Financial Math FM/Annuities

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.

• $a+ar+ar^{2}+\cdots {\overset {\text{ def }}{=}}\sum _{k=0}^{\infty }ar^{k}={\frac {a}{1-r}},\quad |r|<1$;
• $\underbrace {a+ar+ar^{2}+\cdots +ar^{n}} _{n+1{\text{ terms }}}{\overset {\text{ def }}{=}}\sum _{k=0}^{n}ar^{k}=a\left({\frac {1-r^{n+1}}{1-r}}\right),\quad r\neq 1$.

Level annuities

Definition. (Annuity) An annuity is a series of payments made at equal time intervals.

Remark.

• The length of each time interval is arbitrary, but it is usually one year here.
• An 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.

Example.

• A fund provides payments at June 1st for each year.

• Then, it is an annuity, since the time intervals for payments are all one year .
• (Rough) time diagram for illustration:

     ↓         ↓         ↓  
-----*---------*---------*----------
          1         2         3
|---------|---------|---------|
  1st yr    2nd yr     3rd yr

• A fund provides payments at the end of $n$th year if $n$ is odd.

• Then, it is an annuity, since the time intervals for payments are all two years.
• Time diagram:

     ↓             ↓  
-----*-------------*----------
     1      2      3     
|----|      |------|
1st yr       3rd yr

• A fund provides payments at the beginning of $k$th year if $k$ is prime.

• Then, it is 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:

     ↓     ↓           ↓   
-----*-----*-----*-----*-----*-----
     1     2     3     4     5
     |-----|-----|     |-----|
     2nd yr 3rd yr      5th yr

Exercise.

Select all annuity (or annuities).

	A series of payments of $k$ at the end of $k$th year, in which $k$ is a positive integer.
	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 $m$th year if $m$ is a multiple of 2 or 3.

Annuity-immediate

Definition. (Annuity-immediate) An annuity-immediate is an annuity under which payments are made at the end of each period for $n$ periods ($n$ is a positive integer).

Remark.

• The present value of $n$-period annuity-immediate with payments of 1 is denoted by $a_{{\overline {n}}|}$ (${\overline {n}}|$ is read "angle-n").

• If the (effective) interest rate per period is $i$, then it can also be denoted by $a_{{\overline {n}}|i}$. This holds for other similar notations.

• The accumulated value (or future value) at time $n$ of $n$-period annuity-immediate with payments of 1 (i.e. at the end of $n$th period) is dentoed by $s_{{\overline {n}}|}$.
• Annuity-immediate is "immediate" in the sense that the payments start at the end of first year, without deferring to later year, instead of "start at the beginning of each year, without delay".

Time diagram:

     ↓     ↓           ↓
*----*-----*-----------*------
0    1     2    ...    n      t

Proposition. (Formula of present value of annuity-immediate) $a_{{\overline {n}}|i}={\frac {1-v^{n}}{i}}$.

Proof.

PV
v^n                  1
...
v^2         1    
v     1    
*-----*-----*---...--*
0     1     2   ...  n

From the time diagram, we have $a_{{\overline {n}}|i}=\underbrace {v+v^{2}+\dotsb +v^{n}} _{n{\text{ terms}}}={\frac {v(1-v^{n})}{1-v}}={\frac {(1-v^{n})/1+i}{i/1+i}}={\frac {1-v^{n}}{i}}$.

$\Box$

Remark.

• We can calculate the value of $a_{{\overline {n}}|i}$ using BA II Plus.

• If the (annual effective) interest rate is $k\%$, then press k I/Y.
• Since the amount of each payment is 1 by definition, press 1 PMT (in general, if the amount is $m$, 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 receives payments).
• If the annuity lasts for $n$ years, then press n N.
• Finally, press CPT PV to compute the present value, i.e. the value of $a_{{\overline {n}}|i}$. 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 $n$ (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 $n$ of payments from this annuity, we have $s_{{\overline {n}}|i}=a_{{\overline {n}}|i}(1+i)^{n}$ from the relationship $FV=PVa(n),\quad a(t)=(1+i)^{t}$ (the present value of annuity accumulates with accumulation function $a(t)=(1+i)^{t}$).

• Explicitly, it follows from this proposition that $s_{{\overline {n}}|}={\frac {(1+i)^{n}-1}{i}}$.

