due}} whose payments continue {{colored em|forever}}.
* The present value of perpetuity-immediate with payments of 1 is denoted by <math>a_{\overline\infty |}</math>, and equals <math>\lim_{n\to \infty}a_{\overline n|}=\lim_{n\to \infty}\frac{1-v^n}{i}=\frac{1}{i}</math> (since <math>v^n\to 0</math> as <math>n\to\infty</math>).
* The present value of perpetuity-due with payments of 1 is denoted by <math>\ddot a_{\overline \infty|}</math>, and equals <math>\lim_{n\to \infty}\ddot a_{\overline n|}=\lim_{n\to \infty}\frac{1-v^n}{d}=\frac{1}{d}</math>.
}}
Time diagram:
<pre>
     ↓     ↓    ...     
*----*-----*------------------
0    1     2    ...           t
</pre>
{{colored example|
We can derive the formula of perpetuity-immediate alternatively as follows: <math>a_{\overline \infty|}=\lim_{n\to \infty}a_{\overline n|}=\lim_{n\to \infty}\sum_{j=1}^{n}v^j\overset{\text{ def }}=\sum_{j=1}^{\infty}v^j=\frac{v}{1-v}=\frac{1/(1+i)}{i/(1+i)}=\frac{1}{i}</math>. 
}}
{{colored example|
* Amy purchases an annual perpetuity-immediate at its present value, which is 1200. 
* Suppose the annual interest rate is 5%.
* Then, the annual payment is <math>1200(0.05)=60</math>. 
{{colored proof|
* Let <math>P</math> be the annual payment.
* Then, for this perpetuity, we have
<math display=block>Pa_{\overline \infty|}=\frac{P}{0.05}=1200\Rightarrow P=1200(0.05)=60.</math>
}}
{{colored exercise|
<quiz display=simple>
{Calculate the annual payment if the perpetuity is perpetuity-due instead.
|type="()"}
- {{#expr:1140*0.05}}
+ {{#expr:1200*(0.05/1.05) round2}}
|| The annual discount rate is <math>d=\frac{0.05}{1+0.05}</math>, so the annual payment is <math>1200(0.05/1.05)</math>. 
- 60
|| The present values of perpetuity-immediate and perpetuity-due have different formulas.
- {{#expr:1260*0.05}}
- {{#expr:1200*(0.05/0.95) round2}}
|| <math>d\ne \frac{i}{1-i}</math>. 

{If the annual interest rate increases, is it true that the annual payment from the perpetuity-immediate will be higher than 60?
|type="()"}
+ Yes.
|| Let <math>i</math> be the annual interest rate.
|| Since the annual payment is <math>1200i</math>, which <math>\uparrow</math> when <math>i\uparrow</math>.
- No.
- Uncertain.

