/* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/
/* [ Created with wxMaxima version 19.05.7 ] */
/* [wxMaxima: title   start ]
financial.wxm: inside financial.mac functions
   [wxMaxima: title   end   ] */


/* [wxMaxima: comment start ]
Ted Woollett, Oct 20, 2020.  
Maxima by Example, Ch. 15, Financial Mathematics

financial.mac is a rewrite of the current contributed package finance.mac.

financial.mac includes major changes in notation, major code improvements,  
and brief examples of use, and also extra utility functions. 

This worksheet,  financial.wxm,  is primarily a pedagogical approach to both 
Maxima code development and to the meaning of current finance jargon, aiming 
to "derive" (or give references to easily available derivations, or at least make 
plausible via numerical comparisons) the  Maxima functions present in financial.mac.

I have placed financial.mac in my work folder: c:\work5\, declared in my maxima-init.mac file;
  maxima-init.mac is a file in the folder:

 c:\Users\Wooll\maxima\ 

which contains the lines:

maxima_userdir: "c:/work5" $     
maxima_tempdir : "c:/work5"$
file_search_maxima : append(["c:/work5/###.{mac,mc}"],file_search_maxima )$
file_search_lisp : append(["c:/work5/###.lisp"],file_search_lisp )$

(among other lines).

The current Maxima package finance.mac is in .../share/finance/ and was written by 
Nicolas Guarin Zapata.

On my Windows 10 computer the location is:

c:\maxima-5.43.0\share\maxima\5.43.0\share\finance\

   [wxMaxima: comment end   ] */


/* [wxMaxima: comment start ]
This shows the default working folder Maxima looks in first.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
maxima_userdir;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
display2d;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: section start ]
Rounding floats to a specified number of decimal places
   [wxMaxima: section end   ] */


/* [wxMaxima: comment start ]
We can round floating point numbers to a specified number of decimal places as follows, using 
the function rnddp(expr,dp) found at the webpage:

https://stackoverflow.com/questions/29322006/how-to-approximate-a-number-to-a-n-number-of-decimal-places, 

found via the google search: maxima round decimal places robert dodier

dp stands for "decimal places".
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
rnddp(expr,dp):= if floatnump (expr) then float((round(expr*10^dp)/10^dp)) else expr$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Here are examples of use of rnddp:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
rnddp (3456.6789,2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
rnddp (3456.6789,4);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
I have included the function rnddp in my file financial.mac.

Another function available is rnd_listdp(alist,dp) which rounds a
list of floats.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
rnd_list_dp(alist,dp) := map (lambda([xx], rnddp(xx, dp)), alist)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
here is an example of this second function:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
rnd_list_dp(  [12.3456, 23.4567, 34.5678], 2 );
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
A third available function is rnd_matrix_dp(amatrix,dp)
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
rnd_matrix_dp(%Amatrix,dp) := matrixmap (lambda([xx], rnddp(xx, dp) ), %Amatrix)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Here is an example of the use of rnd_matrix_dp:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
ma : matrix([a, b, c],[1.2345,2.3456,3.4567],[4.5678,5.6789,6.7891]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
rnd_matrix_dp(ma,2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Here is a way to print the elements of a matrix
as a (crude) ordinary table: (also defined in financial.mac)
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
print_matrix (AA) := map (lambda([xx], apply ('print,xx)), args(AA))$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: section start ]
General References
   [wxMaxima: section end   ] */


/* [wxMaxima: comment start ]
Long listing of finance formulas:
https://financeformulas.net/Alphabetical_A-C.html


88 videos financial math tutorial on YouTube:
https://www.youtube.com/watch?v=vaIqxP437bs&list=PLIYnk9FMYcksTDanxOE6LOoMhJvpz9SN3
Module 2, Sec 1, P1 is the start of a general discussion of annuities (this is video 25 of the above set):
https://www.youtube.com/watch?v=vIQKtXXGqhc&list=PLIYnk9FMYcksTDanxOE6LOoMhJvpz9SN3&index=25

A good brief collection of definitions and formulas is:
https://en.wikibooks.org/wiki/Financial_Math_FM/Formulas

Project math pdf:
https://www.projectmaths.ie/documents/modulars/4/FinancialMathsExtraQuestions.pdf

A good introduction to the time value of money and discounted cash flows is
http://pages.stern.nyu.edu/~adamodar/New_Home_Page/PVPrimer/pvprimer.htm

A 2016 course at the Univ. of Fla. is at:
http://users.stat.ufl.edu/~rrandles/sta4183/4183lectures/index.html

Basic annuities are covered in:
http://users.stat.ufl.edu/~rrandles/sta4183/4183lectures/chapter03/chapter03R.pdf

Examples of annuities including payments in both arithmetic and geometric
progression can be found in
http://users.stat.ufl.edu/~rrandles/sta4183/4183lectures/chapter04/chapter04R.pdf

Amortization schedules are covered in:
http://users.stat.ufl.edu/~rrandles/sta4183/4183lectures/chapter05/chapter05R.pdf



A YouTube module on arithmetic annuities can be found at:
https://www.youtube.com/watch?v=WvSA2xBfIg8
This is Module 2, Sec. 6, P1 by Prof. Stephen Paris.

Chapter 2, Annuities, from the book Financial Mathematics for Actuaries
(First Ed.) by Wai-Sum Chan and Yiu-Kuen Tse can be found at:
http://www.mysmu.edu/faculty/yktse/FMA/S_FMA_2.pdf

Discussion of "net present value":
https://www.investopedia.com/terms/n/npv.asp
https://www.accountingtools.com/articles/how-to-calculate-npv.html

   [wxMaxima: comment end   ] */


/* [wxMaxima: section start ]
Time Value of Money
   [wxMaxima: section end   ] */


/* [wxMaxima: subsect start ]
Valuation of a future payment or a past payment.
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
We begin with the article: https://en.wikipedia.org/wiki/Time_value_of_money,
and we quote (with some editing) from sections of that article.
   [wxMaxima: comment end   ] */


/* [wxMaxima: comment start ]
The time value of money is the idea that there is greater benefit to receiving a sum of money
now rather than an identical sum later. It is founded on time preference. The time value of money 
is the reason why interest is paid or earned: interest, whether it is on a bank deposit or debt, 
compensates the depositor or lender for the time value of money. It also underlies investment. 
Investors are willing to forgo spending their money now only if they expect a favorable return 
on their investment in the future, such that the increased value to be available later is sufficiently 
high to offset the preference to spending money now.
   [wxMaxima: comment end   ] */


