Analysis of the Call Feature on Municipal Debt.Including a call option in a municipal bond issue can have a significant impact on interest costs. This article discusses the circumstances CIRCUMSTANCES, evidence. The particulars which accompany a fact. 2. The facts proved are either possible or impossible, ordinary and probable, or extraordinary and improbable, recent or ancient; they may have happened near us, or afar off; they are public or that lead to higher interest rates for bonds with call features, and it attempts to aid issuers in deciding when to include a call feature. The features of a bond issue have a very significant impact on the cost of borrowing, and the list of choices an issuer must make with respect to bond features is long and complex. For example, a bond issue can be variable rate or fixed coupon, and it may or may not contain a sinking fund sinking fund, sum set apart periodically from the income of a government or a business and allowed to accumulate in order ultimately to pay off a debt. A preferred investment for a sinking fund is the purchase of the government's or firm's bonds that are to be paid . Further, some municipal debt is taxable while most is not. This article will focus on the call feature, which allows the issuer to redeem redeem v. to buy back, as when an owner who had mortgaged his/her real property pays off the debt. The term also refers to paying the amount due and all charges after a foreclosure (due to failure to make payments when due) has begun. outstanding bonds prior to their stated date of maturity at a specified price. It will present an analysis of the impact a call feature can have on the interest cost of a municipal bond issue, and based on this analysis, it will attempt to aid issuers in deciding whether or not to include a call option. Recent advances in the modeling of options on debt instruments permit precise calculations to predict the impact of a call option, depending on market conditions and details of the call and the bond itself. The next section provides discussion of the factors that may influence the decision to include a call and some helpful tips for making the decision. The following section will discuss how the value (and yield) of a callable bond Callable Bond A bond that can be redeemed by the issuer prior to its maturity. Usually a premium is paid to the bond owner when the bond is called. Also known as a "redeemable bond". Notes: The main cause of a call is a decline in interest rates. differs from a noncallable bond Noun 1. noncallable bond - a bond containing a provision that the holder cannot redeem the security before a specific date (usually at maturity) bond certificate, bond - a certificate of debt (usually interest-bearing or discounted) that is issued by a government . Using a well-known well-known adj. 1. Widely known; familiar or famous: a well-known performer. 2. Fully known: well-known facts. option pricing model option pricing model A mathematical formula for determining the price at which an option should trade. The model expresses the value of an option as a function of the value of the underlying asset, length of time until maturity, exercise price, yields on , the final section will provide actual computations of how many more basis points a callable Callable Applies mainly to convertible securities. Redeemable by the issuer before the scheduled maturity under specific conditions and at a stated price, which usually begins at a premium to par and declines annually. issuer should expect to pay over a noncallable Noncallable Securities that cannot be called by the issuer prior to maturity. Notes: Noncallable securities include preferred stocks and bonds. These securities usually offer lower yields to investors due to their reduced risk. issuer. It will be shown that when aspects of the call and market conditions lead to a large value for the call option, small changes in any factor can have a large impact on the spread between callables and noncallables. Costs and Benefits of a Call Feature The decision to include or not include a call feature depends on several factors. In the final analysis, the decision is based on the likelihood of a number of events that are difficult, if not impossible, to forecast. The issuer must decide if the probabilistic (probability) probabilistic - Relating to, or governed by, probability. The behaviour of a probabilistic system cannot be predicted exactly but the probability of certain behaviours is known. Such systems may be simulated using pseudorandom numbers. benefits of including the call outweigh out·weigh tr.v. out·weighed, out·weigh·ing, out·weighs 1. To weigh more than. 2. To be more significant than; exceed in value or importance: The benefits outweigh the risks. the costs. The most obvious benefit is the ability to refund TO REFUND. To pay back by the party who has received it, to the party who has paid it, money which ought not to have been paid. 2. On a deficiency of assets, executors and administrators cum testamento annexo, are entitled to have refunded to them legacies the debt at a lower rate if interest rates decline. One can quantify Quantify - A performance analysis tool from Pure Software. the savings if interest rates decline, but the issuer has no sure way of knowing if interest rates will decline and, furthermore, cannot know how much they will decline. Nonetheless, let us start with some of the most obvious and commonly cited reasons for corporate issuers of debt to include a call, where the same logic may be applied to municipal bonds. In corporate finance theory, an issuer will be more likely to include a call if interest rates are comparatively high at issuance and the corporation expects a strong likelihood of a future decline in rates that will make exercising the call profitable. A difficulty with this is that many academics and professionals believe that the market for debt instruments is quite efficient, which implies that the "buy" side of the market has similar expectations that interest rates are likely to decline, and they will demand that the issuer pay out a higher yield for the right to call. That is, the issuer cannot reap the benefits of the strong likelihood of a decline in interest rates and subsequent call without paying for it up-front up-front or up·front Informal adj. 1. Straightforward; frank. 2. Paid or due in advance: up-front cash. adv. in the form of greater interest costs. Another reason for the inclusion of a call feature in corporate finance is to reduce the cost of asymmetric information Asymmetric Information Information available to some people but not others. Notes: In other words, the asymmetric information is held by only one side, meaning someone is keeping a secret. . Assume that a firm has strong positive information about its prospects that it cannot credibly cred·i·ble adj. 1. Capable of being believed; plausible. See Synonyms at plausible. 2. Worthy of confidence; reliable. signal to the market. In such a case, the firm will be forced to issue bonds at a greater interest cost than justified by its true prospects, but it can include a call feature to hedge. Later, when the strong prospects are realized, the firm can call the bonds and replace them with debt that reflects its improved credit quality and allows lower interest costs on the replacement debt. Although the theory of municipal debt call inclusion has not been modeled to the same degree as corporate bonds, the same principles can be applied. Municipal bond issuers may be more inclined to include calls if interest rates are high by historical standards and expected to decline. Also, if the municipal issuer thinks that the market is requiring too great a yield to be consistent with the credit quality and future prospects of the issuer, a call can make favorable fa·vor·a·ble adj. 1. Advantageous; helpful: favorable winds. 2. Encouraging; propitious: a favorable diagnosis. 3. refunding Reimbursing funds in restitution or repayment. The process of refinancing or borrowing money, ordinarily through the sale of bonds, to pay off an existing debt with the proceeds derived therefrom. at a later date quite easy. For example, the issuer may include a call option if they expect development in the area to greatly enhance the tax base and revenue prospects. Beyond these, factors such as size of the bond issue, maturity of the bond, and the total amount of debt outstanding can play a role in the call/no call decision. [1] Smaller issuers should realize that there are considerable fixed flotation costs Flotation Cost The costs associated with the issuance of new securities. Notes: Flotation costs include both the underwriting spread and the costs incurred by the issuing company from the offering. for replacing the debt that must be paid whatever the size of the issue. Savings from replacing the debt increase with issue size as a differentially lower rate is applied to more principal. That is, small issuers may not be able to generate enough savings from a call to justify a call feature. (Flotation costs include accounting and legal fees, registration fees, printing costs, etc.). Issuers are likely to find more benefit in making longer maturity bonds callable. This is because there is a greater likelihood of wanting to replace the bonds the longer they are outstanding due to the potential for declines in interest rates or other factors discussed immediately below. