Mortgage Refinance and Option Pricing

Being a blogger with a contact form, I often receive PR outreach messages. They want me to write about what they are trying to promote. I ignore most of those. Once in a while, I get something worth reading.

Andrew Kalotay Associates is a fixed income analytics and debt management advisory services company in New York. They sent me a special report they wrote for Mortgage Bankers Association, the industry trade group.

A Financial Analysis of Consumer Mortgage Decisions, Andrew J. Kalotay and Qi Fu

The 60-page report attempts to answer three common questions in getting a mortgage. Two of which are of interest to me.

1. How many points should you pay on your (fixed rate) mortgage? No point, 1 point, 2 points, or a no-cost loan with negative points?

Most of the calculators on the web, including ones created by The Mortgage Professor, use break-even analysis. They show you how long you will break even between different loans. Then they ask you “how long will you keep the house?” That’s actually the wrong question to ask. The correct question should be “how long will you keep the loan?” because you can refinance if the rate goes down.

The Kalotay and Fu paper takes a different approach. It applies “industrial strength” option pricing model to these loan decisions.

A mortgage loan in the U.S. is basically a callable bond. When a borrower gets a loan, the borrower issues a bond to the lender. The borrower is also free to call the bond at par at any time. The call option embedded in the bond has value. The value of the call option differs depending on the interest rate, interest rate volatility, and the remaining term of the loan.

Comparing different loans means comparing the APR for the loan payments, the upfront cost, and the value of the call option. Kalotay and Fu call it option-adjusted APR or APRPlus.

They created a mortgage points calculator for comparing APRPlus. I got some current quotes for a 30-year fixed rate loan from a lender and I calculated the regular APR and the APRPlus:

Rate Points & Fees as % of loan APR APRPlus
4.500% 2.03% 4.601% 4.07%
4.625% 1.61% 4.720% 4.09%
4.875% 0.42% 4.973% 4.08%
5.000% 0.04% 5.099% 4.09%

The 5% loan is a no-cost loan. There is little upfront cost, but it has the highest rate and the highest APR. The 4.5% loan has a high upfront cost with a low APR. After adjusting for the value of the call option, the option-adjusted APRs for all these loans are practically identical.

Even after adjusting for the value of the call option, I think it still biases toward the loan with a higher upfront cost, because the typical holding period for a 30-year loan is much shorter than 30 years even without refinancing.

2. Should you refinance now?

Kalotay and Fu apply the same option pricing framework to the mortgage refinancing decision. When you refinance to a lower rate, you also reduce the value of the embedded option in the loan. The cost savings from refinancing must be sufficient to cover the loss of the option value. This is the same decision framework corporate treasurers use when they decide when to call their bonds.

Because homeowners can’t easily calculate the value of the call option in their loan, Andrew Kalotay Associates developed a mortgage refinancing calculator for us. You enter the information about your current loan and the refinancing offer. The calculator will calculate a Kalotay Refi Score (the higher the better), together with a recommended action. You will get “Don’t Even Think About It!”, “Not Yet!”, “OK, But Not Optimal”, or “Go For It!”

For example, if someone has a $200k mortgage at 5.25% with 28 years to go, refinancing to a 5.0% loan with $2,000 closing cost will get a “Not Yet!”. Cutting down the closing cost to $1,500 will get a “OK, But Not Optimal”. If the closing cost can be reduced to $950 or less, the calculator will say “Go For It!”

I like these option-aware calculators. The mortgage rates are low once again. Try them and see if you should refinance. I already sent out an e-mail to the loan officer I used last time. When the rate hits my target, they will let me know.

Refinance Your Mortgage

Mortgage rates hit new lows. I saw rates as low as 3.25% for 30-year fixed, 2.625% for 15-year fixed, with no points and low closing cost. Let banks compete for your loan. Get up to 5 offers at


  1. Chuck says

    That’s an interesting idea, however you can sell (or buy to close) an exchange traded option for its intrinsic + extrinsic (time or option) value. But you can’t refinance a mortgage and recover the time value, you get intrinsic value (principal) only. When you pay up-front costs for a mortgage, that money is gone forever.

  2. Raghu says

    Looks like 5 ARM is 3.875. I have 4.875% fixed right now. Do you think going to 5 ARM from fixed is good idea?

  3. Harry Sit says

    Raghu – Unless I’m certain I won’t keep the loan for more than five years (or shortly thereafter), I won’t do the 5/1 ARM. I prefer to step down when the rates go lower and lock in. If I’m close to the end of my mortgage term though, 5/1 ARM is a great deal. If the rates are high after five years, I just pay it all off.

  4. AM says

    Regarding the refinance calculator, I wonder why it doesn’t let you change to a shorter term. Like if I have 20 years left and I want to switch to a 15-year mortgage.

  5. enonymous says

    I tried this calculator since I think that it addresses one of the most important and overlooked aspects of mortgages – i.e., the ‘call’ option for the borrower.

    Yet, I was really surprised by the results, and don’t know whether to believe them. In particular, it does not seem to adjust for the fact that the new mortgage may not be held for all 30 years (or 15 years) either.