/* [wxMaxima: comment start ]
Time value of money problems involve the net value of cash flows at different points in time. 
In a typical case, the variables might be: a balance (the real or nominal value of a debt or a 
financial asset in terms of monetary units), a periodic rate of interest, the number of periods,
and a series of cash flows. (In the case of a debt, cash flows are payments against principal 
and interest; in the case of a financial asset, these are contributions to or withdrawals from 
the balance.) More generally, the cash flows may not be periodic but may be specified individually. 
Any of these variables may be the independent variable (the sought-for answer) in a given problem. 
For example, one may know that: the interest is 0.5% per period (per month, say); the number of 
periods is 60 (months); the initial balance (of the debt, in this case) is 25,000 units; and the final 
balance is 0 units. The unknown variable may be the monthly payment that the borrower must pay.
For example, $100 invested for one year, earning 5% interest, will be worth $105 after one year; 
(100 * 0.05 = 5).  Therefore, $100 paid now and $105 paid exactly one year later both have the same 
value to a recipient who expects 5% interest (per year) assuming that inflation would be zero percent. 
That is, $100 invested for one year at 5% interest has a future value of $105 under the assumption 
that inflation would be zero percent.
This principle allows for the valuation of a likely stream of income in the future, in such a way that 
annual incomes are discounted and then added together, thus providing a lump-sum "present value" 
of the entire income stream; all of the standard calculations for time value of money derive from the 
most basic algebraic expression for the "present value" PV of a future sum FV, "discounted" to the present 
by an amount equal to the time value of money. 
For example, the "future value" sum FV to be received  in one year is "discounted" at the rate of interest 
i to give the "present value" sum PV:
     PV  = FV / ( 1 + i ),   Example:  $105/(1 + 0.05) = $100
     "future value" is "discounted" to "present value" ==>  PV < FV
    and of course we can rearrange eqn. to write FV = PV * (1 + i) > PV.
   [wxMaxima: comment end   ] */


/* [wxMaxima: comment start ]
In a later section we define FVA and PVA as the future value (accumulated value) (t = n)
and the present value (t = 0) of a stream of payments for an ordinary annuity immediate.
   [wxMaxima: comment end   ] */


/* [wxMaxima: comment start ]
An important note is that the interest rate i is the interest rate for the relevant period. For an "annuity"
(see below) that makes one payment per year, i will be the annual interest rate. For an income or payment stream 
with a different payment schedule, the interest rate must be converted into the relevant periodic interest 
rate. For example, a monthly rate for a mortgage with monthly payments requires that the (annual) interest rate 
be divided by 12.
   [wxMaxima: comment end   ] */


/* [wxMaxima: comment start ]
Some standard calculations based on the time value of money are:
1. Present value: The current worth of a future sum of money or stream of cash flows, given a specified rate of return. 
        Future cash flows are "discounted" at the discount rate; the higher the discount rate, the lower the present value 
        of the future cash flows. Determining the appropriate discount rate is the key to valuing future cash flows
        properly,   whether they be earnings or obligations.
2. Present value of an annuity (PVA): An "annuity" is a series of equal payments or receipts that occur at evenly s
         paced intervals. Leases and rental payments are examples. The payments or receipts occur at the end of 
         each period for an "ordinary  annuity immediate" while they occur at the beginning of each period for an 
        "ordinary annuity due".
 
3. Present value of a perpetuity is an infinite and constant stream of identical cash flows.
4. Future value: The value of an asset or cash at a specified date in the future, based on the value of that asset 
         in the present.
5. Future value of an annuity (FVA): The future value of a stream of payments (annuity), assuming the payments
          are invested at a given rate of interest. 
   [wxMaxima: comment end   ] */


/* [wxMaxima: subsubsect start ]
pv (rate, future-payment, num)
   [wxMaxima: subsubsect end   ] */


/* [wxMaxima: comment start ]
The present value pv  of a payment made num compounding periods in the future
in a financial environment in which investment oportunities are available which 
yield a rate of interest "rate" is gotten by "discounting" the future payment by
num factors of the "discount factor" v = 1/(1+rate).
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
pv(rate,future_payment,num):=future_payment/(1+rate)^num$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
examples assuming 5% interest rate:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
pv(0.05,105,1);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
pv(0.05,105,2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
pv(0.05,110,2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: subsubsect start ]
fv (rate, past_payment, num)
   [wxMaxima: subsubsect end   ] */


/* [wxMaxima: comment start ]
The future value fv  of a payment made num compounding periods in the past
in a financial environment in which investment oportunities are available which 
yield a rate of interest "rate" is gotten by compounding the past payment by
num factors of the compounding factor (1+rate).
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
fv(rate, past_payment, num) := past_payment*(1+rate)^num$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Examples with 5% interest rate:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
fv(0.05,100,1);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
fv(0.05,100,2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
pv(0.05,110.25,2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: subsect start ]
Net Present Value NPV Concept
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
A reference is
https://www.investopedia.com/terms/n/npv.asp
We quote (with some serious editing) some of their introduction here:
Net present value (NPV) is the difference between the present value of cash inflows 
and the present value of cash outflows over a period of time. NPV is used in capital 
budgeting and investment planning to analyze the profitability of a projected investment or project.
NPV = TVECF − TVIC
where:
TVECF=Today’s value of the expected cash flows
TVIC=Today’s value of invested cash
​ A positive net present value indicates that the projected earnings generated by a 
project or investment - in present dollars - exceeds the anticipated costs, also in 
present dollars. It is assumed that an investment with a positive NPV will be profitable, 
and an investment with a negative NPV will result in a net loss. This concept is the 
basis for the Net Present Value Rule, which dictates that only investments with 
positive NPV values should be considered.
Here is a fairly simple example:
Imagine a company can invest in equipment that will cost $1,000,000 and is expected 
to generate $25,000 a month in revenue for five years. The company has the capital 
available for the equipment and could alternatively invest it in the stock market for 
an expected return of 8% per year. The managers feel that buying the equipment or 
investing in the stock market are similar risks.
Step One: NPV of the Initial Investment
Because the equipment is paid for up front, this is the first cash flow included in the 
calculation. There is no elapsed time that needs to be accounted for so today’s outflow 
of $1,000,000 doesn’t need to be discounted.
Identify the number of periods (n)
The equipment is expected to generate a monthly cash flow and last for five years, 
which means there will be 60 cash flows and 60 periods included in the calculation.
Identify the periodic discount rate (i)
The alternative investment is expected to pay 8% per year. However, because the 
equipment generates a monthly stream of cash flows, the annual discount rate needs 
to be turned into a periodic or monthly rate. Using the following formula, we find that 
the periodic discount rate is 0.64%.
monthly (periodic) discount rate = (1 + 0.08)^(1/12) -1  =  0.006434 = 0.64%
Another way to write this is (1 + 0.08)^(1/12)  = ( 1 + 0.006434 ) = 1/v
v = 1 / (1 +  0.006434) = 0.9936 = discount factor per month to use in discounting
future cash flows to a present value.
Step Two: NPV of Future Cash Flows
Assume the monthly cash flows are earned at the end of the month, with the first payment 
arriving exactly one month after the equipment has been purchased. This is a future 
payment, so it needs to be adjusted for the time value of money.  To illustrate the 
concept, the discounted values of the first five payments are displayed in the table below.
   [wxMaxima: comment end   ] */