{If the perpetuity-immediate stops payment at the end of <math>n</math>th year (<math>n</math> is a positive integer) (then it becomes annuity-immediate), is it true that the annual payment will be higher than 60?
|type="()"}
+ Yes.
|| <math>Pa_{\overline n|}=\frac{P(1-1.05^{-n})}{0.05}=1200\Rightarrow P=\frac{60}{1-1.05^{-n}}</math>. 
|| Since <math>1.05^{-n}<1</math> for each positive (integer) <math>n</math>, <math>1-1.05^{-n}<1</math> for each positive (integer) <math>n</math>.
|| It follows that <math>P>60</math> for each positive (integer) <math>n</math>.
- No.
- Uncertain.
</quiz>
}}
}}
* For perpetuities payable <math>m</math>thly, the approach for calculating their present values is the same as that for annuities payable <math>m</math>thly, namely adjusting the measurement period and calculating new interest rate correspondingly.
* Also, for perpetuities payable continuously, the approach for calculating their present values is the same, except that we are dealing with improper integrals (upper limits of integrals involved are <math>\infty</math>).
==Non-level annuities==
* In general, the present value of non-level annuities can be calculated by summing up the present value of each payment.
* However, this approach can take a lot of time, and thus may not be the most efficient approach.
* We will discuss several special cases of non-level annuities, for which the present value can be calculated in an efficient way.
===Arithmetic varying annuities===
* For some annuities, payments vary (increase or decrease) in arithmetic sequence.
* We will develop a formula for calculating their present values in this subsection.
{{colored theorem|
(Present value of arithmetic varying annuities)
Suppose payments in an <math>n</math>-period annuity-immediate begin at <math>P</math> and increase by <math>D</math> per period thereafter <ref><math>D</math> stands for "difference"</ref>.
Then, the present value of payments is <math>\left(P+\frac{D}{i}\right)a_{\overline n|i}-\frac{Dn}{i}\cdot v^n</math> with effective interest rate <math>i</math> during each of <math>n</math> periods. 
}}
{{colored proof|
* Consider the following time diagram:
<pre>
                                 Row
                           D     1st
                     D     D     2nd
                     .     .
                     .     .
                     .     .
         D           D     D    n-1 th
   P     P           P     P    nth
---*-----*----...----*-----*
0  1     2    ...   n-1    n    t
</pre>
* For payments in the <math>n</math>th row, the present value is <math>Pa_{\overline n|i}</math>; 
* for payments in the <math>n-1</math>th row, the present value is <math>Da_{\overline {n-1}|i}v=Dv\cdot\frac{1-v^{n-1}}{i}</math>; 
* ...
* for payments in the <math>n-j</math>th row, the present value is <math>Da_{\overline {n-j}|i}v^j=Dv^j\cdot\frac{1-v^{n-j}}{i}</math>; 
* for payments in the 2nd row, the present value is <math>Da_{\overline 2|i}v^{n-2}=Dv^{n-2}\cdot\frac{1-v^2}{i}</math>; 
* for payments in the 1st row, the present value is <math>Da_{\overline 1|i}v^{n-1}=Dv^{n-1}\cdot\frac{1-v}{i}</math>. 
* So, the present value of all payments is
<math display=block>
\begin{align}
Pa_{\overline n|i}+\sum_{k=1}^{n-1}Dv^k\cdot\frac{1-v^{n-k}}{i}
&=Pa_{\overline n|i}+\frac{D}{i}\sum_{k=1}^{n-1}(v^k-v^n)\\
&=Pa_{\overline n|i}+\frac{D}{i}\left(\sum_{k=1}^{n-1}(v^k)-(n-1)v^n\right)\\
&=Pa_{\overline n|i}+\frac{D}{i}\bigg(\underbrace{\sum_{k=1}^{n-1}(v^k)+v^n}_{=\sum_{k=1}^{{\color{blue}n}}v^k\overset{\text{ def }}=a_{\overline n|i}}-nv^n\bigg)\\
&=Pa_{\overline n|i}+\frac{D}{i}(a_{\overline n|i}-nv^n)\\
&=\left(P+\frac{D}{i}\right)a_{\overline n|i}-\frac{Dn}{i}\cdot v^n.\\
\end{align}
</math>
}}
{{colored remark|
* Using BA II Plus, press '''<math>P+\frac{D}{i}</math> PMT <math>-\frac{Dn}{i}</math> FV <math>n</math> N <math>100i</math> I/Y CPT PV'''.
:* In particular, <math>-\frac{Dn}{i}</math> is inputted into '''FV''', since it can be regarded as cash outflow, or negative of cash inflow, at time <math>n</math> (because of the factor <math>v^n</math>), if we regard <math>P+\frac{D}{i}</math> as cash inflow.
* If <math>D</math> is negative, then payments decrease in arithmetic sequence.
* It follows from this theorem that the present value of the arithmetic varying {{colored em|perpetuities}} having the same properties (i.e. only the term of the annuity changes) is 
<math display=block>\lim_{n\to \infty}\left(\left(P+\frac{D}{i}\right)a_{\overline n|}-\frac{Dn}{i}\cdot v^n\right)=\left(P+\frac{D}{i}\right)\cdot\frac{1}{i}=\frac{P}{i}+\frac{D}{i^2}.</math>
:* In particular, <math>\lim_{n\to \infty}nv^n=0</math>, which can be shown by L'Hospital rule. Intuitively, the limit equals zero since <math>n\mapsto v^n</math> decreases "much faster" than <math>n\mapsto n</math>. 
* When <math>P=D=1</math>, the annuity-immediate is called {{colored em|increasing annuity}}, whose present value is denoted by <math>(Ia)_{\overline n|}</math>.
:* Its accumulated value at the end of <math>n</math>th period is denoted by <math>(Is)_{\overline n|}</math>, which equals <math>(Ia)_{\overline n|}(1+i)^n</math>. 
* When <math>P=n</math> and <math>D=-1</math>, the annuity-immediate is called {{colored em|decreasing annuity}} (payments decrease from <math>n</math> to 1 with common difference 1), whose present value is denoted by <math>(Da)_{\overline n|}</math>. 
:* Its accumulated value at the end of <math>n</math>th period is denoted by <math>(Ds)_{\overline n|}</math>, which equals <math>(Da)_{\overline n|}(1+i)^n</math>. 
* For annuity-due and annuity payable <math>m</math>thly with payments varying in arithmetic sequence, we can use similar approach discussed previously, in addition to this theorem, to calculate their present values.
}}
{{colored example|
* An annuity has payments of 100, 120, 140, 160, 180, 200, 200 at the end of 1st, 2nd, 3rd, 4th, 5th, 6th and 7th year respectively.
* The annual effective interest rate is 10%.
* Consider the time diagram:
<pre>
   100   120   140   160   180   200|  200
----*-----*-----*-----*-----*-----*-|---*----
    1     2     3     4     5     6 |   7     t
</pre>
* For the first six payments, they increase in arithmetic sequence with first term 100 and common difference 20, and they are made at the end of year.
* So, we can apply the above theorem (<math>P=100,D=20,n=6,\text{ and }i=0.1</math>) to get their present value, which is approximately 629.21.
* For the 7th payment, its present value is <math>200v^7_{0.1}\approx 102.63</math>. 
* So, the present value of these seven payments is approximately 731.84.
{{colored exercise|
<quiz display=simple>
{Suppose the annuity keeps paying 200 at the end of each year from the end of 6th year onward. Calculate the present value of the annuity.
|type="()"}
- 11289.48
|| How about the first 6 payments?
+ 11918.69
|| For the first 6 payments, the present value is still approximately 629.21.
|| For the remaining payments, their present value is <math>\frac{2000(1.1)^{-6}}{0.1}\approx 11289.48</math>. 
|| Adding up these two present values yield the answer.
- 19370.79
- 20000 
- 20629.21

