Here's another way to look at it. Suppose you want to borrow $100,000. You choose a 30-year fixed rate loan at 7.5%, and pay one discount point ($1,000), a 1% origination fee ($1,000), and $350 in other fees. Although the lender is giving you a loan for $100,000, you have paid $2,350 to the lender. Your payments are based on a loan of $100,000, but your net proceeds are only $97,650. Hence your APR is 7.75%. In other words, if you selected a rate of 7.75% and did NOT pay the discount point, the origination fee, or the $350 in other fees, the APR says you would have the same overall value as the 7.5% loan with the $2,350 in expenses.

There are substantial limitations to the APR. In the previous example it was noted that the two loans (7.5% with $2,350 in expenses and the 7.75% with no expenses) were the same value with respect to the APR. However, the APR assumes both loans go to the full term. If the loan is paid off in five to seven years (the average life of a mortgage), the two loans are NOT the same value. The higher-rate loan is the better value. Suppose the borrower with the 7.5% loan sells the house in five years. That borrower has made 60 payments of $694.87 plus the initial expenses ($2,350) for a total of $44,042.20. The borrower with the 7.75% loan has made sixty payments of $711.82 with no initial expenses for a total of $42,709.20. The higher-rate loan costs $1,333 LESS than the lower rate loan. Clearly, in this case using APR to make a decision would be unwise. There are similar cases when a loan with the lower APR may actually cost the borrower more when all factors are considered.

1. In paragraph 1: Calculate:
 The monthly payments on the mortgage
 The monthly IRR (use PMT )
 Show that the APR in the paragraph = monthly IRR * 12
 Compute the EAIR

2. In paragraph 2:
 Compute the monthly IRR, APR and EAIR for both loans (word of warning: the numbers on the Web page are slightly off).
 Which loan is preferable?

3. The Web page claims that loan 2 is preferable because the total payments over 60 months on this loan are lower than that on loan 1. Show that this is wrong.

4. Despite its higher initial cost, you might think that loan 2 is preferable because it has a higher initial loan amount. Calculate how much you would need to borrow with loan 1 in order to receive a net amount of $100,000 (you can do this calculation using Goal Seek, but it can also be done using the formulas of Chapter 2). Doing it this way, show that Loan 1 is preferable.

1. In paragraph 1:

To calculate the monthly payments on the mortgage, we need to use the loan amount, interest rate, and loan term. In this case, the loan amount is $100,000, the interest rate is 7.5% (or 0.075 as a decimal), and the loan term is 30 years.

Using the formula for the monthly payment on a fixed-rate mortgage:

PMT = P * r * (1 + r)^n / ((1 + r)^n - 1)

where PMT is the monthly payment, P is the loan amount, r is the monthly interest rate, and n is the total number of payments, we can calculate the monthly payment:

PMT = 100,000 * 0.075/12 * (1 + 0.075/12)^(30*12) / ((1 + 0.075/12)^(30*12) - 1)

The monthly payment on the mortgage equals $699.21 (rounded to the nearest cent).

Next, to calculate the monthly Internal Rate of Return (IRR), we can use the PMT function in Excel or similar software. IRR calculates the discount rate at which the net present value (NPV) of cash flows is zero.

Using the PMT function, we enter the following parameters:
Rate: 7.5%/12 (monthly interest rate)
Nper: 30*12 (total number of payments)
Pmt: -$699.21 (negative because it represents outgoing cash flow)

The result is an IRR of 0.59324 (rounded to five decimal places). Note that the IRR is expressed as a decimal.

To confirm that the APR in the paragraph is equal to the monthly IRR multiplied by 12, we calculate:
APR = 0.59324 * 12 = 7.11888 (approximately)

Lastly, to compute the Effective Annual Interest Rate (EAIR), we can use the following formula:

EAIR = (1 + IRR)^(12/n) - 1

Here, n is the total number of payments per year. In this case, there are 12 payments per year.

EAIR = (1 + 0.59324)^(12/1) - 1 = 0.80685 (approximately)

2. In paragraph 2:

To compute the monthly IRR, APR, and EAIR for both loans, we would need the relevant loan parameters such as interest rates, fees, and loan terms. The paragraph does not provide specific values for these parameters. Therefore, we cannot calculate the requested values without additional information.

Please note that the paragraph mentions that the numbers on the webpage are slightly off, so caution should be exercised when using them.

Regarding which loan is preferable, it would typically depend on individual preferences and financial goals. Different loans offer varying trade-offs between upfront costs and long-term savings. The choice between loans would require a thorough evaluation of factors such as the borrower's financial situation, future plans, and priorities.

3. The statement claiming that loan 2 is preferable because the total payments over 60 months are lower is not correct as mentioned in the paragraph. The calculation comparing the total payments of both loans over 60 months shows that loan 1 actually costs $1,333 less than loan 2. Therefore, loan 1 would be more cost-effective in this scenario.

4. To calculate how much you would need to borrow with loan 1 in order to receive a net amount of $100,000, you can use the formula for present value (PV):

PV = FV / (1 + r)^n

Here, PV is the present value (loan amount), FV is the future value (net amount), r is the interest rate per period (monthly rate), and n is the number of periods (loan term in months).

We rearrange the formula to solve for PV:

PV = FV / (1 + r)^n

Plugging in the provided values:
FV = $100,000
r = 0.075/12 (monthly interest rate)
n = 30*12 (total number of payments)

PV = 100,000 / (1 + 0.075/12)^(30*12)

The calculated present value (loan amount) is approximately $93,184.67.

By comparing the loan amounts required to achieve a net amount of $100,000, we can see that loan 1 is more favorable as it requires a lower initial loan amount.