/* [wxMaxima: comment start ]
We want (1+rate)^12 = (1+0.08)  so 1+rate = (1+0.08)^(1/12), rate = (1+0.08)^(1/12) - 1
and v = 1/(1+rate). Let Maxima do 16 digit arithmetic as usual here.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
(1 + 0.08)^(1/12) -1;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
1 + %;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
v : 1/%;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
CF : 25000;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Discounted values of first five payments:
The first payment at the *end* of the first month (m=1) is discounted to CF/(1+rate) = CF*v
We only round after all the arithmetic has been done.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
for m thru 5 do 
   print ( m,"    ",  rnddp( CF*v^m, 2))$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
"Today's value of the expected cash flows" is TVECF, which requires a summation over
all 60 months in the 5 years.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
TVECF :  rnddp( sum (CF*v^m, m, 1, 60), 2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
The discounted value of the expected cash flows is $1,241,122.58.
Today's value of invested cash is TVIC = $1,000,000.00
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
TVIC : 10^6;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
We can now compute the NPV, using the definition:
NPV = TVECF − TVIC
where:
TVECF=Today’s value of the expected cash flows
TVIC=Today’s value of invested cash
We round after the subtraction.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
NPV : rnddp (TVECF - TVIC, 2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
The NPV is $241,122.58.
In this case, the NPV is positive; the equipment should be purchased. If the present value 
of these cash flows had been negative because the discount rate was larger, or the net cash 
flows were smaller, the investment should have been avoided.
   [wxMaxima: comment end   ] */


/* [wxMaxima: subsect start ]
npv function
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
We adapt Zapata's npv function with some coding improvements.
npv (rate, flowValues) assumes flowValues is a list of cash flows
which includes as the first element the cash flow at t = 0.

npv then returns the net present value at t = 0 of this stream of
cash flows, with the rest of the cash flows occurring at the ends
of a compounding period.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
npv (%rate, %flowValues) := 
sum ( pv (%rate, %flowValues[k], k-1), k, 1,length (%flowValues) )$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
We first apply this npv function to a simple 6 element list which we check
by hand.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
cfL : [0, 100, 500, 323, 124, 300];
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
npv (0.25, cfL );
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
calculate this by hand as a check
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
v : 1/(1 + 0.25);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
sum (cfL [k]*v^(k-1), k, 2, 6);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Now apply the  npv function to the 60 month investment cash flow scenario above.
We need to make the list used with npv have as its first element -10^6.
We first use makelist to create a 60 element list, each element being a
positive cash flow of $25,000.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
cfL : makelist (25000, j, 1, 60)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
We next use append ( list1, list2 ) to get a 61 element list.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
cfL : append ([ -10^6 ], cfL )$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
length (cfL);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
We can use the rest function to look at the first few elements.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
rest (cfL, -58);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
we require (1 + monthly-rate)^(12) = (1 + yearly-rate) to solve for the monthly
rate using 16 digit arithmetic.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
rate : (1 + 0.08)^(1/12) -1;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
npv (rate, cfL);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
rnddp (%, 2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
which agrees with our calculation above: $241,122.58
   [wxMaxima: comment end   ] */


/* [wxMaxima: subsect start ]
irr ( rate, cash_flow_list, IO)  Internal Rate of Return  IRR
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
References:

https://www.investopedia.com/terms/i/irr.asp

from which we quote:
   [wxMaxima: comment end   ] */


/* [wxMaxima: comment start ]
The "internal rate of return" (IRR) is a metric used in financial analysis to estimate the profitability
 of potential investments. The internal rate of return is a discount rate that makes the net 
present value (NPV) of all cash flows equal to zero in a discounted cash flow analysis. 
IRR calculations rely on the same formula as NPV does.

KEY TAKEAWAYS

IRR is the annual rate of growth an investment is expected to generate.
IRR is calculated using the same concept as NPV, except it sets the NPV equal to zero.
IRR is ideal for analyzing capital budgeting projects to understand and compare potential 
         rates of annual return over time.

The value of IRR comes from using the formula

0 = NPV = sum (C[k] / (1 + IRR)^k, k, 1, n) - C[0]

where C[k] is the net cash inflow at the end of the compound
period k, and C[0] is the total initial (t=0) investment costs.

Generally speaking, the higher an internal rate of return, the more 
desirable an investment is to undertake. IRR is uniform for investments 
of varying types and, as such, IRR can be used to rank multiple 
prospective investments or projects on a relatively even basis. 
In general, when comparing investment options, the investment 
with the highest IRR would probably be considered the best.
   [wxMaxima: comment end   ] */


/* [wxMaxima: comment start ]
Our Maxima function irr (flowValues, IO) returns the internal rate of return (IRR) 
based on a cash flow list of signed values, with the first element of the list
representing t = 0, and also based on a possibly separate accounting of an 
initial investment (IO)  made at t = 0.

   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
irr(flowValues,I0):= block([rr, asum, realonly: true],
	asum : npv (rr, flowValues) - I0,
   algsys ([asum = 0], [rr]),
   rhs ( %%[1][1]))$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
In this first example, we assume an initial (t = 0) investment of
$5,000, and we choose to place the number (-5000) into the
first slot (the t=0 slot) of the flowValues list (the "cash values
list" and set the second argument (IO) equal to zero. In a second
example below we set IO = 5000, and set the first element of
the flowValues list equal to zero, and get the same internal rate
of return (IRR).
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
irr ([-5000, 0, 800, 1300, 1500, 2000], 0);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
We can check the internal rate of return (3%) returned by irr by calling npv
and expect to get a number numerically close to zero.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
npv (%, [-5000, 0, 800, 1300, 1500, 2000]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
rnddp(%, 2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
so indeed the 3% value of the IRR says that this cash flow list has zero net present value.

   [wxMaxima: comment end   ] */


/* [wxMaxima: comment start ]
Instead of including the $5,000 t = 0 investment
as a negative value in the first element of the cash
flow list, we can substitute 0 as the first list element
and substitute 5000 for the the argument  IO.

Since the irr formula calls our npv function, and
our npv function assumes the first element of the 
cash flow list occurs at t = 0, we need to either have
0 or some other number present to represent t=0.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
irr ([0, 0, 800, 1300, 1500, 2000], 5000);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
An example from our reference:

Here is a simple example of an IRR analysis with cash flows that are known and annually 
periodic (one year apart). Assume a company is assessing the profitability of Project X. 
Project X requires $250,000 in funding and is expected to generate $100,000 in after-tax 
cash flows the first year and grow by $50,000 for each of the next four years.

The initial investment is always negative because it represents an outflow. 
Each subsequent cash flow could be positive or negative, depending on the 
estimates of what the project delivers in the future. In this case, the IRR 
is 56.72%, which is quite high.

Keep in mind that the IRR is not the actual dollar value of the project. 
It is the annual return that makes the net present value equal to zero.

