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					                                            Lecture 2. Annuity (finance theory)

The term annuity is used in finance theory to refer to any terminating stream of fixed payments over a specified
period of time. This usage is most commonly seen in discussions of finance, usually in connection with the valuation of
the stream of payments, taking into account time value of money concepts such as interest rate and future value.

Examples of annuities are regular deposits to a savings account, monthly home mortgage payments and monthly
insurance payments. Annuities are classified by payment dates. The payments (deposits) may be made weekly,
monthly, quarterly, yearly, or at any other interval of time.

An ordinary annuity (also referred as annuity-immediate) is an annuity whose payments are made at the end of each
                                                                                                          [2]
period (e.g. a month, a year). The values of an ordinary annuity can be calculated through the following:
Let:
         r = the yearly nominal interest rate.
         t = the number of years.
         m = the number of periods per year.
         i = the interest rate per period.
         n = the number of periods.
Note:


            n = tm
Also let:
            P = the principal (or present value).
            S = the future value of an annuity.
            R = the periodic payment in an annuity (the amortized payment).




Also:




Clearly, in the limit as n increases,



Thus, even an infinite series of finite payments (perpetuity) with a non-zero discount rate has a finite present value.

Proof
The next payment is to be paid in one period. Thus, the present value is computed to be:




.


We notice that the second term is a geometric progression of scale factor 1 and of common ratio                 . We can
write



                                             .
Finally, after simplifications, we obtain



                                                                                       .


                                                                                                                          1
Similarly, we can prove the formula for the future value. The payment made at the end of the last year would
accumulate no interest and the payment made at the end of the first year would accumulate interest for a total of
(n−1) years. Therefore,

                                                                                                             .
Hence:


                                   .




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Annuities and Annuity Payments
Many people add to a retirement account at regular intervals through payroll withholding. You may have a savings
goal. Solving for an annuity payment is one way to figure out how much you should be saving in order to meet your
goal. First, let's define our terms.
An annuity is a stream of payments made through time. A stream of equal payments at equal time intervals is a fixed
annuity. If those payments are made at the end of each time period (month, quarter, year, etc...) it is an Ordinary
Annuity. If the payments are due at the beginning of each period, it is an Annuity Due.
The payment amount, interest rate, and number of payments all contribute to the future value of the annuity. Any
annuity calculation has these four variables, and with any three you can find the fourth.
The formula and example below calculates the periodic payment required for an ordinary annuity, with a given
interest rate and number of payments (periods), in order to achieve a desired future value. This assumes a starting
value of zero.
This formula can also be manipulated to calculate the payments from an existing sum, which would find the payouts
that would draw the annuity down to an ending value of zero.

Formula to Calulate the Payment Amount of an Ordinary Annuity
                        n
PMT = FV(OA) / [((1 + i) - 1) / i ] where:
     FV(OA), or Future Value of Ordinary Annuity: the value of the annuity at time t=0
     PMT: Payment amount (value) of the individual payments in each period
     i: periodic interest rate that gets compounded for each period of time
        (periodic rate may be determined by dividing an annual rate by the number of periods in a year)
     n: number of peroids (same as the number of payments)

Ordinary Annuities
An annuity is a stream of payments made through time. A stream of equal payments at equal time intervals is a fixed
annuity. If those payments are made at the end of each time period (month, quarter, year, etc...) it is an Ordinary
Annuity. If the payments are due at the beginning of each period, it is an Annuity Due.
The payment amount, interest rate, and number of payments all contribute to the future value of the annuity. Any
annuity calculation has these four variables, and with any three you can find the fourth.
The formula and example below calculates the future value of an ordinary annuity from the interest rate, payment
amount, and number of payments (periods).
Calculate the Future Value of an Ordinary Annuity with a step by step example using your values for the periodic
interest rate, number of periods, and periodic payment amount.


Calcuating the Future Value of an Annuity Due
You can convert the valuation of an ordinary annuity to an annuity due by compounding the resulting future value for
one more period. (Since the payments for an annuity due are all due at the beginning of the period, instead of the end
like with an ordinary annuity, this makes it effectively one period sooner, but not one more payment.) The Future
Value of an Annuity Due is:
FV(AD) = FV(OA) • (1 + i)




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Annuities - a stream of payments
An annuity is defined as a stream of payments made over time. An annuity is typically an investment in which one
party puts money in with the promise of the other paying it back. There are several categories of annuities:
     Fixed Annuity
         The time between payments doesn't vary, the interest rate stays the same, and the amount of the payments
         is always the same. You know what you are getting with a fixed annuity.
     Variable Annuity
         If any of the time, interest rate, or payment amounts are not fixed, it becomes a variable annuity. Variable
         annuities sold as investments are subject to securities regulations.
     Equity Indexed Annuity
         A specialized variable annuity where interest/investment returns are indexed to equities (stock market).
Each of these categories of annuities can come in two flavors - ordinary, and annuity due:
     Ordinarily, annuity payments are due at the end of each period, so we call those an ordinary annuity.
     Sometimes payments are due at the start of each period and we call those an annuity due. Lease payments
         usually work like an annuity due.

