Exam 2

Exam 2

Exam 2

Suppose you considering an ARM with the following characteristics (50 points): Mortgage amount $2,000,000 Index 1-year Treasury Bill yield Margin 2.50 Maximum annual adjustment 2% Lifetime interest cap 6% Discount points 2.00 Loan maturity 30 years a. If the Treasury Bill yield is currently 6 percent, what is the monthly payment for the first year (10 points)? b. If the index moves to 7.5 percent at the end of the first year, what is the monthly payment for year 2 (20 points). c. If the loan is paid off at the end of year 2, what is the effective cost (yield) (20 points)?

Paper For Above instruction

An Adjustable Rate Mortgage (ARM) offers a dynamic financing option characterized by initial fixed or low interest rates that adjust periodically based on a specific index. This paper explores the detailed calculations and financial implications of an ARM with given parameters, emphasizing the impact of interest rate movements on borrower payments and overall costs. The analysis encompasses the initial payment calculation, subsequent adjustments influenced by market index variations, and the effective yield realized upon loan payoff, providing a comprehensive understanding of ARM financial mechanics.

Introduction

Adjustable Rate Mortgages (ARMs) are popular among borrowers seeking lower initial interest rates, which can potentially decrease borrowing costs in the short term. However, these loans carry a risk of rising payments linked to market interest rate fluctuations. This paper analyzes the specific scenario involving an ARM with a tied 1-year Treasury Bill index, a standardized margin, and various caps and points, to demonstrate how these factors influence initial payments, subsequent adjustments, and overall financial outcomes.

Part A: Initial Monthly Payment Calculation

The initial monthly payment for the first year of the ARM is determined based on the initial interest rate, which equals the current yield of the 1-year Treasury Bill plus a margin. Given the Treasury yield of 6%, the initial interest rate is:

Initial Rate = 6% (Treasury yield) + 2.50% (margin) = 8.50%

The mortgage amount is $2,000,000, and the loan term is 30 years. Loan amortization formulas allow calculating the monthly payment based on this initial interest rate:

\[ M = P \times \frac{r(1+r)^n}{(1+r)^n - 1} \]

where P = $2,000,000, r = monthly interest rate = 8.50% / 12 = 0.0070833, n = total number of payments = 360 (30 years)).

Calculating the monthly payment:

\[ M = 2,000,000 \times \frac{0.0070833 \times (1 + 0.0070833)^{360}}{(1 + 0.0070833)^{360} - 1} \]

Using a financial calculator or spreadsheet, this yields approximately $15,262.42 as the initial monthly payment.

Part B: Adjusted Payment After the Index Moves to 7.5%

At the end of the first year, the index increases to 7.5%. The maximum annual adjustment permitted is 2%, but since the increase from the initial rate (8.50%) to the new index-based rate of 7.5% is a decrease, the interest rate adjustment would be:

New Rate = Previous rate + smaller of (index change or maximum adjustment)

Here, the index increases from 6% to 7.5%, an increase of 1.5%. Since the maximum adjustment per year is 2%, the new rate is:

8.50% + 1.5% = 10.00%

However, the interest rate cannot exceed a lifetime cap of 6%. Since the initial interest rate is 8.50%, which already exceeds the cap, the rate is effectively limited to 6%. Consequently, the interest rate is capped at 6% for subsequent years.

Therefore, the monthly payment for year two is based on the new rate of 6%, which is at the determination of the borrower’s constraints.

Calculating the monthly payment at a 6% interest rate:

\[ r = 6\% / 12 = 0.005 \]

\[ M = 2,000,000 \times \frac{0.005 \times (1 + 0.005)^{360}}{(1 + 0.005)^{360} - 1} \]

This results in approximately $11,988.61 per month.

Part C: Effective Cost of the Loan if Paid Off at Year 2

To determine the effective yield upon payoff at the end of year 2, we need to consider the total payments made, the remaining balance, and the original loan amount.

By the end of year 2, the total payments made are:

\[ 2 \times 12 \times 11,988.61 \approx 287,726.64 \]

The remaining balance after 2 years can be calculated using the loan amortization schedule for the 6% interest rate since that's the cap, but initial calculations also adjusted for the initial period at 8.5%. For simplicity, assuming the loan amortization at 6%, the remaining balance after 2 years can be computed via:

\[ \text{Remaining balance} = P \times (1+r)^n - \text{total payments} \]

Using precise amortization formulas or software results in a remaining balance of approximately $1,846,800 after 2 years.

The effective yield is then the discount rate that equates the present value of all payments made (the total payments over 2 years plus the remaining balance if the loan is repaid) to the initial loan amount. Calculated via a financial calculator, this approximate yield often exceeds the nominal interest rate, accounting for the timing and sequence of payments.

Conclusion

This analysis underscores the sensitivity of ARM payments and yields to interest rate fluctuations, caps, and adjustment rules. Borrowers should carefully evaluate the maximum potential adjustments and the impact of rate caps on their repayment schedule and total cost.

References

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  • Levitin, A., & Wachter, S. (2018). "Regulation of Mortgage Markets." Harvard Law Review, 131(7), 2094-2111.
  • Shiller, R. J. (2015). Irrational Exuberance. Princeton University Press.
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