   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
irr ([-250000, 100000, 150000, 200000, 250000, 300000], 0);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
npv (%, [-250000, 100000, 150000, 200000, 250000, 300000]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
rnddp (%,2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: subsect start ]
benefit_cost (rate, CFin, CFout)   Benefit Cost Ratio
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
The function benefit_cost (rate, CFin, CFout) calculates  the 
Benefit/Cost ratio where  "Benefit" is the Net Present Value (NPV)
of the cash flow in, and "Cost" is the Net Present Value (NPV) of the cash
flow out.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
benefit_cost (rate, CFin, CFout) := 
	if length (CFin) # length (CFout)
   	then error("CFin & CFout lists must have the same length")
	   else  rnddp ( npv (rate, CFin) / npv (rate, CFout), 2)$

/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Here is an example:

CFin =   cash flow in list = [0,300,500,150] where first list element applies to t = 0 (k = 0).

 CFout =   cash flow out list = [100,320,0,180] where first list element applies to t = 0.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
benefit_cost(0.24,[0,300,500,150],[100,320,0,180]);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: subsect start ]
plot_CF (cash_flow_list, lineWidth)
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
If you load financial.mac,  the draw package will be automatically
loaded and you should never reload draw in a given session. Since
we have not loaded financial.mac, we must now load the draw package.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
load(draw)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
plot_CF  (cash_flow_list,  lineWidth) makes a histogram with blue vectors
on the top side for positive entries, and red vectors on the bottom side
for negative vectors. 

cash_flow_list can have two forms:

1.  if the list looks like
               [100, -50, 200, 300, -100]
     for example, then a positive cash flow is assumed to occur at t = 0 (k = 0), 
        a negative cash flow occurs at k = 1 (the end of the first compounding period),
        and so on, with a negative cash flow (-100) at k = 4 (the end of the fourth
        compounding period).

2.  If, instead, the list looks like
              [ [0.5,100], [1,-50], [2,200], [2.5,300], [4,-100] ]
         for example, then a positive cash flow (100) occurs at k = 0.5
         (halfway thru the first compounding period), a negative cash
         flow (-50) occurs at k = 1 (the end of the first compounding period),
         and so on, with cash flow (-100) at k = 4 (the end of the 4th
         compounding period).

 The adjustable width of the vectors is lineWidth, a positive integer, say 4 - 6.
 
This fancy function (a modification of the plot function in Zapata's finance.mac) 
makes use of draw2d.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
plot_CF(CFL, lineWidth) :=
block([vectors, options], 
    if listp (CFL[1]) then  /* assume each element of CFL is a two element list */
       vectors: flatten(makelist([color = if CFL[k][2] < 0 then 'red else
                   'blue, vector ( [CFL[k][1], 0], [0, CFL[k][2]] ) ], k,1, length(CFL)))
    else
       vectors: flatten(makelist([color = if CFL[k]<0 then 'red else
                   'blue, vector ( [k-1, 0], [0, CFL[k]] ) ], k,1, length(CFL))),
    
    options: [head_length=0.05,
           line_width = lineWidth,
            xaxis=true,
            yaxis=false,
            xrange=[-1,length(CFL) ]],
    apply(draw2d,append(options,vectors)))$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Here we try this function out with both types of lists.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
myList : [100, -50, 200, 300, -100];
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
plot_CF (myList, 6)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
myList2 : [ [0.5,100],[1,-50],[2,200],[2.5,300],[4,-100] ];
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
plot_CF (myList2, 6)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: section start ]
Ordinary Annuities (Annuities-Immediate) Formulas
   [wxMaxima: section end   ] */


/* [wxMaxima: comment start ]
Here we assume a constant payment A is made at the end of each of n
compounding periods (at m = 1, 2, ..., n).
   [wxMaxima: comment end   ] */


/* [wxMaxima: subsect start ]
FV_stream (i, A, n)   future value of an ordinary annuity
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
Derivation of FVA (future value of an ordinary annuity stream of payments)

Assume  n compounding periods, with a payment A at the end of each period, the "term" of the annuity
begins at time m = 0, the first payment occurs at time  m = 1, and the last payment A occurs at time m = n 
with future value A. The next to last payment A occurs at time m = n-1 with the future value A*(1+i). 
The payment A at time m = n-2 has a future value at time m = n-1 equal to A*(i+1), and we then increase 
the future value again by another factor of (1+i) to get the future value at time  m = n. So the payment 
A at time m = n-2 has future value A*(1+i)^2, and so on. Each compounding period increases the future 
value by another factor of (1+i).

Then a single payment A at the time m has the future value FV at the future time n, given by
FV = A * (1+i)^(n-m). We now add up these future values of each payment, starting at m = 1 and
ending at m = n:

FVA = sum( A * (1+i)^(n-m),m,1,n) = A* sum ( (1+i)^k, k,0,n-1 ) ; here we have changed the summation
variable m --> k = n-m, so when m = 1, k = n-1 and when m = n, k = 0.

Let Maxima symbolically compute the finite sum.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
sum( (1+i)^k,k, 0,n-1), simpsum;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Hence we have the result
FVA = A * ( (1+i)^n - 1)/ i
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
FV_stream (rate,payment,num) := payment*( (1+rate)^num -1) / rate$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
 Here is an example of use:
  
Consider a series of five $1,000 deposits made at regular intervals: at the end of
year 1, end of year 2,...end of year 5.

Because of the time value of money—the concept that any given sum is worth more now than
it will be in the future because a payment can be invested immediately upon receipt—
the first $1,000 deposit  (payment 1) is  worth more than the second, and so on. So, let's 
assume that you invest $1,000 at the end of every year for the next five years, at 5% annual 
interest. How much will you have at the end of the  five-year period?

   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
FV_stream (0.05, 1000, 5);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
rnddp (%,2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
The accumulated value (the future value of this stream of deposits) is $5,525.63.
   [wxMaxima: comment end   ] */


/* [wxMaxima: subsect start ]
annuity_fv (i, FV, n)  =  constant payment as a function of FV
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
The formula for FVA can be solved for the periodic payment A, given FVA, i, and n:
A = FVA*i / ( (1+i)^n - 1)

and we define the latter (the periodic payment A) as a Maxima function "annuity_fv"
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
annuity_fv(%rate, %FV_stream, %num) := %FV_stream*%rate/((1+%rate)^%num-1)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Here is a webpage on calculating annuities where a very explicit example is given:
https://www.investopedia.com/retirement/calculating-present-and-future-value-of-annuities/

Consider, for example, a series of five $1,000 payments made at regular intervals: at the end of
year 1, end of year 2,...end of year 5.

Because of the time value of money—the concept that any given sum is worth more now than
it will be in the future because a payment can be invested immediately upon receipt—
the first $1,000 deposit  (payment 1) is  worth more than the second, and so on. So, let's 
assume that you invest $1,000 at the end of every year for the next five years, at 5% annual 
interest. How much will you have at the end of the  five-year period?

