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Total Cost of Borrowing

*** READ ME FIRST BEFORE YOU CONTINUE ***

TCOB over the entire Amortization Period

TCOB over the Term

Interest Accruing

Interest Only

Discounting a Loan

Calculating YIELDS on a Discounted Mortgage


TCOB over the entire Amortization Period

A Broker arranges for a Lender to hand you a cheque for $45,000 and you hand your Broker a cheque for $3,500 to cover the cost of the total fees. The blended monthly payments (principal and interest) for a loan of $45,000 are $464.35 based on an annual interest rate of 12% with semi-annual compounding and amortized for 25 years. If the borrower is making payments for the next twenty five years based upon a 12% rate (constant 12% rate for the twenty five years) what is the impact of the total fees of $3,500?

1.. Set the Dollar Discount Calculation Option to Standard Speed High Precision. It is always recommended to use this setting unless your computer is really slow.

(Screenshot 1)

The new interest rate due to the impact of the total fees is 13.233 % which translates into an effective interest rate of 13.6708 % due to semi-annual compounding. Thus the TCOB is 13.67 % at two decimal places.

It is interesting to note that the American, Truth in Lending legislation would have the value of 13.23% quoted as the APR, rather than quoting the new EIR as in Ontario, Canada.

This TCOB calculation could also have been handled by the MORTGAGE2 PRO program as follows;

(Screenshot 2)

This is the same loan information in the CALCULATOR. Subtract the total fees of $3,500 from the principal and recalculate the annual Interest Rate as shown below;

(Screenshot 3)

The new EIR because of the total fees is 13.670034 thus the TCOB is 13.67% at two decimal places. The logic is that you are making 300 payments based upon a $45,000 loan but in effect you only really have $41,500 thus the interest must be higher.

 

 

TCOB over the Term

A Broker arranges for a Lender to hand you a cheque for $45,000 and you hand your Broker a cheque for $3,500 to cover the cost of his total fees. The blended monthly payments (principal and interest) for a loan of $45,000 are $464.35 based on an annual interest rate of 12% with semi-annual compounding and amortized for 25 years. The term of the mortgage is 5 years. What is the impact of the brokers total fees of $3,500 spread over the term of 60 months?

Set the Dollar Discount Calculation to High Precision

(Screenshot 4)

The new interest rate due to the brokers total fees is 14.327 % which translates into an effective interest rate of 14.8402 % due to semi-annual compounding. Thus the TCOB is 14.84 % at two decimal places.

In Standard Precision mode which is selected only for slow or old computers the same results are achieved at two decimal places, that is TCOB = 14.84%

(Screenshot 5)

 

 

Interest Accruing

A mortgage has a face value of $45,000. The interest is to be 12% with semi-annual compounding, not in advance. The loan is to be repaid in three years. No payments of principal or interest are to be made during the three years. At the end of the three years all the outstanding principal and interest will be repaid in one payment. The total cost of all the fees is $3,500 and thus the borrower is actually receiving only $41,500 because the broker is getting his $3,500.

A financial calculator could be used for this calculation, .. providing you know how to calculate the "12 % semi-annual compounding, monthly interest factor" which is 0.009758794179192. This factor is required to calculate the Future value (FV) of $45,000 in 36 months. One of the unique features of MORTGAGE2 PRO software is its ability to do PV/FV calculations without requiring you to understand how to calculate this interest factor, 0.009758794179192.

1.. You make the MORTGAGE2 PRO amortization schedule a negative amortization schedule by making all the payments equal to zero for the first 36 months. The balance at 36 months is the Future Value of $45,000 growing at 12% per year semi-annually compounded, which is $63,833.35 What you have is a series of FV results in a grid format from one month to 36 months which is more informative than the regular PV/FV financial calculator format.

(Screenshot 6)

The factor 0.009758794179192 can also be obtained by using the MORTGAGE PRO Utilities Menu.

(Screenshot 7)

2.. Using a financial calculator the new monthly interest factor needs to be determined. In other words, what does the new interest really have to be on a monthly basis in order to grow $41,500 to $63,833.35 over 36 months?

(Screenshot 8)

The new monthly interest factor is 0.01203244 as a decimal. To calculate the effective interest rate (EIR) add 1.0 to this new monthly interest factor and multiply by itself 12 times. Shown mathematically as;

(1.01203244)^12 = 1.154338

Subtract 1.0 and multiply by 100 and the effective interest rate is 15.43 % at two decimal places.

Thus the TCOB is 15.43%

 

 

Interest Only

A Broker arranges for a Lender to hand you a cheque for $45,000 and you hand your Broker a cheque for $3,500 to cover the cost of his total fees. The blended monthly payments (principal and interest) for a loan of $45,000 are $464.35 based on an annual interest rate of 12% with semi-annual compounding and amortized for 25 years.