Annuity phases: Accumulation and Distribution
We said an annuity is usually an investment where one party puts money in with the promise of the other paying it
back. The time when money is going into the annuity is the accumulation phase. The money comes back out during
the distribution phase.
Either phase could be a single payment, and there may or may not be much time between the last payment in and the
first payment out. If either phase is more than a single payment, an annuity may exist. (If each is a single payment,
there is no annuity, and you can calculate present value or future value of a lump sum.)
Figuring the present value or future value of a series of payments (annuity) can be done just like figuring PV or FV of a
single amount, but doing it again and again for each payment and adding them together. That works, but it is
cumbersome. Some math genius figured out a formula for doing it all at once for fixed annuities.

Present Value of Annuity with Fixed Payments for n periods (Ordinary or Annuity Due)
To calculate the present value for an ordinary fixed annuity (payment and interest rate don't change during life of
annuity), there are four variables. With any three we can solve for the fourth:
      PV(OA), or Present Value of Ordinary Annuity: the value of the annuity at time t=0
      PMT: Payment amount (value) of the individual payments in each period
      i: interest rate compounded for each period of time
      n: number of payment periods
                                    n
PV(OA) = (PMT/i) · [1 - (1 / (1 + i) )]
The difference between an ordinary annuity (above) and an annuity due, is the annuity due had the payment at the
beginning of each period, so it should get one more period of compounding than an ordinary annuity. All you have to
do to get the PV of an annuity due is multiply the above equation by (1 + i) to calculate the value for one period
sooner.
PV(AD) = PV(OA) · (1 + i)
As alternatives to these formulas, tables in the back of finance textbooks provide factors for calculating present and
future values of annuities and single amounts. Also, financial calculators and electronic spreadsheets include financial
functions and allow for entering the three variables you know and solving for the fourth.

Future Value of Annuity with Fixed Payments for n periods (Ordinary or Annuity Due)
To calculate the future value for an ordinary fixed annuity (payment and interest rate don't change during life of
annuity), there are four variables. With any three we can solve for the fourth:
     FV(OA), or Future Value of Ordinary Annuity: the value of the annuity at time t=n
     PMT: Payment amount (value) of the individual payments in each period
     i: interest rate compounded for each period of time
     n: number of payment periods
                        n
FV(OA) = PMT · [((1 + i) - 1) / i ]
The difference between an ordinary annuity (above) and an annuity due, is the annuity due had the payment at the
beginning of each period, so it should get one more period of compounding than an ordinary annuity. All you have to
do to get the FV of an annuity due is multiply the above equation by (1 + i) to calculate the value for one period
sooner.
FV(AD) = FV(OA) · (1 + i)


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As alternatives to these formulas, tables in the back of finance textbooks provide factors for calculating present and
future values of annuities and single amounts. Also, financial calculators and electronic spreadsheets include financial
functions and allow for entering the three variables you know and solving for the fourth.

Annuities - An example homework problem
On Marion's 35th birthday, her insurance company told her she is expected to live until age 85. She wants to retire at
age 60. Many of her expenses will be eliminated by then, so she estimates she will only need 15,000 per year to live
comfortably.
Marion has a family history of disease, so she plans to have home health care starting at age 70 which will cost 45,000
per year.
She doesn't want to outlive her income, so she allows another 3 years of life beyond the actuary's estimate.
She wants 40,000 to be left to cover her final expenses, including cremation.
Long term interest rates suggest that her opportunity cost of cash approximates the 20-year treasury bond rate of 8%
per annum.
She has not started saving yet, but wants to start right away.
    1. How much money does she need to have when she retires to achieve her goals?
    2. How much money does she need to save each year from now until the time she retires in order to have
         enough money when she retires to achieve her goals?

Annuities - laying out the example homework problem
This problem tests your understanding of present and future values of sums and annuities, not your ability to do
financial planning, since it ignores things like inflation.
With present and future value problems we need to understand the stages of accumulation and distribution. (You may
want to draw a timeline to make the problem easier to visualize.) This one starts with an accumulation phase starting
now and continuing for 25 years until age 60. Then the distribution phase kicks in, with distributions continuing for 28
years, with an increase along the way, and then a final distribution.




