Financial Calculators: Deferred Mortgages: Solving for Payment and Price

Financial Calculators:  Deferred Mortgages: Solving for Payment and Price

Finance Now, Pay Later!

When a customer financing a loan or mortgage, and the customer can enjoy the benefits for a period of months prior to being required to make the first payment, we are dealing with a deferred mortgage.   Interest is accrued from the beginning of the mortgage.

The procedures outlined on this blog entry can be applied to any financial calculator* with time value of money keys, such as the HP 12C and the HP 10bII+ (see source below).  

* The procedure should be fine for most financial calculators.   Check your manual for details.  

Deferred Mortgage:  Finding The Monthly Payment

Part 1:

Set the calculator to END mode.

N:  Enter the number of deferred payments minus 1

I:   Enter the interest rate of the loan

PV:  Enter the borrowed amount as a negative number

PMT:  Set to 0

Solve for FV

Part 2:

PV = FV from part 1   ( RCL FV, STO PV )

N:  Enter the number of payments of the term

I = interest rate of the loan

FV:  Enter the balloon payment as a negative number.  If there is no balloon payment, enter 0.

Solve for PMT, this is your payment of the deferred mortgage.  

Example 1:  

A homeowner finances a house for $455,000.00.   The mortgage lasts for 30 years at 2.86%.   No payment is due for the first six months of the mortgage.  

Part 1:

N: 6 -1 = 5

I:  2.86% annual rate, about 0.24% per month

PV:  -455000.00

PMT:  0.00

FV = 460447.99

Part 2:

PV:  460447.99

N:  30* 12 = 360

I:  2.86% annual rate, about 0.24% per month

FV:  0.00

PMT = -1906.67

The monthly payment is $1,906.67.

Example 2:

A homeowner purchases a fixer-up property for $149,000.00.  The buyer secures a 15-year mortgage at 4%.  The end of the loan requires a $1,000.00 balloon payment but allows the buyer to defer the first payment for five months.

Part 1:

N:  5 - 1 = 4

I:  4% annual rate, about 0.33% per month

PV:  -149000.00

PMT: 0.00

FV = 150996.62

Part 2:

PV:  150996.62

N:  15 * 12 = 180

I:  4% annual rate, about 0.33% per month

FV: -1000.00

PMT = -1112.84

Deferred Payment:  Calculating the Price of the Mortgage

Part 1:

Set the calculator to END mode.

N:  Enter the number of payments of the term

I:   Enter the interest rate of the loan

PMT: Enter the payment as a positive number

FV: Enter the balloon payment as a positive number.  If there is no balloon payment, enter 0.   

Solve for FV

Part 2:

FV = PV from part 1   ( RCL PV, STO FV )

N:  Enter the number of deferred payments minus 1

I = interest rate of the loan

Set PMT to 0

Solve for PV, this is your price of the deferred mortgage.  

Example 3:

An owner enters a lease which requires at monthly payment of $495.00.  The lease lasts for 8 years at 6.69%.  There is no balloon payment at the end of the lease.  The lease allowed the owner to defer the first required payments for three months.  What is the value of the lease?

Part 1:

N:  8 * 12 = 96

I:  6.69% annual rate, about 0.56% per month

PMT:  495.00

FV:  0.00

PV = -36721.17

Part 2:

FV:  -36721.17

N:  3 - 1 = 2

I:  6.69% annual rate, about 0.56% per month

Set PMT to 0

PV = 36315.13

The value of the lease is $36,315.13.

Example 4:

An owner executes a 30-year, 3% mortgage with the required payment of $1,005.85.  The mortgage allows for the first payment to be deferred for six months.  A $1,000.00 balloon payment is required.  What is the price of the mortgage?

Part 1:

N:  30 * 12 = 360

I:  3% annual rate, 0.25% per month

PMT:  1005.85

FV:  1000.00

PV = -238983.97

Part 2:

FV:  -238983.97

N:  6 - 1 = 5

I:  3% annual rate, 0.25% per month

Set PMT to 0

PV = 236018.94

The value of the lease is $236,018.94.

And that is how we work with deferred mortgages.  

Source:

Greynolds Jr., Elbert B. and Aronofsky, Julius S.  Practical Real Estate Financial Analysis: Using The HP 12C Calculator:  A Step-by-Step Approach   Real Estate Education Company: Dearborn Financial Publishing, Inc.  1983.  ISBN 0-88462-378-5

Eddie

All original content copyright, © 2011-2020.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Discounts and Sales Tax: What is the Final Price?

SALE! GET THIS WONDERFUL ITEM FOR A SUBSTANTIAL DISCOUNT! GOOD FOR THIS WEEKEND ONLY!

