FINITE MATH 100% ORIGINAL & A+ QUALITY WORK

Finite Math

Introduction

This work is an attempt to develop various mathematical models that can be applied in determining various financial variables in a mortgage loan which is an annuity by virtue of its installment payments. It uses mathematical models, applies mathematical formulas by giving step-by-step explanations, illustrates using an example which is run through an excel sheet by applying the formula derived and ends by giving interpretation of excel results.

Models Derivation

A continuous repayment is a mortgage loan which is paid by way of a continuous annuity. Generally, settling of mortgages occur over a period of years by a series of annuities with each payment accumulating compound interest from the time the deposit is made till the end of the mortgage life. With a loan, per period interest rate, periods to maturity and regular per period payment, the end of period balancing equation can be given as:

In a continuous repayment mortgage of this nature it would be interesting derive a time- continuous equation. The classical formulation of the present value of annuities of fixed monthly payments with a mount invested to attract monthly interest rate is given as

This formula can be reorganized to get the monthly payment on the principal given and.

Replacing with where is the annual interest rate, and the annual frequency of periods of compounding is represented by. We again replace with with T being the total loan period. In this more general form we can obtain -the fixed payment corresponding to the stated frequency. If, for instance 365, corresponds to a daily annuity (Lowenstein, 2009).

As becomes large, becomes smaller as the product approaches a limiting value.

It is good to note that is simply the amount paid every year given the annual payment rate.

This reasoning well argues that

With the application of the same principle to the formula for annual repayment we are able to determine the limiting value:

At this juncture in the orthodox formulation for PV is better presented as a function of N and time t:

We can use this derivation to get quite a number of solutions to day-to-day mathematical and financial problems.

Calculation of Accumulated Mortgage Interest and Principal Payments

Given the annual interest payment as

we can integrate both sides of this expression to get

Where

The accumulated interest payments from to ;

The accumulated principal payments over the loan life; and

The cumulated fixed payments at the end of time t

If we evaluate, we get an expression for which is the interest paid:

It can be observed that the second integral evaluates to hence:

This algebraic expression is identically the same thing with the one above.

Period of a loan

A borrower who can afford an annual repayment rate can have his repayment period determined by rearranging the equation as follows:

Calculation of Interest rate on Mortgage

In a discrete time interval model, determination of mortgage based interest rate with the rest of the parameters is not possible with the use of many analytical methods.

At first look this would pass off for the case in the continuous model.

Given:

We can do a little simplification as follows:

To increase the apprehensiveness of the equation above as a function of, we can respectively apply hypothetical values to, and as 10000, 6000 and 3. The minimum value of the resulting function can be determined by differentiation:

By nature this function is parabolic between the roots at thus we can estimate the required value as:

The Present Value and the Future value Formulae

Corresponding to the classical PV formula given as:

We can implore this to our already established time-continuous analogue:

Example

The average cost of a new home in State College, PA is $135,000. With our decent credit score PNC Bank would approve us for a 3-year mortgage at 6.1% compounded monthly. However in an attempt to reduce total interest paid we will calculate assuming; no money down, $10,000 down and, $25,000 down.

Solution from the Model

Using the formula

We get that the monthly payments would be $4,113.08.

We will construct a spread sheet in excel which will show remaining balance per year, total interest paid for each scenario assuming all payments made on time, and payment. Then to find balance per year will we change n accordingly in our excel model (Lowenstein, 2009).

