### A CD Optimization Problem

My kids opened up one-year "bump CDs" earlier this year at a local bank. These are traditional certificates of deposit with an added twist--a "bump" feature that gives the holder the one-time right during the one-year term of the CD to bump the initial interest rate higher to the current market interest rate for the same product if rates rise during the holding period. Basically, there's an option embedded in the CD that allows the holder to benefit from a rising interest rate environment. Because the holder must proactively request a rate increase, my kids (with dad's help, of course) have a decision to make: When should we exercise our right to bump the rate of the CD higher--early on, in the middle of the term, later on? Is there some optimal exercise point during the one-year term?

So far, here is how one-year bump CD rates at our bank have moved since we invested:

March 2005: 3.0% (our initial base rate)

April: 3.1%

May: 3.2%

June: 3.2%

July: 3.2% (current market rate)

Interest on our CDs has been accruing at the initial rate of 3.0% for the five months we have been invested so far. The bump feature currently gives us the right to bump the accrual rate higher to 3.2% for the remaining seven months of the one-year term, but if we do so now we forego the opportunity to benefit from any further rise in interest rates.

The 20 b.p. rise in market rates (from 3.0% to 3.2%) we have seen to date has been the result of a general rise in short-term interest rates, driven by Fed policy. If we thought that short-term rates would remain the same or fall from here, we would exercise the bump option immediately to lock in the extra 20 b.p. and start accruing interest at 3.2%. However, since Chairman Greenspan has signaled that the Fed will continue to notch short-term rates higher over the remainder of the year, our inclination is to wait a little longer before we bump. But how much longer should we wait?

We can put some math to this optimization problem to gain a little more insight (at least for those who are analytically inclined):

Let "t" represent time, ranging from t = 0 (when we opened the CDs) to t = 1 (i.e., one year out, when the CDs mature). Suppose (for simplicity) that market CD rates rise following a linear path: Initial rate + (a x t), where "a" is a constant indicating the speed of increase (e.g., 50 b.p. per year). Then, the incremental benefit we realize over the remainder of the term of our CD from exercising the bump option at time t is given by:

Benefit = (a x t) x (1 - t) [linear scenario]

We would like to maximize our benefit by selecting the best time to exercise the bump option, i.e., the value of t (between 0 and 1) that produces the highest benefit. This can be done either by inspection (using some intuition) or by applying elementary calculus: d(Benefit)/dt = a x (1 - 2t), which equals 0 when t = 0.5, i.e., half way into the one-year holding period.

So, if interest rates rise linearly, the optimal exercise point is six months into our one-year term, i.e., during the next couple of months for the CDs my kids own. However, as is often the case with optimization problems, the result depends on how we choose to model the situation--more specifically, what's important in our problem is the path that interest rates follow. Beyond the simple linear path scenario, another appropriate scenario is to assume that interest rates move higher through a diffusion process typical of random walk modeling:

Benefit = b x (t ^ 0.5) x (1 - t) [diffusive scenario]

Again, differentiating to find the maximum: d(Benefit)/dt = b x [0.5 /(t ^ 0.5) - 1.5 x (t ^ 0.5)]. Setting the first derivative to zero and solving for t, we have: t = 1/3, indicating that one-third of the way into the one-year holding period is the optimal exercise point.

The actual path that market rates follow through March 2006 when the CDs mature is subject to Fed policy, general market conditions, internal decisions at the bank, international events, etc. I believe that short-term interest rates will gradually head higher on the back of Greenspan's indications. I also note that our bank appears to respond to rising interest rates in a stairstep fashion and, since the one-year bump CD rate has been stuck at 3.2% for the past three months, we are probably due for an increase soon. Consequently, I am advising my kids to wait for the next boost in the market rate of the one-year bump CD that our bank offers (possibly to 3.3%?) before exercising their bump option. This will allow them to receive more interest accrual during the latter half of their CD term--hopefully with optimal benefit over the entire one-year holding period. Of course, come next March when the CDs mature we will know whether or not we made the right decision.

So far, here is how one-year bump CD rates at our bank have moved since we invested:

March 2005: 3.0% (our initial base rate)

April: 3.1%

May: 3.2%

June: 3.2%

July: 3.2% (current market rate)

Interest on our CDs has been accruing at the initial rate of 3.0% for the five months we have been invested so far. The bump feature currently gives us the right to bump the accrual rate higher to 3.2% for the remaining seven months of the one-year term, but if we do so now we forego the opportunity to benefit from any further rise in interest rates.

The 20 b.p. rise in market rates (from 3.0% to 3.2%) we have seen to date has been the result of a general rise in short-term interest rates, driven by Fed policy. If we thought that short-term rates would remain the same or fall from here, we would exercise the bump option immediately to lock in the extra 20 b.p. and start accruing interest at 3.2%. However, since Chairman Greenspan has signaled that the Fed will continue to notch short-term rates higher over the remainder of the year, our inclination is to wait a little longer before we bump. But how much longer should we wait?

We can put some math to this optimization problem to gain a little more insight (at least for those who are analytically inclined):

Let "t" represent time, ranging from t = 0 (when we opened the CDs) to t = 1 (i.e., one year out, when the CDs mature). Suppose (for simplicity) that market CD rates rise following a linear path: Initial rate + (a x t), where "a" is a constant indicating the speed of increase (e.g., 50 b.p. per year). Then, the incremental benefit we realize over the remainder of the term of our CD from exercising the bump option at time t is given by:

Benefit = (a x t) x (1 - t) [linear scenario]

We would like to maximize our benefit by selecting the best time to exercise the bump option, i.e., the value of t (between 0 and 1) that produces the highest benefit. This can be done either by inspection (using some intuition) or by applying elementary calculus: d(Benefit)/dt = a x (1 - 2t), which equals 0 when t = 0.5, i.e., half way into the one-year holding period.

So, if interest rates rise linearly, the optimal exercise point is six months into our one-year term, i.e., during the next couple of months for the CDs my kids own. However, as is often the case with optimization problems, the result depends on how we choose to model the situation--more specifically, what's important in our problem is the path that interest rates follow. Beyond the simple linear path scenario, another appropriate scenario is to assume that interest rates move higher through a diffusion process typical of random walk modeling:

Benefit = b x (t ^ 0.5) x (1 - t) [diffusive scenario]

Again, differentiating to find the maximum: d(Benefit)/dt = b x [0.5 /(t ^ 0.5) - 1.5 x (t ^ 0.5)]. Setting the first derivative to zero and solving for t, we have: t = 1/3, indicating that one-third of the way into the one-year holding period is the optimal exercise point.

The actual path that market rates follow through March 2006 when the CDs mature is subject to Fed policy, general market conditions, internal decisions at the bank, international events, etc. I believe that short-term interest rates will gradually head higher on the back of Greenspan's indications. I also note that our bank appears to respond to rising interest rates in a stairstep fashion and, since the one-year bump CD rate has been stuck at 3.2% for the past three months, we are probably due for an increase soon. Consequently, I am advising my kids to wait for the next boost in the market rate of the one-year bump CD that our bank offers (possibly to 3.3%?) before exercising their bump option. This will allow them to receive more interest accrual during the latter half of their CD term--hopefully with optimal benefit over the entire one-year holding period. Of course, come next March when the CDs mature we will know whether or not we made the right decision.

## 4 Comments:

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Good information. Also be weary of bump-up CDs. I've seen many banks offer them as teasers, but then they don't adjust the rates on the terms that people purchased. As a result, there is no opportunity to bump.

Step-ups are much better, but I haven't seen many of these offered, of late.

Very useful material, thank you for the article.

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