{Suppose the annuity has payments of 220, 240, 260, 280, 300 at the end of 8th, 9th, 10th, 11st, and 12nd year respectively. Calculate the present value of the annuity.
|type="()"}
- 1177.43
- 1548.86
- 1559.12
- 1661.75
|| The payment of 200 repeats at the end of 6th and at the end of 7th year.
|| So, the whole series of payments is not in arithmetic sequence
+ 1693.95
|| Consider payments from the end of 7th year to that of 12th year.
|| They are in arithmetic sequence, and so their value at the end of 6th year is <math>(200+20/0.1)a_{\overline 6|0.1}-\frac{20(6)}{0.1}(v_{0.1}^{-6})</math>. 
|| Multiplying this value by <math>v_{0.1}^6</math> yields the present value of these payments, which is approximately 1064.74.
|| Adding this with the present value of first 6 payments, which is approximately 629.21, yields the answer.
</quiz>
}}
}}

===Geometric varying annuities===
* Since the expression for present value of annuity is essentially geometric series <ref>For example, <math>a_{\overline n|}=v+v^2+\dotsb+v^n</math>, and <math>\ddot a_{\overline \infty|}=1+v+v^2+\dotsb</math>, which are geometric series</ref>, even with payments varying in geometric sequence, the expression is still geometric series, and thus we can use the geometric series formula to calculate the present value.
* So, in general, for geometric varying annuities, we use "first principle" to calculate their present value, in the sense that we use geometric series formula to evaluate the expanded form of the present value.
{{colored example|
* An annuity has payments at the beginning of each year without stopping, which begin at 100 at the beginning of first year, and then increase by 10% per year.
* Suppose the interest rate is 20% payable quarterly.
* Calculate the present value of the annuity.
'''Solution''':
* The nominal interest rate of 20% implies the quarterly effective interest rate is 5%.
* So, the annual effective interest rate is <math>1.05^4-1\approx 21.551\%</math>. 
* Consider the time diagram:
<pre>
100     100(1.1)  100(1.1)^2   100(1.1)^3
*---------*---------*------------*------
0         1         2            3
</pre>
* It follows that the present value of the annuity is
<math display=block>
100+100(1.1)(1.21551)^{-1}+100(1.1)^2(1.21551)^{-2}+\dotsb
=100+100\cdot\frac{1.1}{1.21551}+100\cdot\left(\frac{1.1}{1.21551}\right)^2+\dotsb
=\frac{100}{1-\frac{1.1}{1.21551}}
\approx 1052.33.
</math>
{{colored exercise|
<quiz display=simple>
{Suppose the given interest rate is fixed and payments of the annuity increase by <math>x\%</math> per year (<math>x</math> is positive). Calculate the maximum value of <math>x</math> such that the present value of the annuity is finite.
|type="()"}
- 20
+ 21.55
|| To ensure the present value to be finite, the absolute value of the common ratio of the geometric series needs to be strictly less than 1.
|| So, <math>|\underbrace{\frac{1+x/100}{1.21551}}_{>0}|<1\Rightarrow \frac{1+x/100}{1.21551}<1\Rightarrow 1+x/100<1.21551\Rightarrow x<21.551</math> 
- 120
- 121.55
- The present value is finite for each positive <math>x</math>.

{Suppose payments of the annuity lasts for 10 years only. Calculate the present value of the annuity.
|type="()"}
- 564.61
- 578.92
- 601.45
|| What is the first term of the geometric series for this annuity?
- 623.90
|| There are 10 terms in the geometric series for this annuity.
+ 664.61
|| The present value is <math display=block>100+100(1.1)(1.21551)^{-1}+\dotsb+100(1.1)^9(1.21551)^{-9}=\frac{100(1-(1.1/1.21551)^{10})}{1-1.1/1.21551}\approx 664.61</math>
</quiz>
}}
}}
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