• This formula provides a mnemonic: for annuity-immediate, the present value $a_{{\overline {n}}|}={\frac {1-v^{n}}{\color {darkgreen}i}}$ ("$i$" in the denominator matches with "i" in immediate).

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 $1000a_{{\overline {10}}|0.05}\approx 7721.73$ (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 $n$ years instead (and other conditions are the same). Then, the least value of $n$ 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 $n$ is integer).
• Suppose the annuity pays $k$ at the end of each year, such that the present value of payments is 4500. Then, $k\approx 582.77$ (by pressing 10 N CPT PMT without clearing memory)

Exercise.

(Calculator exercise) Let $x_{1},x_{2},\dotsc ,x_{6}$ be the present value of payments if the effective interest rate is 1%, 2%, ..., 6% respectively. Calculate $x_{1}+x_{2}+\dotsb +x_{6}$ (Hint: you may utilize the memory in BA II Plus calculator, i.e. use the STO and RCL keys (check calculator manual for details))

	50176.81
	42816.72
	4054.08

Example. Calculate $i$ such that $a_{{\overline {10}}|i}=a_{{\overline {5}}|0.03}$.

Solution:

• Using BA II Plus and pressing 5 N 3 I/Y 1 PMT CPT PV, $a_{{\overline {5}}|0.03}\approx 4.58$.
• After that, pressing 10 N CPT I/Y without clearing memory (so that the number stored in PV is still the previous answer) yields $i\approx 17.47\%$.

Exercise.

Calculate the minimum value of $n$ such that $a_{{\overline {n}}|0.05}\geq a_{{\overline {10}}|0.1}$.

	6
	7
	8
	9
	10

Example. Show that $a_{{\overline {n}}|i}$ is a decreasing function of $i$ for each nonnegative $i$.

Proof.

• It is a decreasing function of $i$ if and only if ${\frac {d}{di}}a_{{\overline {n}}|i}\leq 0$, for each nonnegative $i$.
• Since $a_{{\overline {n}}|i}=v+v^{2}+\dotsb +v^{n}$ (it is more convenient to use this form here, rather than $a_{{\overline {n}}|i}={\frac {1-v^{n}}{i}}$),

$${\begin{aligned}{\frac {d}{di}}a_{{\overline {n}}|i}&={\frac {d}{di}}((1+i)^{-1}+(1+i)^{-2}+\dotsb +(1+i)^{-n})\\&=-\underbrace {(1+i)^{-2}} _{\geq 0}-2\underbrace {(1+i)^{-3}} _{\geq 0}-\dotsb -n\underbrace {(1+i)^{-n-1}} _{\geq 0}\\&=-{\big (}\underbrace {(1+i)^{-2}+2(1+i)^{-3}+\dotsb +n(1+i)^{n-1}} _{\geq 0}{\big )}\\&\leq 0.\end{aligned}}$$

$\Box$

Exercise.

Select all correct expression(s) of ${\frac {a_{{\overline {2n}}|i}}{a_{{\overline {n}}|i}}}$ in terms of $i$ and $n$ ($i\neq 0$).

	$1+(1+i)^{-n}$
	$1-(1+i)^{-n}$
	$(1+i)^{-n}-1$
	$(1+i)^{-n}$
	$1-(1+i)^{n}$

Annuity-due

Definition. (Annuity-due) An annuity-due is an annuity under which payments are made at the beginning of each period for $n$ periods ($n$ is a positive integer).

Remark.

• The present value of $n$-period annuity-due with payments of 1 is denoted by ${\ddot {a}}_{{\overline {n}}|}$.
• The future value at time $n$ (i.e. at the end of $n$th year) of $n$-period annuity-due with payments of 1 is denoted by ${\ddot {s}}_{{\overline {n}}|}$.
• 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:

↓    ↓     ↓           ↓
*----*-----*-----------*------
0    1     2    ...   n-1      t

Proposition. (Relationship between present values of annuity-immediate and annuity-due) ${\ddot {a}}_{{\overline {n}}|i}=a_{{\overline {n}}|i}(1+i)$.

Proof.