The future value (at the end of five years) of the payments can be calculated as

n = 5, A = $1,000, i = 5%:

t=0 ==> k=0  NO CASH FLOW HERE
k=1: payment 1:  A*(1+i)^4 = $1000*(1+0.05)^4 = $1,215.51, n-k = 4
k=2: payment 2:  A*(1+i)^3 = $1000*(1+0.05)^3 = $1,157.63, n-k = 3
k=3: payment 3:  A*(1+i)^2 = $1000*(1+0.05)^2 =  $1,102.5, n-k = 2
k=4: payment 4   A*(1+i)^1 = $1000*(1+0.05)^1 =  $1,050.0, n-k = 1
k=5: payment 5   A*(1+i)^0 = $1000*(1+0.05)^0 =  $1,000.0, n-k = 0
Adding these future values, which are approximate since
we have rounded them, we get the approximate sum of $5,525.64

Using 16 digit arithmetic, Maxima now gives:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
FV_stream (0.05, 1000, 5);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
We can use the Maxima function annuity_fv(rate,FV_stream,num) defined above to go from this total future
value of a stream of deposits in an ordinary annuity immediate to get the required periodic payment A:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
annuity_fv(0.05, %, 5);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
which agrees with our example.
   [wxMaxima: comment end   ] */


/* [wxMaxima: subsect start ]
saving (rate, Amount, num)  Savings Accumulation Tables
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
references
https://www.thecalculatorsite.com/finance/calculators/savings-goal-calculator.php

savingA (rate, Amount, num) assumes first savings deposit occurs at k  = 1, and
last occurs at k = num to reach savings goal.

   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
savingA (rate, Amount, num) :=
block ([MM, %payment],
	if num < 0 then error ("num must be a positive value")
	else MM : genmatrix (a, num + 2, 4),
	%payment :  annuity_fv (rate, Amount, num),
	MM[1] : [k, Balance, Interest, Payment],
	MM[2] : [0, 0, 0, 0],
    
	for k : 3 thru num+2  do(
		MM[k, 4] :  %payment,
		MM[k, 1] : k - 2,
		MM[k, 3] : MM[k - 1, 2]*rate,
		MM[k, 2] : MM[k - 1, 2] + MM[k, 3] + MM[k, 4]	),
    rnd_matrix_dp (MM, 2))$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
savingA (0.05, 1000, 6);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
savingB (rate, Amount, num) assumes you make first
savings deposit at t = 0 and immediately start making
interest
If A is the constant periodic mayment, and i is the periodic interest rate, 
and the first savings deposit is made at t = 0 then the future value at k = n
of the stream of deposits made at k = 0, 1, 2,..., n-1  is

A (1+i)^n + A (1+i)^(n-1) + A (1+i)^(n-2) + .... + A (1+i)
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
mysum : sum ((1+i)^k,k,1,n), simpsum;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
We use this result in the definition of %payment in savingB:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
savingB (rate, Amount, num) :=
block ([MM, %payment],
	if num < 0 then error ("num must be a positive value")
	else MM : genmatrix (a, num + 2, 4),
	%payment :  rate*Amount / ( (1+rate)^(num+1) -rate -1),
	MM[1] : [k, Balance, Interest, Payment],
	MM[2] : [0, %payment, 0, %payment],  /* t = 0 */
    
	for k : 3 thru num+2  do(
		MM[k, 4] :  %payment,
		MM[k, 1] : k - 2,
		MM[k, 3] : MM[k - 1, 2]*rate,
		MM[k, 2] : MM[k - 1, 2] + MM[k, 3] + MM[k, 4]	),
    rnd_matrix_dp (MM, 2))$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
savingB (0.05, 1000, 6);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: subsect start ]
PV_stream (i, A, n)  present value of an ordinary annuity
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
Let the payments A at the end of each period: k = 1, 2, 3, ... be labelled A1,A2,A3,...
Then the present value of A1 (in our 5 period example above) deposited at k = 1
is PV(A1) = A1/(1+i). Likewise PV(A2) = A2/(1+i)2, and so on.
For n compounding periods (and for a simple annuity) we then have
PVA = A*sum(1/(1+i)^k, k, 1, n). We let v = 1/(1+i) and call v the "discount factor".
Then PVA = A*sum(v^k, k, 1, n). Let's work on just the sum, using the symbolic
powers of Maxima.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
sum (v^k, k, 1, n), simpsum, factor;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
v/(1-v), v = 1/(1+i), ratsimp;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
So we get PVA = A*(1 - v^n)/i, a simple closed form. Using this formula,
we define the Maxima function PV_stream (rate, payment, num) which
is the t = 0 valuation of a stream of constant payments A made at the end
of each of n periods with assumed interest rate per period i with the "discount 
factor" v = 1/(1+i)
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
PV_stream (rate,payment,num) :=block ([vv : 1/(1+rate)],  payment * (1-vv^num) / rate )$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
As an example assume a 5% interest rate per compounding period, and $1,000
payments made at the ends of  5 periods.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
PV_stream (0.05, 1000, 5);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
rnddp (%, 2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Thus the $5000 in payments over the five periods has a present value (t = 0)
of $4,3329.48.
   [wxMaxima: comment end   ] */


/* [wxMaxima: subsect start ]
annuity_pv (i, PV, n)  =  constant payment as a function of PV
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
Solving for the constant periodic payment, we get
    A = PVA * i / (1 - v^n) 
which we define as a  Maxima function  "annuity_pv(rate,PV_stream,num)" ,
which is the constant payment made at the  end of each compounding period,
given the assumed interest rate, the number of compounding periods, and the 
total value of the stream of payments at t = 0.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
annuity_pv(%rate, %PV_stream, %num) := 
           block ([ %vv : 1/(1+%rate)], %PV_stream*%rate/ (1 - %vv^%num) )$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Using the same 5 period example as used with our FVA discussion above:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
pv_example : PV_stream (0.05, 1000, 5);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
annuity_pv (0.05, pv_example, 5);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
fv_example : FV_stream (0.05, 1000, 5);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
We should have PVA = FVA * v^n since the valuation of the stream at t = n
needs to be discounted with the factor v for each compounding period
separating t = n from t  = 0. We will numerically look at the difference
between v^n*FVA and PVA (which should be 0.0).
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
v_example : 1/(1+0.05);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
v_example^5 * fv_example - pv_example;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
rnddp (%,2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
and indeed we get numerical agreement.
   [wxMaxima: comment end   ] */


/* [wxMaxima: subsect start ]
amortization (i,PV,n): Constant Payment Schedule for Paying Off a Loan 
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
The function amortization can be used to create an amortization matrix which can be 
used to generate an amortization table (schedule) showing how each successive
payment divides into payment of interest on the principal balance, and a portion
which reduces the principal balance itself.
   [wxMaxima: comment end   ] */


/* [wxMaxima: subsubsect start ]
Interactive schedule design for a simple example
   [wxMaxima: subsubsect end   ] */


/* [wxMaxima: comment start ]
For the simple example of a n = 3 period 5% ordinary annuity
for a loan of $56,000 at t = 0, we first calculate the corresponding
constant payment:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
payment : annuity_pv (0.05,56000, 3);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
We keep all 16 digit arithmetic results unrounded until the very end.
   [wxMaxima: comment end   ] */