Marion has given us several goals to include in the solution. It is possible to solve for each part separately and have
annual savings goals for each. Instead, we are going to first determine how much Marion will need to have
accumulated at retirement (question 1), and then calculate the accumulation phase only once, to meet that (question
2).
To calculate the amounts each part will require be available at Marion's retirement, I set up a separate section for:
     ordinary retirement income
     home health care
     final expenses
Then we come back together, combine these requirements, and figure the savings requirement to solve the two
problems.

Present Value of an Annuity: solving for retirement income
Marion says she needs 15,000 annually from age 60 through age 88. That will be 28 years of 15,000 payments and we
will use her 8% interest factor.
The Excel formula for present value of an annuity looks like this:
=PV(0.08,28,-15000)
=165,766.18 required at Age 60
We will add this sum to her other requirements later to determine her total required savings.

Present Value of an Annuity: solving for home health care
Marion needs 45,000 additional annually from age 70 through age 88. We will first solve for how much Marion will
need when that distribution starts.
This is a "Present Value of an Annuity" problem. There will be 18 periods, 8% interest, and 45,000 annual payments.
The Excel formula looks like this:
=PV(0.08,18,-45000) and it resolves to

                                                                                                                      5
=421,734.92 required at Age 70 (when this distribution begins)
To find the PV of that amount at Age 60, we discount it back 10 years by taking the Present Value of a sum. Again, I
use Excel, and the formula looks like this:
=PV(0.08,10,0,-421734.92)
=195,344.87 required at Age 60, which we will combine with the other values later.

Present Value of an Annuity: solving for final expenses
Marion has a fixed amount (40,000 for final expenses) she needs at age 88.
This is a straightforward "present value of a sum" problem. I am using the =PV financial function in an Excel
spreadsheet to calculate the the value needed at retirement (age 60, which is 28 years before she expects to die.)
I use her 8% interest rate, 0 payment amounts, start with a present value of 0, since she hasn't started saving for this
yet, and the future value of 40000 since that is the goal.
The format in Excel looks like this:
=PV(i,n,Payment,FV) so my input is:
=PV(0.08,28,0,-40,000) which results in:
=4,646.55 required at Age 60, which we will combine with the other values in the next section.

Present Value of an Annuity: solving for annuity value
If we combine all of the required amounts at age 60, we get:
 4,646.55 for final expenses
 165,766.18 for ordinary retirement income
 195,344.87 for home health care
+__________
 365,747.60 accumulation required at age 60

Present Value of an Annuity: solving for annuity payment
Now we need to know the annual saving required to accumulate 365,747.60. This time we know the future value and
are solving for the Payment.
There are 25 years to save, and we still use the 8% interest rate. The Excel formula for solving for a payment is:
=PMT(int,n,pv,fv).
We will use 0 as the present value since she hasn't started saving yet. Here is my entry:
=PMT(0.08,25,0,365747.60)                                           which                                  equals:
=-5,002.98
She needs to save 5,002.98 each year until she retires.

Annuity Example - recap of annuity payments and value
We just figured out that Marion needs to save $5,002.98 per year for 25 years. Thus, she is paying $125,074.50 for her
retirement plan.
Since she will receive 8% interest all along the way (good luck with that, Marion) she will really have $365,747.60 set
aside when she retires (at least that is the goal we calculated). That means she will make $240,673.10 in interest by
the time she retires. That is more than she will put in herself!
Let's see how much she is going to receive from the $365,747.60 she is saving. She expects to receive retirement
income of $15,000 per year plus another $45,000 per year for home health care for 18 years, and still have $40,000
left for final expenses.
       $15,000 per year x 28 years = $420,000
       $45,000 per year x 18 years = $810,000
       And she willl provide $40,000 for final expenses.
She will get $1,270,000 for investing $125,000.
If she had more years to invest, it would work even better. I love compounding. Are you ready to start your
retirement fund now?