• Consider the time diagram for annuity-immediate with payments of 1:

     1     1     1     1
*----*-----*-----*-----*------
0    1     2    ...    n   ... t

• The present value of this annuity is $a_{{\overline {n}}|i}$.
• So, the value of this annuity at $t=1$ is $a_{{\overline {n}}|i}(1+i)$.
• Then, if we regard $t=1$ as present (i.e. $t=0$) by changing time labels, the time diagram becomes:

     1     1     1     1
*----*-----*-----*-----*------
-1   0     1    ...   n-1  ... t

• We can observe that this is the time diagram for annuity-due with payments of 1, and the value at $t=0$ (i.e. present value) is ${\ddot {a}}_{{\overline {n}}|i}$.
• It follows that ${\ddot {a}}_{{\overline {n}}|i}=a_{{\overline {n}}|i}(1+i)$, since these two expressions tell the value at the same time point (with different labels only), with the same series of payment.

$\Box$

Remark.

• It follows from this proposition that ${\ddot {a}}_{{\overline {n}}|}=(1+i)a_{{\overline {n}}|}=(1+i)\cdot {\frac {1-v^{n}}{i}}={\frac {1-v^{n}}{d}}$ ($d$ is equivalent to $i$).

• This formula provides a mnemonic: for annuity-due, the present value ${\ddot {a}}_{{\overline {n}}|i}={\frac {1-v^{n}}{\color {darkgreen}d}}$ ("$d$" in the denominator matches with "d" in due).

• Because of this relationship, we can calculate the value of ${\ddot {a}}_{{\overline {n}}|i}$ using BA II Plus, by calculating the value of $a_{{\overline {n}}|i}$ first, and then divide it by $1+i$ to get ${\ddot {a}}_{{\overline {n}}|i}$.
• 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 $a_{{\overline {n}}|i}$ before to compute ${\ddot {a}}_{{\overline {n}}|i}$ instead.

• Warning: However, we should press 2ND PMT 2ND SET to change back to the default calculation mode, "END", to compute the value of $a_{{\overline {n}}|i}$, otherwise the computed value will be wrong when we are computing $a_{{\overline {n}}|i}$ 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 ${\ddot {a}}_{{\overline {n}}|i}$.

• Similarly, since the future value at time $n$ is given by $FV=PVa(n),\quad a(t)=(1+i)^{t}$, we have ${\ddot {s}}_{{\overline {n}}|i}={\ddot {a}}_{{\overline {n}}|i}(1+i)^{n}$.

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

$$500a_{{\overline {5}}|}+2000{\ddot {a}}_{{\overline {5}}|}v^{5}\approx 6174.99.$$

Proof.

• Consider the time diagram:

             2000
  500         500   2000        2000
---*----...----*------*----...----*-----*  
   1           5      6           9     10  t

• For payments of 500, their present value is $500a_{{\overline {5}}|}$.
• For payments of 2000, their value at $t=5$ is $2000{\ddot {a}}_{{\overline {5}}|}$.

• It follows that their present value (i.e. value at $t=0$) is $2000{\ddot {a}}_{{\overline {5}}|}v^{5}$.

• Then, using BA II Plus, pressing 500 PMT 5 N 10 I/Y CPT PV, and we compute that $500a_{{\overline {5}}|}\approx 1895.39$.
• Similarly, pressing 2000 PMT CPT PV ÷ 1.1 = ÷ (1.1 y^x 5) = yields $2500a_{{\overline {5}}|}v^{5}\approx 4279.60$.
• Adding these two numbers up yields the desired result.

$\Box$

Exercise.

1 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).

	1
	2
	3
	6
	7

2 Calculate the annual effective interest rate such that the present value of payments is 5000.

	4.18%
	18.83%
	21.89%
	25.18%
	30.26%

Annuities payable mthly

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.

Example.

• The annual interest rate is 12%.
• Then, the equivalent monthly interest rate is $12(1.12^{1/12}-1)$ .
• Since there are 120 months in 10 years, the present value of 10-year annuity-immediate with payments of 1 payable monthly is $a_{{\overline {120}}|12(1.12^{1/12}-1)}\approx 8.78$.

Time diagram:

      1 1 1 1 1 1 1 1 1 1 1 1
----*-----------------------*------
    |-----------------------|
            12 %
    |-|   
12(1.12^{1/12}-1)

Exercise.

Calculate the present value of the annuity with payments of 1 payable per two years instead.

	2.67
	3.52
	3.60
	3.93
	4.86

Example.

• A fund provides payments with 4000 quarterly in arrears (i.e. at the end of each quarter) for 36 months.
• The monthly discount rate is 4%.
• Calculate the present value of these payments.

