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Refunding Efficiency
By Jerry Hoberg
Jul 17, 2002, 18:03

Savings Comparison between Threshold Savings and Option Valuation Refunding Methods

Introduction
Historically, issuers have picked their refunding candidates primarily using the percent savings threshold to determine their refundability. On the surface, this method is fairly attractive because it allows issuers to realize savings immediately. On the other hand, even some of the best refunding candidates, with the highest percentage of PV savings, could have potential for much higher savings in the future. In order to quantify these potentially forgone savings and to help municipal issuers to reduce and restructure their debt more efficiently, Ferrand Jordan & Co. has developed a sophisticated model for picking the best refunding candidates based on the option valuation concept.
The first section of the paper will outline the main concepts behind the option valuation model and will briefly explain how these concepts were addressed in our model. The second part of the paper will address the advantages of the option valuation model over the refunding models currently used by most states. Finally, we will present the results of testing our model on the outstanding debt of a major state issuer.

I. Option Valuation Model
The option valuation model is developed on the assumption of interest rate volatility and the option value is calculated using the concept of the binomial tree of future forward rates. This binomial tree allows us to consider a number of interest rate scenarios and to calculate the value of PV savings given each of these scenarios. Namely, the binomial tree is developed based on today’s yield curve and the semi-annual interest rate volatility (estimated at 11%), so each of the nodes in the tree represents a possible future half-year forward rate. PV savings are computed at each node of the binomial tree, and discounted back to present. Thus, a probability-weighted average of these results provides the mathematical expectation of savings in each period. The highest of these expectations is the value of the call option.
The efficiency of the refunding is then computed as the ratio of the PV savings (if refunding is performed now) and the option value. If efficiency is greater than 90%, the refunding should be carried out immediately, and otherwise it is better to wait.

II. Advantages of Option Valuation Model Over Threshold Savings Refunding Model
In this section we will outline the advantages of option valuation based refunding over the percentage savings threshold based refunding that most issuers currently use.
· Flexibility of refunding threshold:
The danger of using rigid refunding threshold comes from two sources: 1) forgoing savings because they are below a certain threshold and letting bonds mature without realizing any savings at all, and 2) forgoing even bigger future savings on issues that satisfy the threshold. In both cases the option valuation model allows issuers to capture more savings.
Varying the threshold over time can alleviate the problem of using a fixed threshold. Some municipal issuers employ this strategy, but only to a limited extent . Since the option value consists of the intrinsic value and the time value, the efficiency changes over time – as the issue’s maturity gets closer, the time value of the option reduces and the efficiency rises. Because the efficiency of refunding changes over time even if everything else is held constant, our model varies the refunding threshold in an organized manner.
· Timing of refunding:
In addition to computing the efficiency of refunding today, the option value model gives issuers the sense of when is the best time to refund bonds (if not today). Asking only a simple question if PV savings are above or below a certain threshold, the issuers using the threshold model are not able to evaluate if they are possibly forgoing higher savings in future.

III. Results of Testing the Option Valuation Model
We tested our model on the outstanding debt of a major issuer as of June 1993. Lacking a reliable database of the issuer’s outstanding debt in 1993, we tried to reproduce the database, which would be as close to the real one as possible. In order to do this, we performed the refunding with 3% savings threshold of all bonds dated before 1/1/93 based on June 1992 interest rates. The bonds outstanding after this refunding were included in the database on which we performed our comparison between the threshold and the efficiency model.
The setup of our analysis is as follows: in June of 1993 both solution methods start with the same database. At that time, two separate databases are created for keeping the results of both refunding strategies. The refunding threshold for the optimal solution is taken to be 3%, while the threshold for the option valuation method is 90% efficiency. Each year the qualifying candidates are refunded based on the yield curve that the issuer faced in June of the given year . The next year’s analysis is then carried out on the remainders in their respective databases. The analysis ends with the final refunding in June of 1998.
Figure 1: PV savings in each period are discounted back to 1993 (the discount rate that is used is the 1993 bond arbitrage yield). The discounted total is $38,814,255 for the optimal solution and $53,116,745 for the efficient solution. The yellow color for 1999 depicts the refunding potential still remaining under both models. Note that the refunding of 1999 is performed on different databases, but with the same threshold requirement of 0% (efficient solution is not ran in 1999).

Our solution indicates that the efficient solution yields 36.84% more savings than the optimal solution. In addition the efficient solution leaves $1,591,484 of unrealized savings vs. $3,771,225 for optimal solution that could be captured if another refunding is done in 1999.
The reason for extending our analysis to 1999 is to measure the amount of savings that could be still extracted after the five-year run of both refunding methods. For that purpose, we could have chosen some other refunding threshold or even some efficiency threshold, but we believe that 0% threshold gives us the best sense of the future refunding possibilities . So, even though the threshold method leaves a debt portfolio with slightly more refunding opportunities than the efficient solution, the efficient solution is ahead of the game by far.
Of course, before we conclude that the efficient solution is better, we must look at the sizes of our refunding issues.
Figure 2: Actual refunding issue size is graphed for each period, and these values are not discounted back to 1993. The reason for not discounting them is to give issuers a sense of how much money they would have to raise each year to make the refundings possible. The gross amount of refunding bonds is $1,141,400,000 for optimal and $1,425,925,000 for efficient solution. Again, the yellow bars in the graph indicate the required sizes of refunding issues with 0% of required threshold.

The difference between required size for efficient and the threshold solution is substantial – 24.93%. One could argue that such a difference in required bond sizes is offsetting some of the benefits of higher savings provided by the efficient solution. However, a better measure of the refunding issue sizes, and that is their PVs discounted back to 1993, gives us a much different picture: the efficient solution requires $1,289,588,100 of 1993 dollars, vs. $1,092,675,676 for the threshold solution. This difference is only 18.02% and is far from offsetting the 36.84% difference in savings. When gross savings from the efficient model and gross savings from the threshold model are compared with the gross refunding bond sizes, the savings percentages are 4.12% and 3.55% respectively, putting our efficient model far ahead of the savings threshold model that most states use now. The breakdown of savings percentages by year is given in Fig. 3.

Figure 3: The graph depicts the percentage savings that both models yield in each of the 6 years of our analysis. Except for the first year when the optimal solution is slightly ahead of the efficient solution, the efficient solution is yielding higher savings percentages in other years. Yellow bars are again savings percentages that can be stilled extracted from the outstanding debt if the 0% savings threshold is employed.

IV. Summary of the Analysis
The study we have performed suggests strong advantages of using the option-pricing model in determining the refunding candidates. We concluded that the issuer could have saved 36.84% more if it had used our model instead of the regular threshold model in picking bonds to refund. In addition to yielding higher savings, our model has proved to yield higher percentage savings (4.12% vs. 3.55% for the threshold model).
In the summary, an issuer choosing between employing the option valuation method and the regular savings-threshold method will often have to choose between realizing savings immediately or waiting for better refunding opportunities in future. While generating interest rates using the binomial tree will be correct on average and in long run, some savings opportunities can be forgone in the short-run due to sudden hikes of interest rates. Our binomial forward rate tree accommodates both increases and decreases of interest rates at each node of the tree with equal probability.


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