Refinancing a Commercial Mortgage: Break-Even and Horizon
A commercial borrower has a $300,000 mortgage at 6.0% nominal annual interest convertible monthly, amortized over 30 years with level monthly payments. After exactly 60 payments, the outstanding balance is $279,166 and the current payment is $1,798.65. The lender offers a refinance at 5.25% nominal convertible monthly, again amortized over 30 years. The refinance costs 2 points on the new balance plus $4,000 in closing costs, both paid at closing and added to the principal. The borrower expects to hold the property for seven more years, then sell and repay the loan. The borrower's marginal tax rate is 30%, but for simplicity the analysis is pre-tax. A broker argues that because the monthly payment falls by about $204, the deal pays for itself in under four years. The borrower wants a time-value break-even and an assessment of whether the expected holding period justifies the refinance.
1. Compute the new loan principal, the new monthly payment, and the simple undiscounted payback period in months.
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The new principal includes the outstanding balance, the points and the closing costs: P_new = 279,166 + 0.02 x 279,166 + 4,000 = 279,166 + 5,583.32 + 4,000 = 288,749.32.
The new monthly rate is 0.0525 / 12 = 0.004375. The 360-month annuity factor is a_360 = (1 - (1.004375)^-360) / 0.004375. Now (1.004375)^360 = 4.814, so (1.004375)^-360 = 0.2077 and a_360 = (1 - 0.2077) / 0.004375 = 181.10.
New payment = 288,749.32 / 181.10 = 1,594.40. Monthly saving = 1,798.65 - 1,594.40 = 204.25. Upfront cost = 5,583.32 + 4,000 = 9,583.32. Simple payback = 9,583.32 / 204.25 = 46.9 months.
So the broker's claim of under four years is close but slightly optimistic; the undiscounted payback is about 47 months, or 3.9 years.
2. Compute the true break-even horizon using the new monthly rate as the discount rate, and calculate the NPV of the refinance over the expected 7-year holding period. Should the borrower refinance?
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The true break-even solves 204.25 x a_n = 9,583.32 at i = 0.004375, so a_n = 46.92. Using n = -ln(1 - i x a_n) / ln(1 + i), n = -ln(1 - 0.004375 x 46.92) / ln(1.004375) = -ln(1 - 0.2053) / 0.004365 = -ln(0.7947) / 0.004365 = 0.2298 / 0.004365 = 52.7 months. The time-value break-even is about 53 months, not 47, because the savings arrive over time.
Over 84 months, a_84 = (1 - (1.004375)^-84) / 0.004375. Since (1.004375)^84 = 1.443, the inverse is 0.693 and a_84 = (1 - 0.693) / 0.004375 = 70.17. PV of savings = 204.25 x 70.17 = 14,333. NPV = 14,333 - 9,583 = 4,750.
The refinance is worthwhile if the borrower действительно holds for seven years: the break-even is about 53 months and the expected horizon is 84 months. The margin is positive but not enormous. If the borrower sells earlier than about 4.4 years, the refinance destroys value. The decision should also consider whether the borrower may need to prepay or whether rates could fall further.
3. The lender offers a no-cost refinance at 5.60% instead, with no points or closing costs. Compare this with the 5.25% option over 7 years and recommend one, considering the borrower may sell after 3 years.
Read the worked answer
For the no-cost option, assume the new principal is the outstanding balance, 279,166. The monthly rate is 0.056 / 12 = 0.0046667. The 360-month annuity factor is approximately 174.19, so the payment is 279,166 / 174.19 = 1,602.8. The monthly saving versus the current payment is 1,798.65 - 1,602.8 = 195.85, with zero upfront cost.
For a common comparison, discount both savings streams at 5.0% nominal monthly, i = 0.0041667. Over 84 months, a_84 = (1 - (1.0041667)^-84) / 0.0041667 = 70.75. 5.25% option: PV savings = 204.25 x 70.75 = 14,449; NPV = 14,449 - 9,583 = 4,866. No-cost option: PV savings = 195.85 x 70.75 = 13,856; NPV = 13,856.
The no-cost option is clearly better over seven years because the extra 8.40 per month from the 5.25% loan does not repay 9,583 of upfront cost. Over a three-year horizon, a_36 = 33.39. The 5.25% option has PV savings 204.25 x 33.39 = 6,819, so NPV = 6,819 - 9,583 = -2,764. The no-cost option has PV savings 195.85 x 33.39 = 6,539, which is positive. Recommendation: take the no-cost 5.60% refinance unless the borrower is certain to hold for well beyond seven years and has no prepayment risk.