Solution:

• Let $i$ be the equivalent quarterly interest rate.
• Then, $(1+i)^{4}=(1-0.04)^{-12}\Rightarrow i\approx 0.13028$.
• Also, there are 9 quarters in 36 months.
• Thus, the present value is

$$4000a_{{\overline {9}}|0.13028}\approx 20505.21.$$

Exercise.

Suppose it is given that the monthly constant force of interest is 0.02 instead. Calculate the present value.

	11104.02
	24649.58
	32617.60
	35114.08
	45330.70

Annuities payable continuously

• Recall from the motivation of force of interest that "payable continuously" is essentially "payable $\infty$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 rate to describe the behaviour of continuous payment.

Example.

• A $n$-year annuity has payment continuously at a (uniform and constant) rate of 1 per year .
• Suppose the annual constant force of interest is $\delta$ (and thus the annual interest rate is $e^{\delta }-1$).
• The present value of this type of annuity is denoted by ${\overline {a}}_{{\overline {n}}|}$, and equals $\int _{0}^{n}e^{-\delta t}\,dt={\frac {1-e^{-n\delta }}{\delta }}$.

Proof.

• The present value is $\underbrace {\int _{0}^{n}} _{\text{summing up}}\underbrace {a^{-1}(t)} _{\text{for PV}}\overbrace {(\underbrace {1} _{\text{rate}})\,\underbrace {dt} _{\text{`time'}}} ^{\text{`amount'}}$.
• Since $a^{-1}(t)=e^{-\delta t}$, we get the desired integral.
• Also, $\int _{0}^{n}e^{-\delta t}\,dt={\frac {-1}{\delta }}{\big [}e^{-\delta t}{\big ]}_{0}^{n}={\frac {-(e^{-n\delta }-1)}{\delta }}={\frac {1-e^{-n\delta }}{\delta }}$.
• In the integral, we can interpret $t$ is in years (so the lower limit is $t=0$ (at the beginning of first year), and the upper limit is $t=n$ (at the end of $n$th year).

$\Box$

Exercise.

Select all expression(s) for the present value if the payment rate is $k$ per year instead ($v={\frac {1}{1+i}}$, in which $i$ is the annual interest rate that is equivalent to the force of interest $\delta$).

	$k\int _{0}^{k}v^{n}\,dt$
	$k\int _{0}^{n}v^{t}\,dt$
	$k\int _{0}^{n}e^{-\delta t}\,dt$
	$\int _{0}^{k}e^{-\delta t}\,dt$
	$\int _{0}^{n}e^{-k\delta t}\,dt$

Remark.

• For variable force of interest $\delta _{t}$ ($t$ is in years), the present value (with payment rate $k$) is $\int _{0}^{n}k\exp \left(\int _{0}^{t}\delta _{s}\,ds\right)\,dt$, since $a^{-1}(t)=\exp \left(\int _{0}^{t}\delta _{s}\,ds\right)$.
• In general, the present value (with payment rate $k$) is $\int _{0}^{n}ka^{-1}(t)\,dt$.

Example.

• Fund A provides payment continuously at a rate of 100 per year, at the force of interest $\delta _{t}=(1+t)^{-1}$.
• 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

$$\int _{0}^{10}100\exp \left(\int _{0}^{t}(1+s)^{-1}\,ds\right)\,dt=\int _{0}^{10}100e^{\ln(1+t)}\,dt=\int _{0}^{10}100(1+t)\,dt=[50t^{2}+100t]_{0}^{10}=6000.$$

• For fund B, the monthly interest rate that is equivalent to the constant force of interest is $(e^{0.08})^{1/12}-1\approx 0.0067$.
• Also, the monthly payment is $1200/12=100$ (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

$$100{\ddot {s}}_{{\overline {120}}|0.0067}\approx 18200.17.$$

• It follows that the difference is approximately $18200.17-6000=12200.17$ .

Exercise.

Calculate the difference between the present value of all payments of fund A and all payments of fund B.

	2232.56
	7938.07
	7992.77
	8047.84
	12119.11

Perpetuities

Definition. (Perpetuity) A perpetuity is an annuity whose payments continue forever.

Remark.