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Amortization calculator

An amortization calculator is used to determine the periodic payment amount due on a loan (typically a mortgage),
based on the amortization process.
The amortization repayment model factors varying amounts of both interest and principal into every installment,
though the total amount of each payment is the same.
An amortization schedule calculator is often used to adjust the loan amount until the monthly payments will fit
comfortably into budget, and can vary the interest rate to see the difference a better rate might make in the kind of
home or car one can afford. An amortization calculator can also reveal the exact dollar amount that goes towards
interest and the exact dollar amount that goes towards principal out of each individual payment. The amortization
schedule is a table delineating these figures across the duration of the loan in chronological order.
The formula
The calculation used to arrive at the periodic payment amount assumes that the first payment is not due on the first
day of the loan, but rather one full payment period into the loan.
While normally used to solve for A, (the payment, given the terms) it can be used to solve for any single variable in the
equation provided that all other variables are known. One can rearrange the formula to solve for any one term, except
for i, for which one can use a root-finding algorithm.
The annuity formula is:



Where:
        A = periodic payment amount
        P = amount of principal, net of initial payments, meaning "subtract any down-payments"
        i = periodic interest rate
        n = total number of payments
         For              a          30-year             loan            with          monthly                payments,

Note that the interest rate is commonly referred to as an annual percent (e.g. 8% APR), but in the above formula,
since the payments are monthly, the rate i must be in terms of a monthly percent. Converting an annual interest rate
(that is to say, annual percentage yield or APY) to the monthly rate is not as simple as dividing by 12, see the formula
and discussion in APR. However if the rate is stated in terms of "APR" and not "annual interest rate", then dividing by
12 is an appropriate means of determining the monthly interest rate.




                                                                                                                       7
Amortization schedule

An amortization schedule is a table detailing each periodic payment on an amortizing loan (typically a mortgage), as
generated by an amortization calculator.
Amortization refers at the process of paying off a debt (often from a loan or mortgage) over time through regular
payments. A portion of each payment is for interest while the remaining amount is applied towards the principal
balance. The percentage of interest versus principal in each payment is determined in an amortization schedule.
While a portion of every payment is applied towards both the interest and the principal balance of the loan, the exact
amount applied to principal each time varies (with the remainder going to interest). An amortization schedule reveals
the specific monetary amount put towards interest, as well as the specific amount put towards the principal balance,
with each payment. Initially, a large portion of each payment is devoted to interest. As the loan matures, larger
portions go towards paying down the principal.
Many kinds of amortization exist, including:
      Straight line (linear)
      Declining balance
      Annuity
      Bullet (all at once)
      Increasing balance (negative amortization)
Amortization schedules run in chronological order. The first payment is assumed to take place one full payment period
after the loan was taken out, not on the first day (the amortization date) of the loan. The last payment completely
pays off the remainder of the loan. Often, the last payment will be a slightly different amount than all earlier
payments.
In addition to breaking down each payment into interest and principal portions, an amortization schedule also reveals
interest-paid-to-date, principal-paid-to-date, and the remaining principal balance on each payment date.

Example amortization schedule
(To run your own numbers, try an amortization calculator.)
This amortization schedule is based on the following assumptions:
Note: Rounding errors mean that, depending how the lender accumulates these errors, the blended payment (principal
+ interest) may vary slightly some months to keep these errors from accumulating; or, the accumulated errors are
adjusted for at the end of each year, or at the final loan payment.
There are a few crucial points worth noting when mortgaging a home with an amortized loan. First, there is
substantial disparate allocation of the monthly payments toward the interest, especially during the first 18 years of
the mortgage. In the example above, Payment 1 allocates about 80-90% of the total payment towards interest and
only $67.09 (or 10-20%) toward the Principal balance. The exact percentage allocated towards payment of the
principal depends on the interest rate. Not until payment 257 or 21 years into the loan does the payment allocation
towards principal and interest even out and subsequently tip the majority of the monthly payment toward Principal
balance pay down.
Second, understanding the above statement, the repetitive refinancing of an amortized mortgage loan, even with
decreasing interest rates and decreasing Principal balance, can cause the borrower to pay over 500% of the value of
the original loan amount. 'Re-amortization' or restarting the amortization schedule via a refinance causes the entire
schedule to restart: the new loan will be 30 years from the refinance date, and initial payments on this loan will again
be largely interest, not principal. If the rate is the same, say 8%, then the interest/principal allocation will be the same
as at the start of the original loan (say, 90/10). This economically unfavorable situation is often mitigated by the
apparent decrease in monthly payment and interest rate of a refinance, when in fact the borrower is increasing the
total cost of the property. This fact is often (understandably) overlooked by borrowers.
Third, the payment on an amortized mortgage loan remains the same for the entire loan term, regardless of Principal
balance owed. For example, the payment on the above scenario will remain $733.76 regardless if the Principal balance
is $100,000 or $50,000. Paying down large chunks of the Principal balance in no way affects the monthly payment, it
simply reduces the term of the loan and reduces the amount of interest that can be charged by the lender resulting in
a quicker payoff. To avoid these caveats of an amortizing mortgage loan many borrowers are choosing an Interest-
only loan to satisfy their mortgage financing needs. Interest-only loans have their caveats as well which must be
understood before choosing the mortgage payment term that is right for the individual borrower.




