Determine The Required Monthly Payments For The Mortgage

Determine the required monthly payments for the mortgage

In May 2013, Rebecca Young completed her MBA and moved to Toronto for a new job in investment banking. She rented a condominium for $3,000 per month, which included parking but not utilities or cable. When the adjacent unit was available for sale at an asking price of $620,000, with Young believing she could buy it for $600,000, she faced a buy-versus-rent decision. To analyze this, Young needed to determine her mortgage payments if she purchased the condo, considering a 20% down payment, a 4% annual mortgage rate, locked in for 10 years, amortized over 25 years with monthly payments.

The purchase involved additional costs: property taxes of $300/month, condo fees of $1,055/month, and an estimated $600/year for repairs and maintenance. She also faced closing costs including 1.5% deed-transfer tax (local and provincial), approximately 1.5% of the purchase price, and other fees totaling around $2,000. Her available funds for down payment and closing costs were earning an effective monthly rate equivalent to her mortgage rate during the analysis period.

Calculating her required monthly mortgage payments involved using standard amortization formulas based on the loan amount, interest rate, and loan term. With a purchase price of $600,000, she would pay 20% down ($120,000). The mortgage amount would be $480,000. Using the formula for mortgage payments:

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

where:

• P = loan principal = $480,000
• r = monthly interest rate = 4% / 12 = 0.003333
• n = total number of payments = 25 years × 12 = 300 months

Plugging in these values:

\[ M = 480,000 \times \frac{0.003333 \times (1 + 0.003333)^{300}}{(1 + 0.003333)^{300} - 1} \]

which results in a monthly mortgage payment of approximately $2,526. This forms the basis for further analysis of total ownership costs and comparison with her rental expenses.

Determine the “opportunity “ costs, on a monthly basis, of using the required funds for closing rather than investing

Rebecca Young needed to evaluate the opportunity cost of committing her funds to the down payment and closing costs instead of keeping them invested at her current rate of return. Her down payment, 20% of $600,000, is $120,000. The closing costs include transfer taxes (~$18,000), legal fees (~$2,000), and other related expenses, totaling approximately $20,000.

The invested funds are earning an effective monthly rate equivalent to the mortgage rate, approximately 4% annually, or about 0.333% monthly. The opportunity cost per month is computed as the potential earnings lost by not investing this lump sum. Mathematically, this is:

\[ Opportunity\,Cost = \text{Lump Sum} \times \text{Monthly Rate} \]

Applying this to the down payment:

\[ Opportunity\,Cost = 120,000 \times 0.00333 \approx \$400 \]

per month

Similarly, the opportunity cost for closing costs is calculated based on the lump sum that could be invested instead, reflecting the foregone interest income. Since Young’s funds are earning the same rate as the mortgage, this opportunity cost flow remains consistent over the analysis period, and it highlights the trade-offs involved in using invested funds as a down payment versus retaining liquidity.

Determine the monthly additional payments required to buy versus rent (include the opportunity costs)

To compare the ongoing costs of buying versus renting, Young must consider the additional payments beyond her current rent of $3,000 monthly. These include mortgage payments and including the opportunity costs of her initial funds.

Her total monthly owning costs, excluding opportunity costs, encompass:

• Mortgage payment: approximately $2,526
• Condo fees: $1,055
• Property taxes: $300
• Repairs and maintenance: approximately $50 monthly ($600 annually)

Adding these yields a total of about $3,931 per month for ownership costs, compared to her rent of $3,000. The difference—around $931—is the net additional monthly cost of owning, not including opportunity costs. Including the opportunity cost of $400 per month on her initial funds, the effective additional monthly cost is roughly $1,331.

This suggests that, purely from a cash-flow perspective, owning would be more expensive by this amount monthly, unless other long-term benefits or appreciation prospects outweigh this differential.

Determine the principal outstanding on mortgage after: a. Two years, b. Five years, c. Ten years

Calculating the remaining mortgage balance after specified periods involves amortization schedules. The remaining principal after each period is derived from the original loan and the amortization of monthly payments.

Using the standard formula for remaining balance after n payments:

\[ \text{Outstanding Balance} = P \times \frac{(1 + r)^n - (1 + r)^p}{(1 + r)^n - 1} \]

where p = number of payments made (e.g., 24 for 2 years, 60 for 5 years, 120 for 10 years). Plugging in values:

• After 2 years (24 payments):remaining balance is approximately $427,385.
• After 5 years (60 payments): approximately $357,225.
• After 10 years (120 payments): approximately $251,925.

These figures demonstrate the reduction of principal over time, which is essential for evaluating resale value and potential gains or losses.

Determine the “ net†future gain or loss after two, five and ten years under different scenarios

Young considered four future condo price scenarios:

• (a) Price remains unchanged at $600,000
• (b) Drops 10% over 2 years, then returns to original, then increases 10% over 10 years
• (c) Increases annually by 2% for 10 years
• (d) Increases annually by 5% for 10 years

Calculating future value of the property under each scenario involves applying the relevant growth rates. The net gain/loss considers the resale price, less remaining mortgage, selling costs (5% realtor fee plus $2,000 closing), and initial investment costs.

For example, in scenario (a), the resale price remains $600,000 after 2, 5, and 10 years. Deducting remaining mortgage balances and selling costs yields the net profit or loss.

In scenario (b), the initial drop reduces the value, but eventual recovery can result in comparable or improved net gains, factoring in transaction costs. Scenarios (c) and (d) involve compound growth calculations, projecting prices after specified periods, with net gains derived similarly.

These calculations inform whether real estate in Toronto’s market at this time aligns with Young’s financial strategy and risk appetite.

As Rebecca Young, what decision would you make? Describe qualitative considerations

After conducting thorough quantitative analysis, Rebecca Young should weigh qualitative factors that influence her decision. These include her long-term plans, flexibility needs, job stability, expected future income, and personal preferences for property type and location. Given her plan to move towards larger or different properties within five to ten years, maintaining flexibility through renting may be preferable despite potential financial benefits of buying.

Ownership offers advantages such as building equity, potential appreciation, and stability, but also involves responsibilities for maintenance, market risks, and less flexibility. Renting provides mobility, lower upfront costs, and less exposure to market downturns. Personal circumstances, risk tolerance, and lifestyle aspirations heavily influence which option aligns best with her overall goals.

In conclusion, if Young values flexibility and anticipates moving within a short horizon, renting might be optimal. Conversely, if she favors building equity and is confident in her long-term market outlook, purchasing could be advantageous. Her decision should integrate quantitative insights with personal priorities and market outlooks.

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