• "continue forever" means the term of the annuity is infinite, or mathematically, tends to be infinity.
• A perpetuity-immediate is an annuity-immediate whose payments continue forever.
• A perpetuity-due is an annuity-due whose payments continue forever.
• The present value of perpetuity-immediate with payments of 1 is denoted by $a_{{\overline {\infty }}|}$, and equals $\lim _{n\to \infty }a_{{\overline {n}}|}=\lim _{n\to \infty }{\frac {1-v^{n}}{i}}={\frac {1}{i}}$ (since $v^{n}\to 0$ as $n\to \infty$).
• The present value of perpetuity-due with payments of 1 is denoted by ${\ddot {a}}_{{\overline {\infty }}|}$, and equals $\lim _{n\to \infty }{\ddot {a}}_{{\overline {n}}|}=\lim _{n\to \infty }{\frac {1-v^{n}}{d}}={\frac {1}{d}}$.

Time diagram:

     ↓     ↓    ...     
*----*-----*------------------
0    1     2    ...           t

Example. We can derive the formula of perpetuity-immediate alternatively as follows: $a_{{\overline {\infty }}|}=\lim _{n\to \infty }a_{{\overline {n}}|}=\lim _{n\to \infty }\sum _{j=1}^{n}v^{j}{\overset {\text{ def }}{=}}\sum _{j=1}^{\infty }v^{j}={\frac {v}{1-v}}={\frac {1/(1+i)}{i/(1+i)}}={\frac {1}{i}}$.

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 $1200(0.05)=60$.

Proof.

• Let $P$ be the annual payment.
• Then, for this perpetuity, we have

$$Pa_{{\overline {\infty }}|}={\frac {P}{0.05}}=1200\Rightarrow P=1200(0.05)=60.$$

$\Box$

Exercise.

1 Calculate the annual payment if the perpetuity is perpetuity-due instead.

	57
	57.14
	60
	63
	63.16

2 If the annual interest rate increases, is it true that the annual payment from the perpetuity-immediate will be higher than 60?

	Yes.
	No.
	Uncertain.

3 If the perpetuity-immediate stops payment at the end of $n$th year ($n$ is a positive integer) (then it becomes annuity-immediate), is it true that the annual payment will be higher than 60?

	Yes.
	No.
	Uncertain.

• For perpetuities payable $m$thly, the approach for calculating their present values is the same as that for annuities payable $m$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 $\infty$).

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.

Theorem. (Present value of arithmetic varying annuities) Suppose payments in an $n$-period annuity-immediate begin at $P$ and increase by $D$ per period thereafter . Then, the present value of payments is $\left(P+{\frac {D}{i}}\right)a_{{\overline {n}}|i}-{\frac {Dn}{i}}\cdot v^{n}$ with effective interest rate $i$ during each of $n$ periods.

Proof.

• Consider the following time diagram:

                                 Row
                           D     1st
                     D     D     2nd
                     .     .
                     .     .
                     .     .
         D           D     D    n-1 th
   P     P           P     P    nth
---*-----*----...----*-----*
0  1     2    ...   n-1    n    t

• For payments in the $n$th row, the present value is $Pa_{{\overline {n}}|i}$;
• for payments in the $n-1$th row, the present value is $Da_{{\overline {n-1}}|i}v=Dv\cdot {\frac {1-v^{n-1}}{i}}$;
• ...
• for payments in the $n-j$th row, the present value is $Da_{{\overline {n-j}}|i}v^{j}=Dv^{j}\cdot {\frac {1-v^{n-j}}{i}}$;
• for payments in the 2nd row, the present value is $Da_{{\overline {2}}|i}v^{n-2}=Dv^{n-2}\cdot {\frac {1-v^{2}}{i}}$;
• for payments in the 1st row, the present value is $Da_{{\overline {1}}|i}v^{n-1}=Dv^{n-1}\cdot {\frac {1-v}{i}}$.
• So, the present value of all payments is

$${\begin{aligned}Pa_{{\overline {n}}|i}+\sum _{k=1}^{n-1}Dv^{k}\cdot {\frac {1-v^{n-k}}{i}}&=Pa_{{\overline {n}}|i}+{\frac {D}{i}}\sum _{k=1}^{n-1}(v^{k}-v^{n})\\&=Pa_{{\overline {n}}|i}+{\frac {D}{i}}\left(\sum _{k=1}^{n-1}(v^{k})-(n-1)v^{n}\right)\\&=Pa_{{\overline {n}}|i}+{\frac {D}{i}}{\bigg (}\underbrace {\sum _{k=1}^{n-1}(v^{k})+v^{n}} _{=\sum _{k=1}^{\color {blue}n}v^{k}{\overset {\text{ def }}{=}}a_{{\overline {n}}|i}}-nv^{n}{\bigg )}\\&=Pa_{{\overline {n}}|i}+{\frac {D}{i}}(a_{{\overline {n}}|i}-nv^{n})\\&=\left(P+{\frac {D}{i}}\right)a_{{\overline {n}}|i}-{\frac {Dn}{i}}\cdot v^{n}.\\\end{aligned}}$$

$\Box$

Remark.