    For instance, I put in a current term of 28 years remaining, a rate of 5.75% and a new loan with a rate of 5.25% and a term of 30 yrs, all for a 625,000 loan. Even putting in 5,000 of ‘upfront costs,’ it continues to rate the refinance at 100% – go for it!

    I’m actually nearing in on closing this exact scenario, but for $0 out of pocket and 0 points. If I had to pay $5,000 the hurdle would be rather high, and it would take around 2 years just to break even, ignoring the optionality of the mortgage to me the borrower. If rates drop in the next two years, it is the loss of that optionality that is critical – and a ‘hidden cost.’ Thus, what matters most is the length of time that the new mortgage is held. At present, the calculator will only let us use the same length of time (or longer) than the current mortgage. Without being able to adjust this critical detail, the optionality issue is not addressed.

  6. Harry Sit says

    enonymous – I would use the no-cost offer as my baseline and compare it against other offers with an upfront cost, because refinancing to a no-cost loan is a no-brainer. There is no cost to recover. If I enter $625k 30-year @ 5.25% as the current loan and compare against say 5% with $5,000 as the new loan, the calculator says “OK but no optimal”. That means I should choose the no-cost loan.

  7. enonymous says

    what I’m referring to is that with 5.75% as my current loan (28 years left) and 5.25% as the new loan (30 years left), even with $5,000 in upfront costs it gives me a 100% go for it!

    Now my scenario is obviously a total no brainer – zero costs and major savings (other than the headache of going through the refi). But I’m rather surprised that this calculator would indicate that with $5,000 in upfront costs, a $230/month savings is given an obvious green light. Clearly, if the loan is held for 30 years it is a no brainer. But I thought this more sophisticated calculator would be able to have a different input for the length of time that you actually hold the new loan.

    For instance, with 5k in upfront costs, I have to hold the loan for 20+ months before I ‘win’ (in a very unsophisticated analysis – ignoring tax consequences, interest on savings). The optionality of my old loan and new loan have to enter into this analysis, but I don’t see how this calculator is accounting for the fact that 5k in upfront costs can actually be a very bad financial decision in some circumstances.

    I guess what I’m asking is, how does this calculator actually help? I just don’t see it, just like I don’t see how the DecisionAide calculators can ignore the option value of the mortgage…

  8. Harry Sit says

    enonymous – I understand what you are saying. I proposed using the calculator this way:

    (1) Find a no-cost offer with a monthly payment not more than what you are currently paying. Going from 5.75% 28-year to 5.25% 30-year with no upfront cost is a no brainer. If you keep paying what you currently pay, you will pay less interest and pay off faster, with nothing out of pocket for the refi.

    (2) Use that offer as your baseline. Forget about your 5.75% 28-year. Suppose you already have a 5.25% 30-year. Now see if it makes sense to pay some cost for a lower rate.

  9. enonymous says

    ah – I think I see your point now.

    so to truly use this calculator, one should input a no cost refi at the term and rate being offered, then compare it to the same term and refi with the points and closing costs included and the presumably better rate being offered with having to pay those various annoying closing costs and discount points.

    so sort of a two step process I guess…

  10. Qi Fu says

    > Your refinance calculator limits the new loan’s term to equal or longer
    > than the remaining term on the existing loan. Is there a good reason for
    > it? What if someone has a loan with 20 years remaining but wants to
    > refinance to a 15-year loan? Can you fix it? Thanks!

    There are two reasons for this. One, our calculator is designed for
    the most common scenarios, it may not apply to everyone. Of course we
    would like to add more bells and whistles (cash-out refis, etc.) as
    this calculator becomes more widely distributed.

    Two, from a financial theory point of view, when a institution makes
    bond refunding decisions, it doesn’t matter if a 30-yr bond with 28
    years remaining is refunded with a 28- or a 5-yr bond, as long as
    there is sufficient savings, the savings is quantified by the
    difference the fair value between the two bonds given the underlying
    interest rate environment. But in reality, the refunding is more often
    done with a maturity-matched (28-yr) bond, because the cashflows can
    be more easily compared.

    In the mortgage market this is tricky because a 28-year mortgage is
    not available. So to make a maturity-matched cashflow comparison, we
    assume that the mortgagor pays off the 30-yr mortgage after 28 years
    (of course the mortgagor doesn’t have to, but from a valuation
    standpoint this is treated identically). This amount is indicated in
    the “Principal remaining after 28 years ($):” box, and the mortgagor
    can see the month-to-month cashflow comparison (or “Savings per month
    ($):”) for the 28 years.

    Because many people are focused on the “break-even period”, the
    “Savings per month ($):” is misleading. One of the readers commented
    that he/she would need to hold the loan for 20 months to “win”, this
    is disregarding (among other things) the principal remaining after 28
    years. You may need longer than 20 months to “win”, but the
    expected life of a 30-yr loan is 7-10 years.

    The refi calculator does of course take into account the optionality
    and the expected life of the new mortgage. The denominator of the
    efficiency formula is “loss of option value”, meaning the difference
    in the option value of the outstanding and new mortgage. In the case
    of this reader, $5,000 in closing costs may seem like a lot, but it is
    less than 1% of the principal.

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