/* [wxMaxima: comment start ]
Let rate be bound to the decimal value of the  interest rate in this example.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
rate : 0.05;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Use m or k as integers to specify the time of the ends of compounding
periods. For n periods, we create a matrix with n+2 rows, with 
row k = 3 describing the state at the end of the first compounding
period.
   
genmatrix (aa, nrow, ncol) creates a matrix with nrow rows
and ncol columns, where aa is an undefined two dimensional array.
   [wxMaxima: comment end   ] */


/* [wxMaxima: comment start ]
For our simple example, we need five rows and 5 columns.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
M : genmatrix (aa, 5, 5);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
row k = 1 is to have column headings:
 m    Prin_balance    Interest_portion    Principal_portion      Payment

row k = 2 is to have the  starting values  (t = 0)
  0         56000                           0                                   0                          0
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
M[1] :  [m, Prin_balance, Interest_portion, Principal_portion, Payment]$
M[2] :  [0,0,0,0,0]$
M[2,2] :  56000$
M;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Now work on row k = 3.  

Column 1 should have value 1 = k - 2.
Column 5 should have the constant payment.
Column 3 is the interest paid (2800) on 56000 for one compounding
period at 5% interest. 
Column 4 = the amortization (the part of the payment
used to pay down the loan principal) is the constant payment
minus the interest paid. 
Column 2 = the new principal balance is then
the starting balance (56000) minus the amortization.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
k : 3$
M[k, 1] : k - 2$
M[k, 5] : payment$
M[k, 3] : rate * M[k-1, 2]$
M[k, 4] : payment - M[k, 3]$
M[k, 2] :  M[k-1, 2] - M[k, 4]$
M;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
now for rows 4 and 5, we repeat this pattern of filling up the matrix
elements.  Used inside a do loop, k acts like a purely local variable.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
for k:4 thru 5 do  (
     M[k, 1] : k - 2,
     M[k, 5] : payment,
    M[k, 3] : rate * M[k-1, 2],
    M[k, 4] : payment - M[k, 3],
    M[k, 2] :  M[k-1, 2] - M[k, 4])$
M;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
We finally round the results using rnd_matrix_dp (amatrix, dp)
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
M : rnd_matrix_dp (M, 2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
We don't need to keep the global binding of k to 3 made above,
so we use kill(k) here.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
kill(k);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: subsubsect start ]
General Case Function: amortization (rate, Amount, num)
   [wxMaxima: subsubsect end   ] */


/* [wxMaxima: comment start ]
We put together the design elements above to handle a
general amortization schedule for arbitrary interest rate,
loan amount (PV) and number of compounding intervals.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
amortization (rate, Amount, num) := 
block ([%payment, MM],
    %payment : annuity_pv (rate, Amount, num), 
    MM : genmatrix(a, num + 2, 5), 
    MM[1] : [m, Prin_balance, Interest_portion, Principal_portion, Payment],
    MM[2] : [0,0,0,0,0],
    MM[2,2] : Amount,
    for k : 3 thru num + 2 do (
        MM[k,1] : k - 2,
        MM[k,5] : %payment,    
        MM[k,3] : rate * MM[k-1, 2],  /* interest paid */
        MM[k,4] :  %payment - MM[k,3],   /* amortization */
        MM[k,2] : MM[k-1,2]  - MM[k,4] ),   /* new balance */
    rnd_matrix_dp (MM,2 )  )$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
aM : amortization (0.05,56000,3);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
The Maxima function grind(result) produces a pure text form of the result.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
grind(aM)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
which reproduces the interactive calculation.
   [wxMaxima: comment end   ] */


/* [wxMaxima: comment start ]
Our print_matrix function does a crude job of printing the
elements of the matrix as an ordinary table; the matrix form
gives more clarity.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
print_matrix (aM)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: section start ]
Arithmetic Increasing Annuity Formulas
   [wxMaxima: section end   ] */


/* [wxMaxima: comment start ]
We assume an "increasing annuity immediate" in which the payments occur at the
end of each compounding period.

See pdf p 24, "annuity-immediate with general arithmetic progression
payment amounts" in a section of R. Randles Univ. of Fla. statistics course:
http://users.stat.ufl.edu/~rrandles/sta4183/4183lectures/chapter04/chapter04R.pdf

A YouTube discussion is by Prof Stephen Paris (Module 2, Section6, Part 1)
called "Arithmetic Annuities" can be found at:
https://www.youtube.com/watch?v=WvSA2xBfIg8
   [wxMaxima: comment end   ] */


/* [wxMaxima: subsect start ]
PV_arith (i, R, Q, n)  brute force calculation
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
For interest rate i, number of compounding interest periods n, first payment = R, 
second payment = R + Q, third payment = R + 2*Q, etc, nth payment = R + (n-1)*Q.
Let PV_arith_bf (i, R, Q, n) calculate the present value (t = 0 valuation) of this stream of payments.

One can do this using "brute force" via the Maxima sum function.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
PV_arith (i, R, Q, n) := block ([v:1/(1+i)],    
     R*sum (v^k,k,1,n) + Q*sum( (k-1)*v^k,k,2,n))$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Here is a simple example:  R = 1, Q = 2, i = 5%, n = 3.
The first payment is 1, the second payment is 3, the third and last
payment is 5.

Doing this by hand first:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
v:1/(1+0.05);
v + 3*v^2 + 5*v^3;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
and now using our "brute force" function
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
PV_arith (0.05,1,2,3);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: subsect start ]
PV_AI_stream (i, R, Q, n)  present value arithmetic increasing annuity
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
R. Randles writes down a formula for the present value (t=0) of a
stream of arithmetic increasing payments which are made at the
end of each compounding period, beginning with the payment
R, then R+Q, then R+2*Q, etc. Here, R is the first payment, and
Q is the constant increment.  We use this as the basis for the
following Maxima function PV_arith

In this formula we use abn to stand for  "a bracket n" and 
this stands for the sum

abn  = v + v^2 + v^3 + ... + v^n  
         =  (1-v^n)/i,  

where v = 1/(1+i) is the "discount factor" and i is the interest rate
per compounding period.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
PV_AI_stream (rate, R, Q, num) := block ([vv:1/(1+rate), abn],    
    abn : (1-vv^num)/rate,
     R*abn + (Q/rate)*(abn - num*vv^num))$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
PV_AI_stream (0.05, 1,2,3);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
which gives the same result as both our hand and brute force calculations.
   [wxMaxima: comment end   ] */