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Creating an Amortization Schedule
In order to create an amortization schedule, you will need to use the following formula to calculate a periodic
payment, A:



Where P is the principal, i is the periodic interest rate, and n is the number of periods (payments) in which the
principal is to be paid. For monthly payments, the periodic interest rate i is the annual interest rate divided by 12
(number of periods per year), and the number of periods n is the number of years times 12 (again, number of periods
per year).
A warning for Canadian applications: the Canada Interest Act requires that mortgages (as distinct from regular loans)
have interest calculated "annually or semi-annually, not in advance". This means that the banks and financial
institutions have to do some computational gymnastics to compute the effective interest rate. Since semi-annual
calculation is more favorable, they will use
                  (1 / 6)
i = (1 + R / 200)         −1
which needs to be computed quite carefully to avoid rounding and cancellation errors. Weekly and bi-weekly
payments are even more troubling, because the number of weeks in a "half-year" depends on the number of days in
the months. That is, there is no specified rule for how the effective weekly rate should be calculated.
Once you determine the fixed monthly payment using the formula above, you can determine the allocation of each
payment between interest and principal. The amount of principal paid each month is the difference between the
monthly payment amount and the amount of interest due on the balance for that month.
First, determine the amount of interest due for a payment by multiplying the periodic interest rate by the outstanding
principal (for monthly payments, divide the annual rate by 12 to get the periodic rate). For the first payment, the
outstanding principal is the full loan amount. Second, determine the amount of principal paid by subtracting the
interest due from the total monthly payment amount. Finally, subtract the amount of principal paid from the
outstanding loan amount to determine the new principal balance. Repeat the calculation for each following period
(month) using the previous month's ending balance as the next month's outstanding principal in the calculation of
interest due.
As you get near the end of the loan, the loan balance (principal) gets smaller and less interest is due. Since the
monthly payment amount stays the same (at least for a standard 15 or 30 year mortgage) and the interest due
decreases, you apply an increasingly larger amount of each successive payment towards the principal. For your last
few payments, you owe very little interest on the small remaining balance, so you pay off the remaining principal very
quickly.
As a simple example, let's say that we're lending $100 at a 10% a year to be paid back in five years using annual
payments. The payments would be:



We can now create a table detailing the principal, and interest.
Year Outstanding Balance Payment Interest Paid Principal Paid
1    $100                   $26.38     $10.00        $16.38
2    $83.62                 $26.38     $8.36         $18.02
3    $65.60                 $26.38     $6.56         $19.82
4    $45.78                 $26.38     $4.58         $21.80
5    $23.98                 $26.38     $2.40         $23.98

                      Payment           Principal              Interest           Balance
                                                                               $100.00
1                  $26.38               $16.38                $10.00            $83.62
2                  $26.38               $18.02                 $8.36            $65.60
3                  $26.38               $19.82                 $6.56            $45.78
4                  $26.38               $21.80                 $4.58            $23.98
5                  $26.38               $23.98                 $2.40             $0.00
*Tot              $131.90              $100.00                $31.90

As you can see, the amount of interest due each year is 10% of the balance. The amount paid towards the principal is
the difference between the fixed annual payment (determined by the formula) and the annual interest due.
                                                                                                                    9
Outstanding Loan Balance Calculation
The outstanding loan balance at any given time during the term of a loan can be calculated by finding the present
value of the remaining payments at the given interest rate. This amount will consist of principal only.
Example of O/S Loan Balance Calculation:
Loan Amount= $100,000 Term= 20 years Interest Rate = 7% Amortization is monthly
Question: What is the loan balance at the end of year seven?
First, calculate the monthly payments by using the loan amount ($100,000) as present value, term as 240 (20 years x
12 months/year), Interest as .583333% (7%/12 months). This will give you a monthly payment of $775.30. The Present
Value of an Annuity formula should be used here to solve for monthly payment.
Next, in order to find the outstanding loan balance you will need to find the present value of the remaining payments.
Use the monthly payment of $775.30 as the payment function, the term will be 156 ((20-7)x12), and .583333% as the
rate. This will give you an outstanding loan balance of $79,268.02. Again, the Present Value of an Annuity formula
should be used.
This means that at the end of year seven the loan can be paid off in full for the amount of $79,268.02. Typically
mortgage lenders will have a balloon payment clause in the contract that will charge a fee for early payment. This is
because, the lender will not get the same yield if loan balance is not held to maturity.




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