• Using BA II Plus, press $P+{\frac {D}{i}}$ PMT $-{\frac {Dn}{i}}$ FV $n$ N $100i$ I/Y CPT PV.

• In particular, $-{\frac {Dn}{i}}$ is inputted into FV, since it can be regarded as cash outflow, or negative of cash inflow, at time $n$ (because of the factor $v^{n}$), if we regard $P+{\frac {D}{i}}$ as cash inflow.

• If $D$ is negative, then payments decrease in arithmetic sequence.
• It follows from this theorem that the present value of the arithmetic varying perpetuities having the same properties (i.e. only the term of the annuity changes) is

$$\lim _{n\to \infty }\left(\left(P+{\frac {D}{i}}\right)a_{{\overline {n}}|}-{\frac {Dn}{i}}\cdot v^{n}\right)=\left(P+{\frac {D}{i}}\right)\cdot {\frac {1}{i}}={\frac {P}{i}}+{\frac {D}{i^{2}}}.$$

• In particular, $\lim _{n\to \infty }nv^{n}=0$, which can be shown by L'Hospital rule. Intuitively, the limit equals zero since $n\mapsto v^{n}$ decreases "much faster" than $n\mapsto n$.

• When $P=D=1$, the annuity-immediate is called increasing annuity, whose present value is denoted by $(Ia)_{{\overline {n}}|}$.

• Its accumulated value at the end of $n$th period is denoted by $(Is)_{{\overline {n}}|}$, which equals $(Ia)_{{\overline {n}}|}(1+i)^{n}$.

• When $P=n$ and $D=-1$, the annuity-immediate is called decreasing annuity (payments decrease from $n$ to 1 with common difference 1), whose present value is denoted by $(Da)_{{\overline {n}}|}$.

• Its accumulated value at the end of $n$th period is denoted by $(Ds)_{{\overline {n}}|}$, which equals $(Da)_{{\overline {n}}|}(1+i)^{n}$.

• For annuity-due and annuity payable $m$thly with payments varying in arithmetic sequence, we can use similar approach discussed previously, in addition to this theorem, to calculate their present values.

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:

   100   120   140   160   180   200|  200
----*-----*-----*-----*-----*-----*-|---*----
    1     2     3     4     5     6 |   7     t

• 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 ($P=100,D=20,n=6,{\text{ and }}i=0.1$) to get their present value, which is approximately 629.21.
• For the 7th payment, its present value is $200v_{0.1}^{7}\approx 102.63$.
• So, the present value of these seven payments is approximately 731.84.

Exercise.

1 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.

	11289.48
	11918.69
	19370.79
	20000
	20629.21

2 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.

	1177.43
	1548.86
	1559.12
	1661.75
	1693.95

Geometric varying annuities

• Since the expression for present value of annuity is essentially geometric series , 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.

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 $1.05^{4}-1\approx 21.551\%$.
• Consider the time diagram:

100     100(1.1)  100(1.1)^2   100(1.1)^3
*---------*---------*------------*------
0         1         2            3

• It follows that the present value of the annuity is

$$100+100(1.1)(1.21551)^{-1}+100(1.1)^{2}(1.21551)^{-2}+\dotsb =100+100\cdot {\frac {1.1}{1.21551}}+100\cdot \left({\frac {1.1}{1.21551}}\right)^{2}+\dotsb ={\frac {100}{1-{\frac {1.1}{1.21551}}}}\approx 1052.33.$$

Exercise.

1 Suppose the given interest rate is fixed and payments of the annuity increase by $x\%$ per year ($x$ is positive). Calculate the maximum value of $x$ such that the present value of the annuity is finite.

	20
	21.55
	120
	121.55
	The present value is finite for each positive $x$.

2 Suppose payments of the annuity lasts for 10 years only. Calculate the present value of the annuity.

	564.61
	578.92
	601.45
	623.90
	664.61