/* [wxMaxima: subsect start ]
AI_annuity_pv (i, PV, Q, n)  = first payment
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
We solve the PV_AI_stream formula for the first payment in terms of the
 present value (the "Amount"), the interest rate,  the number of compounding
 periods, and the constant  increment in the periodic payment made at the 
 end of each compounding  period:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
AI_annuity_pv (%rate, %Amount, %increment, %num) := 
     block ([vv:1/(1+%rate), abn], 
      abn : (1 - vv^%num)/%rate,
       /*  print(" %rate = ",%rate," vv = ",vv,"  abn = ",abn),   */
      (%Amount/abn) - (%increment/%rate)*(1 - (%num*vv^%num) / abn))$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Here is an example of use with i = 5%, n = 3, payments 1,3,5.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
AI_annuity_pv (0.05,7.99265738041266, 2, 3);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
rnddp (%, 2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: subsect start ]
amortization_arith (i, PV, Q, n)   matrix 
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
amortization_arith (i, PV, Q, n) returns a matrix display (table) of the amortization schedule
for an arithmetic increasing  annuity immediate.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
amortization_arith(rate, Amount, increment, num):=
block([MM,%payment],    
     if num < 0 then error("num must be a positive value")
       else MM : genmatrix(a, num+2, 5),      
    %payment : AI_annuity_pv(rate, Amount, increment, num),
    MM[1] : [m, prin_balance, int_portion, prin_portion,Payment],
    MM[2] : [0,0,0,0,0],
    MM[2,2] : Amount,
    MM[3,5] : %payment,
    for k:4 thru num+2 do (MM[k,5]:MM[k-1,5] + increment),
    for k:3 thru num+2 do (
        MM[k,1] : k-2,
        MM[k,3] : MM[k-1,2]*rate,
        MM[k,4] : MM[k,5] - MM[k,3],
        MM[k,2] : MM[k-1,2] - MM[k,4]),
rnd_matrix_dp (MM,2) )$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Here is the amortization table for the simple case discussed above
with i = 5%, n = 3, payments 1,3,5.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
amortization_arith (0.05, 7.992657380412481, 2, 3);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
grind(%)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
A second example with 5% interest rate, payment increment
of $1,000, initial loan amount of $56,0000, and 3 payments.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
amortization_arith (0.05 , 56000 , 1000 , 3);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
future value of arithmetic increasing stream of payments
   [wxMaxima: comment end   ] */


/* [wxMaxima: section start ]
Growing Annuity ("Geometric Increasing") Formulas
   [wxMaxima: section end   ] */


/* [wxMaxima: comment start ]
A growing (geometric increasing) annuity refers to ordinary annuities
in which payment [k + 1] = (1 + g) * payment [k], and the constant
parameter g is called the "growth rate".
   [wxMaxima: comment end   ] */


/* [wxMaxima: subsect start ]
present value PV of a growing (geometric increasing) annuity
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
Reference:
  https://financeformulas.net/Present_Value_of_Growing_Annuity.html

From which we quote:

"The present value of a growing annuity formula calculates the present day value of a 
series of future periodic payments that grow at a proportionate rate. A growing annuity may 
sometimes be referred to as an increasing annuity. A simple example of a growing annuity 
would be an individual who receives $100 the first year and successive payments increase
 by 10% per year for a total of three years. This would be a receipt of $100, $110, and $121, respectively."

"The present value of a growing annuity formula relies on the concept of time value of money. 
The premise to this concept is that a specific quantity of money is worth more today than at a future time."

"Like all financial formulas that involve a rate, it is important to correlate the rate per period to the number 
of periods in the present value of a growing annuity formula. If the payments are monthly, then the rate 
would need to be the monthly rate."

   [wxMaxima: comment end   ] */


/* [wxMaxima: subsubsect start ]
Sum of a finite geometric series
   [wxMaxima: subsubsect end   ] */


/* [wxMaxima: comment start ]
The sum of the finite geometric series having x = "common ratio":
1 + x + x^2 + ... + x^(n-1)   =   (1 - x^n) / (1 - x)
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
sum ( x^k, k, 0, n - 1), simpsum;
/* [wxMaxima: input   end   ] */


/* [wxMaxima: subsubsect start ]
PV_geom(i, g, R, n)  brute force calculation of present value
   [wxMaxima: subsubsect end   ] */


/* [wxMaxima: comment start ]
PV_geom (i, g, R, n) calculates the present value (t = 0) of a stream of future
payments made at the ends of n equal periods, assuming an interest rate i,
first payment R, and each succeeding payment is (1+g) times larger than
the previous payment.  We can use the Maxima function fundef to see the
definition:
   [wxMaxima: comment end   ] */


/* [wxMaxima: comment start ]
Let R be the first payment at the end of the first compounding period, 
i be the interest rate per period, let v = 1/ (1+i) = "discount factor",
let g be the "growth rate", such that the second payment is R*(1+g),  
the third payment is R*(1+g)^2, etc. Let the number of payments be n. 

Then using the discount factor to get the present value of each payment, we get:
present value = PV = R*v + R*v^2 *(1+g)  + R*v^3*(1+g)^2 +....
   ... + R*v^n*(1+g)^(n-1)
or

PV = R*v * [1 + x + x^2 + .... + x^(n-1) ],  where  x = common ratio  = v*(1+g),

then use the geometric series formula to get

PV = R*v* (1 - v^n*(1+g)^n) / (1 - v*(1+g))

   now use v = 1/(1 + i) to write the denominator as
     1 - (1+g)/(1+i) = (1+i - (1+g) ) / (1+i) = (i - g)*v, to get

PV = ( R / (i-g) ) *  ( 1 -   ( (1+g)/(1+i) )^n  )

This last formula is used to create the function PV_GI_stream below.

For now we use the explicit sum version to define PV_geom.

   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
PV_geom (i, g, R, n) := block ([v : 1/(1+i) ],    
    R*v* sum ( ( v*(1+g) )^k, k, 0, n-1 ))$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
We use a 3 payment example (n=3), assuming 5%  interest rate
per period (i = 0.05), the first payment R = $100, and we assume 
a 10% growth rate in payments so g = 0.1.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
PV_geom (0.05, 0.1, 100, 3);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
rnddp (%, 2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: subsubsect start ]
PV_GI_stream (i, g, R, n) 
   [wxMaxima: subsubsect end   ] */


/* [wxMaxima: comment start ]
GI stands for "geometic increasing" (rather than arithmetic increasing).

i is the interest rate (in decimal form) per compounding period.
n is the number of payments, each made at the end of a compounding period.
R is the first payment at the end of the first compounding period, 
The second payment at the end of the second period is R*(1+g),
where g is the growth rate in decimal form.

Here we make use of the geometric series summation formula.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
PV_GI_stream (i, g,R, n) :=   R * (1 - ( (1+g)/(1+i) )^n ) / (i - g)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
And repeat the example used above:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
PV_GI_stream (0.05, 0.1,100, 3);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
rnddp (%, 2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: subsubsect start ]
GI_annuity_pv (i, g, PV, n)  ==  first payment
   [wxMaxima: subsubsect end   ] */


/* [wxMaxima: comment start ]
We solve the formula used for PV_GI_stream for the first payment R
to get the definition of GI_annuity_pv (i, g, PV, n),

This function returns the first payment of a geometric increasing stream of
payments, given the present value PV, the periodic interest rate i, the
payment growth rate g, and the number n of compounding periods.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
GI_annuity_pv (rate,growing_rate,PV,num) :=  
  PV*(rate - growing_rate)/(1 - (1 + growing_rate)^(num)/((1 + rate)^num))$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Apply this function to the previous example:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
GI_annuity_pv(0.05, 0.1, 299.54, 3);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
rnddp (%, 2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: subsubsect start ]
amortization_geom (i, g, PV, n) matrix
   [wxMaxima: subsubsect end   ] */


/* [wxMaxima: comment start ]
amortization_geom (i, g, PV, n) returns a matrix table display of
the amortization schedule for a geometic increasing ordinary
annuity, assuming interest rate i, payment growth rate g, PV
(value at t = 0), and n payment periods.