(Screenshot 9)

Using the MORTGAGE2 software you can see that the interest for the very first payment is $439.15 per month. With this interest only loan, the borrower is actually receiving $41,500 but is making interest only, monthly payments of $439.15 for the next 36 months based on a loan of $45,000. Using the DISCOUNTING software the TCOB is calculated as follows.

Certain combinations of computers and operating systems will not let you enter the exact interest in the Payment box for an interest only. Just add .001 to the payment ( in this example $439.151) in order to get by this quirk. The payment will still come back as 439.15 and the two interest rates are still correct at two decimal places.

(Screenshot 10)

The new interest rate due to the brokers total fees is 15.417% which translates into an effective interest rate of 16.0112 % due to semi-annual compounding. Thus the TCOB is 16.01 % at two decimal places, for $3,500 fees, using the Standard Speed High Precision.

It is recommended to use the High Precision setting for all calculations unless you have a really slow computer.

UNDERSTANDING THE DISCOUNTING PROGRAM:
Dollar Discount Calculations Options: When you specify a dollar discount amount the program takes a guess at the new rate (% Yield) and performs a calculation of the of the discount. The iterative loop continues until the guess is close to the initial value and then displays the guess at box C. The current program could have taken smaller guess increments and converged on the exact value of $3,500 in box C however it would have taken an unreasonable amount of time. Shown below is the High Speed Standard Precision. The calculation is instantaneous however the last iteration (C) is $3.48 higher than the initial guess of $3,500. You will notice at two decimal places this is not a concern, for total fees of $3,500 the values of 15.42% and 16.01% are the same regardless of which Dollar Discount Option selected.

(Screenshot 11)

 

 

Discounting a Loan

FULL DISCOUNTING TO THE END OF THE AMORTIZATION PERIOD:
Bob signs a ten year loan at 6% and the same day Bill offers to buy the mortgage but wants an 8% interest rate. Another way of looking at these numbers is that Bobs outstanding balance is exactly $100,000 with exactly 120 monthly payments left. If Bill buys this mortgage what discount should Bob give him so that Bill gets his 8%. Using the discounting software input the data as shown below;

(Screenshot 12)

The amortization and the discounting term are both equal to 120 months because we are discounting the mortgage to the end of the amortization period. Comparing the two CALCULATORS screens show the monthly payment remains the same for the next 120 months, the only difference being the 8% interest rate. The PV-FV screens compare within pennies to the two negative amortization schedules. Bill would pay Bob $91,504.74 (B) for the mortgage which represents a discount of $8,495.26 (C).

(Screenshot 13)
(Screenshot 14)

PARTIAL DISCOUNTING ONLY TO THE TERM:
What if Bill only wanted a discount for the first 60 months instead of the full 120 months. This scenario is often because the house may be sold in 5 years or the loan contract (the TERM in Canada) may expire in 60 months.

(Screenshot 15)

The balance at the 60th month, without the discount is $57,425.98 as seen by box A above. It can also be seen from the SPREADSHEET screen below. The difference of two cents is because the value in box A was arrived at by an iterative method. Agreement to the penny is often impossible in most financial calculations.

(Screenshot 16)

Bill would buy this mortgage for $93,298.49 (box B) and the SPREADSHEET below shows the balance at the 60th payment is $57,425.58 which is within one dollar which is close enough for a discount calculation. The payments for the first 60 months are identical, Bills balance is still the same as it would be without a discount, but paid only $93,298 for the mortgage instead of $100,000.

(Screenshot 17)

 

 

Calculating YIELDS on a Discounted Mortgage

The yield of a mortgage (annual rate of return) bought at a discount or premium can be calculated given the original mortgage Principal (Present Value), the annual interest rate, compounding method the periodic payment and the amount paid for the mortgage.

An investor wishes to purchase a $100,000 mortgage taken out at 9% over 20 years. Since the mortgage was issued , 42 monthly payments have been made. The loan is to be paid in full (the balloon payment is the 60th payment plus the balance) at the end of the fifth year (60 months).

QUESTION: What is the yield if the purchase price of the mortgage is $79,000?

First lets find out what the balance is after the 42nd payment. The payment below was automatically calculated after the first three values were typed in the CALCULATOR.

(Screenshot 18)

Basically we are going to calculate a yield by paying $79,000 for a $92,640.29 mortgage for the next 18 months. The $92,640.289 mortgage has an amortization period of 20 years minus 3.5 years which is 16.5 years or 198 months and it is considered to be advanced on 04/10/2006. The discount is $92,640.29 - $79,000 = $13,649.93

Notice on 10/10/2007 the both balances blue and purple lines are the same, $88,706 (within 12 cents).

(Screenshot 19)

ANSWER: The yield achieved is 20.73% and the balloon payment on 10/10/2007 after the 60th payment based upon the initial mortgage is $88,706.78 (in blue) or $88,706.90 (in purple) based upon the 18th payment of the new scenario as shown below by the DISCOUNTING program.

(Screenshot 20)

Another way of visualizing this is to prepare a time line as shown below.

(Screenshot 21)

 

 

 


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