The Excel output is as follows:

Period	Beginning Balance	Payment	Principal	Interest	Cumulative Principal	Cumulative Interest	Ending Balance
1	135,000	$4,113.08	$3,426.83	686.25			131,573
2	131,573	$4,113.08	$3,444.25	668.83	$3,444.25	668.83	128,129
3	128,129	$4,113.08	$3,461.76	651.322	$6,906.01	1320.15	124,667
4	124,667	$4,113.08	$3,479.36	633.725	$10,385.37	1953.88	121,188
5	121,188	$4,113.08	$3,497.04	616.038	$13,882.41	2569.91	117,691
6	117,691	$4,113.08	$3,514.82	598.261	$17,397.23	3168.18	114,176
7	114,176	$4,113.08	$3,532.69	580.394	$20,929.92	3748.57	110,643
8	110,643	$4,113.08	$3,550.64	562.437	$24,480.56	4311.01	107,093
9	107,093	$4,113.08	$3,568.69	544.387	$28,049.25	4855.39	103,524
10	103,524	$4,113.08	$3,586.83	526.247	$31,636.09	5381.64	99,937
11	99,937	$4,113.08	$3,605.07	508.014	$35,241.16	5889.65	96,332
12	96,332	$4,113.08	$3,623.39	489.688	$38,864.55	6379.34	92,709
13	92,709	$4,113.08	$3,641.81	471.269	$42,506.36	6850.61	89,067
14	89,067	$4,113.08	$3,660.32	452.756	$46,166.69	7303.37	85,406
15	85,406	$4,113.08	$3,678.93	434.15	$49,845.62	7737.52	81,728
16	81,728	$4,113.08	$3,697.63	415.448	$53,543.25	8152.97	78,030
17	78,030	$4,113.08	$3,716.43	396.652	$57,259.68	8549.62	74,313
18	74,313	$4,113.08	$3,735.32	377.76	$60,995.00	8927.38	70,578
19	70,578	$4,113.08	$3,754.31	358.772	$64,749.31	9286.15	66,824
20	66,824	$4,113.08	$3,773.39	339.688	$68,522.70	9625.84	63,050
21	63,050	$4,113.08	$3,792.57	320.507	$72,315.28	9946.34	59,258
22	59,258	$4,113.08	$3,811.85	301.228	$76,127.13	10247.6	55,446
23	55,446	$4,113.08	$3,831.23	281.851	$79,958.36	10529.4	51,615
24	51,615	$4,113.08	$3,850.71	262.375	$83,809.07	10791.8	47,764
25	47,764	$4,113.08	$3,870.28	242.801	$87,679.35	11034.6	43,894
26	43,894	$4,113.08	$3,889.95	223.127	$91,569.30	11257.7	40,004
27	40,004	$4,113.08	$3,909.73	203.353	$95,479.03	11461.1	36,094
28	36,094	$4,113.08	$3,929.60	183.479	$99,408.64	11644.6	32,165
29	32,165	$4,113.08	$3,949.58	163.503	$103,358.21	11808.1	28,215
30	28,215	$4,113.08	$3,969.66	143.426	$107,327.87	11951.5	24,245
31	24,245	$4,113.08	$3,989.83	123.247	$111,317.70	12074.7	20,255
32	20,255	$4,113.08	$4,010.12	102.965	$115,327.82	12177.7	16,245
33	16,245	$4,113.08	$4,030.50	82.5805	$119,358.32	12260.3	12,215
34	12,215	$4,113.08	$4,050.99	62.0922	$123,409.31	12322.4	8,164
35	8,164	$4,113.08	$4,071.58	41.4996	$127,480.89	12363.9	4,092
36	4,092	$4,113.08	$4,092.28	20.8024	$131,573.17	12384.7	0

Interpretation of Results

At the end of every month we will be making a constant payment of $4,113.08 to PNC Bank of which a part will be principal and the rest interest. Interest will be reducing by the expiry of the loan term because at any one time the outstanding principal is being offset.

The application would be the same if a down payment were made at the start. The implication of the down payment is that it cannot constitute the starting outstanding balance thus with a down payment of $25000, the beginning balance would be $110000.

References

Kieso, D. E., & Weygandt, J. J. (2007). Intermediate Accounting (12th ed.). New York: John Wiley & Sons. p. 738. ISBN 0-471-74955-9.

Lowenstein, R. (2009). . "Book Review: 'The Myth of the Rational Market' by Justin Fox". Washington Post. Retrieved 21 June 2014.