Chances are that we have all seen this advertisement at least one time in our lives. Chances are that if you are shopping with someone, that person asks what the price is after the discount. And in a lot of instances, depending on where you live, you have to take sales tax into consideration. Today's blog is about how to find the final price after the discount and sales tax.

Note: All the examples presented here are using prices stated in dollars ($). Rest assured the concepts hold for all other units of currency. Also, all dollar amounts are rounded to two decimal places.

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The Math Behind it All

Discounts

Let D be the discount. Since the discount is customarily stated as a percentage, such as "take 20% off of everything on this rack", we'll express the discount as its decimal equivalent: 0.01*D. Discounts are subtracted from the price.

For example, a item normally sold for $59.99 which has a 20% discount has a discounted price of $47.99.

$59.99 - 20%
$59.99 - $59.99 * 0.01 * 20
$59.99 * (1 - 0.20)
$59.99 * 0.80
$47.99

A general formula for subtracting D% from X is:

X - D%
X - X * 0.01 * D
X * (1 - 0.01 * D)

Sales Tax

Let S be the sales tax. Sales tax works like discounts, except that sales tax is added to the price. We'll express the sales tax as its decimal equivalent: 0.01*S.

Where I live, which is located in the County of Los Angeles in California, the sales tax is 9%. For example, if I purchase a taxable item for $59.99, taking sales tax into account, the total cost to me is $65.39.

$59.99 + 9%
$59.99 + $59.99 * 0.01 * 9
$59.99 + $5.40
$65.39

Likewise, we can use a general formula for adding S% to X is:

X + S%
X + 0.01 * S * X
X * (1 + 0.01 * S)

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The Final Price of a Discounted Item After Sales Tax

Let's come up with a general formula where we determine the final price of an item which normally sells for $P (or whatever currency you work with), sold at a discount D%, which the item subject to sales tax of S%.

Customarily, the discount gets applied before the sales tax. Then the final price works out to be:

$P - D% + S%

To simplify this, start with subtracting the discount:
($P - D%) + S%
($P - $P * 0.01 * D) + S%
($P * (1 - 0.01 * D) + S%

Next, add the sales tax:
($P * (1 - 0.01 * D) + ($P * (1 - 0.01 * D)) * S * 0.01
($P * (1 - 0.01*D))(1 + 0.01*S)

So the final cost is:
$P * (1 - 0.01*D) * (1 + 0.01*S)

Taking from our example before, let's say that the item originally priced at $59.99 is discounted at 20%. What is the final cost after 9% sales tax is applied?

P = $59.99
D = 20
S = 9
And the final cost is:

$59.99 * (1 - 0.01*20) * (1 + 0.01*9) = $52.31

In the next section, we will determine a "combined rate".

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Simplified Rate Considering Both Discount and Sales Tax

Here we are going to find a general formula for a "combined rate", or a rate that takes both the discount and sales tax rate into account.

Basically, the "combined rate" is what the final price would be if the original price was $1.00. ($P = $1.00)

The combined rate is (1 - 0.01D)*(1+0.01S)

In our example the combined rate for an item discounted 20%, subject to sales tax of 9% is 0.872. (87.2%)

(1 - 0.01*20)(1 + 0.01*9)
0.872

Under these conditions, you should expect to pay 87.2% of the original price when it is all said and done. For an item with an original price of $59.99, the final cost is:

Price * Combined Rate
$59.99 * 0.872
$52.31

Another example: item has an original price of $18.99. The discount is 10% while the sales tax rate is 9%

Combined rate:
(1 - 0.10 * 10) * (1 + 0.01 * 9)
0.981

Final cost:
$18.99 * 0.981
$18.63

As a check, let's find the final cost if we applied the discount then sales tax step by step.
($18.99 - 10%) + 9%
($18.99 - $18.99 * 0.01 * 10) + 9%
$17.09 + 9%
$17.09 + $17.09 * 0.01 * 9
$18.63

It checks out.

This is a quick table of combined rates using various sales tax and discount rates. The iOS App Numbers is used to generate this table.

Example: What is the final cost for at item originally priced $69.95 that is discounted 35%. The sales tax rate is 10%.

From the table above, the combined rate is 0.715. The final cost is $69.95 * 0.715 = $50.01.

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I hope you find this helpful, and as always I appreciate your comments.

I want to do another mathematical series. I plan to do a short series of ordinary differential equations within the next couple of months, but if there is anything you suggest, subjects, calculators, calculator programming languages, please do. Many thanks as always.

My next blog entry will be within a couple of weeks. I am taking care of family and also going on a short vacation next weekend - hopefully the weather at Landers, CA will be nice as I go on my first astronomical star party.

Eddie



This blog is property of Edward Shore. 2013