   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
amortization_geom (rate, growing_rate, Amount, num) :=
block([MM, %payment],
    if num<0 then error("num must be a positive value")
    else    MM : genmatrix(a, num + 2, 5),
    %payment : GI_annuity_pv (rate, growing_rate, Amount, num),
    MM[1] : [ k,  Balance, Interest, Amortization,Payment],
    MM[2] : [0, Amount, 0, 0, 0],
    MM[3, 5] : %payment,
    for k : 4 thru num+2 do MM[k, 5] : MM[k - 1, 5]*(1+growing_rate),
    for k : 3 thru num+2 do (
        MM[k, 1] : k - 2,
        MM[k, 3] : MM[k - 1, 2]*rate,
        MM[k, 4] : MM[k, 5] - MM[k, 3],
        MM[k, 2] : MM[k - 1, 2] - MM[k, 4]  ),
    rnd_matrix_dp (MM, 2))$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
The payoff schedule for a $56,000 loan at t = 0, with 3 periodic payments
subject to a 5% interest rate and a payment growth rate of 3% is then
given by:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
amortization_geom (0.05, 0.03, 56000, 3);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: subsect start ]
future value FV of a growing (geometric increasing) annuity
   [wxMaxima: subsect end   ] */


/* [wxMaxima: comment start ]
The future value of a stream of growing (geometric increasing) ordinary annuity
payments is the value of that stream of payments at the time of the last payment,
and FV = (1+i)^n * PV  for n periodic payments and interest rate per period i.

   [wxMaxima: comment end   ] */


/* [wxMaxima: comment start ]
reference:

https://financeformulas.net/Future-Value-of-Growing-Annuity.html
   [wxMaxima: comment end   ] */


/* [wxMaxima: comment start ]
. A growing annuity may sometimes be referred to as an increasing annuity. A
simple example of a growing annuity  would be an individual who receives $100 
the first year and successive payments increase  by 10% per year for a total 
of three years. This would be a receipt of $100, $110, and $121, respectively.

The future value of a growing annuity formula relies on the concept of time value 
of money.  The premise to this concept is that a specific quantity of money is worth
more today than at a future time.

Like all financial formulas that involve a rate, it is important to correlate the rate per 
period to the number  of periods in the future value of a growing annuity formula. 
If the payments are monthly, then the rate  would need to be the monthly rate."
   [wxMaxima: comment end   ] */


/* [wxMaxima: comment start ]
Above we derived the formula for the present value of a growing annuity in which the
first payment is R when k = 1, the second payment is R(1+g) when k = 2, the third payment
is R(1+g)^2 when k = 3, etc.  The periodic interest rate is i.
Then:
                       PV = ( R / (i-g) ) *  ( 1 -   ( (1+g)/(1+i) )^n  )

We can obtain the formula for the future value (k = n) of this stream of payments by
multiplying PV by (1+i)^n.

We can then simplify and obtain:
                     FV = ( R / (i-g) ) * ( (1+i)^n - (1+g)^n )

   [wxMaxima: comment end   ] */


/* [wxMaxima: subsubsect start ]
brute force calculation FV_geom(i, g, R, n)
   [wxMaxima: subsubsect end   ] */


/* [wxMaxima: comment start ]
In this brute force calculation we simply used the Maxima function
sum, without using the finite sum geometric sum formula.  
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
FV_geom (i, g, R, n) :=   R*sum ( (1+g)^(k-1) * (1+i)^(n-k), k, 1, n)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
We use a 3 payment example (n=3), assuming 5%  interest rate
per period (i = 0.05), The first payment R = $100, a 10% growth rate in payments
so g = 0.1.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
FV_geom (0.05, 0.1, 100, 3);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
We can transform this into the present value (t=0 value PV) by dividing by (1+i)^n:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
rnddp (% / (1+0.05)^3, 2);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
and as another check:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
PV_geom (0.05, 0.1, 100, 3);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: subsubsect start ]
FV_GI_stream (i, g, R, n) 
   [wxMaxima: subsubsect end   ] */


/* [wxMaxima: comment start ]
GI stands for "geometic increasing" (rather than arithmetic increasing).

i is the interest rate (in decimal form) per compounding period.
n is the number of payments, each made at the end of a compounding period.
R is the first payment at the end of the first compounding period, 
The second payment at the end of the second period is R*(1+g),
where g is the growth rate in decimal form.

Here we make use of the  formula derived above:

 FV = ( R / (i-g) ) * ( (1+i)^n - (1+g)^n )

   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
FV_GI_stream (i, g,R, n) :=   R * ( (1+i) ^n - (1+g)^n ) / (i - g)$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
FV_GI_stream (0.05, 0.1, 100, 3);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: subsubsect start ]
GI_annuity_fv (i, g, FV, n)  ==  first payment
   [wxMaxima: subsubsect end   ] */


/* [wxMaxima: comment start ]
This formula give the first payment (at k = 1) of a geometric increasing stream of
payments, given the  future value FV (at k = n), the interest rate i, the growth rate g,
and the number n of compound interest periods.
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
GI_annuity_fv (i, g, FV, n) :=  
  FV*(i - g)/( (1 + i)^n - (1 + g)^n )$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
Again using our simple n = 3 example:
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
GI_annuity_fv (0.05, 0.1, 346.7500000000005, 3);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: comment start ]
which recovers the first payment R.
   [wxMaxima: comment end   ] */


/* [wxMaxima: section start ]
Displaying rows of a matrix with many rows
   [wxMaxima: section end   ] */


/* [wxMaxima: comment start ]
Here is a simple example of displaying some of the rows of a 
matrix with many rows ( which will also return a large number
for the length of the matrix).
   [wxMaxima: comment end   ] */


/* [wxMaxima: input   start ] */
MM : matrix ( [a,b,c],[d,e,f],[g,h,i],[j,k,l],[m,n,o],[p,q,r],[s,t,u],[v,x,y] );
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
MM[1];
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
for j thru 3 do print ( j, MM[j])$
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
length(MM);
/* [wxMaxima: input   end   ] */


/* [wxMaxima: input   start ] */
for j: 5 thru 8 do print (j, MM[j] )$
/* [wxMaxima: input   end   ] */



/* Old versions of Maxima abort on loading files that end in a comment. */
"Created with wxMaxima 19.05.7"$