# LO 13.3: Describe methods for addressing the possibility o f negative short-term

LO 13.3: Describe methods for addressing the possibility o f negative short-term rates in term structure models.
Note that the terminal nodes in the two-period model generate three possible ending rates: rQ + 2a V dt, rQ, and rQ – 2a V dt. This discrete, finite set of outcomes does not technically represent a normal distribution. However, our knowledge of probability distributions tells us that as the number of steps increases, the terminal distribution at the nodes will approach a continuous normal distribution.
One obvious drawback to Model 1 is that there is always a positive probability that interest rates could become negative. On the surface, negative interest rates do not make much economic sense (i.e., lending $100 and receiving less than$100 back in the future). However, you could plausibly rationalize a small negative interest rate if the safety and/or inconvenience of holding cash were sufficiently high.
The negative interest rate problem will be exacerbated as the investment horizon gets longer, since it is more likely that forecasted interest rates will drop below zero. As an illustration, assume a ten-year horizon and a standard deviation of terminal interest rates of 1.2% x VlO = 3.79%. It is clear that negative interest rates will be well within a two standard deviation confidence interval when centered around a current rate of 6%. Also note that the problem of negative interest rates is greater when the current level of interest rates is low (e.g., 4% instead of the original 6%).
There are two reasonable solutions for negative interest rates. First, the model could use distributions that are always non-negative, such as lognormal or chi-squared distributions. In this way, the interest rate can never be negative, but this action may introduce other non- desirable characteristics such as skewness or inappropriate volatilities. Second, the interest rate tree can force negative interest rates to take a value of zero. In this way, the original interest rate tree is adjusted to constrain the distribution from being below zero. This method may be preferred over the first method because it forces a change in the original distribution only in a very low interest rate environment whereas changing the entire distribution will impact a much wider range of rates.
As a final note, it is ultimately up to the user to decide on the appropriateness of the model. For example, if the purpose of the term structure model is to price coupon-paying bonds, then the valuation is closely tied to the average interest rate over the life of the bond and the possible effect of negative interest rates (small probability of occurring or staying negative for long) is less important. On the other hand, option valuation models that have asymmetric payoffs will be more affected by the negative interest rate problem.
M odel 1 Effectiveness
Given the no-drift assumption of Model 1, we can draw several conclusions regarding the effectiveness of this model for predicting the shape of the term structure:
The no-drift assumption does not give enough flexibility to accurately model basic
term structure shapes. The result is a downward-sloping predicted term structure due to a larger convexity effect. Recall that the convexity effect is the difference between the model par yield using its assumed volatility and the par yield in the structural model with assumed zero volatility.
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Model 1 predicts a flat term structure of volatility, whereas the observed volatility term
structure is hump-shaped, rising and then falling.
Model 1 only has one factor, the short-term rate. Other models that incorporate
additional factors (e.g., drift, time-dependent volatility) form a richer set of predictions. Model 1 implies that any change in the short-term rate would lead to a parallel shift in the yield curve, again, a finding incongruous with observed (non-parallel) yield curve shifts.
T e r m S t r u c t u r e M o d e l w i t h D r i f t (M o d e l 2)
Casual term structure observation typically reveals an upward-sloping yield curve, which is at odds with Model 1, which does not incorporate drift. A natural extension to Model 1 is to add a positive drift term that can be economically interpreted as a positive risk premium associated with longer time horizons. We can augment Model 1 with a constant drift term, which yields Model 2:
dr = \d t + odw
Lets continue with a new example assuming a current short-term interest rate, drift, \ , of 0.24%, and standard deviation, a, of 1.30%. As before, the div realization drawn from a normal distribution (with mean = 0 and standard deviation = 0.2887) is 0.2. Thus, the change in the short-term rate in one month is calculated as:
of 3%,
dr = 0.24% x (1/12) + 1.5% x 0.2 = 0.32%
Hence, the new rate, rv is computed as: 5% + 0.32% = 5.32%. T he monthly drift is 0.24% x 1/12 = 0.02% and the standard deviation of the rate is 1.5% x Vl /12 = 0.43% (i.e., 43 basis points per month). The 2bps drift per month (0.02%) represents any combination of expected changes in the short-term rate (i.e., true drift) and a risk premium. For example, the 2bps observed drift could result from a 1.5bp change in rates coupled with a 0.5bp risk premium.
The interest rate tree for Model 2 will look very similar to Model 1, but the drift term, \dt, will increase by \d t in the next period, 2 \d t in the second period, and so on. This is visually represented in Figure 3. Note that the tree recombines at time 2, but the value at time 2, rQ + 2\dt, is greater than the original rate, rQ, due to the positive drift.
Figure 3: Interest Rate Tree with Constant Drift
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M odel 2 Effectiveness
As you would expect, Model 2 is more effective than Model 1. Intuitively, the drift term can accommodate the typically observed upward-sloping nature of the term structure. In practice, a researcher is likely to choose Hence, the term structure will fit better. The downside of this approach is that the estimated value of drift could be relatively high, especially if considered as a risk premium only. On the other hand, if the drift is viewed as a combination of the risk premium and the expected rate change, the model suggests that the expected rates in year 10 will be higher than year 9, for example. This view is more appropriate in the short run, since it is more difficult to justify increases in expected rates in the long run.
and X based on the calibration of observed rates.
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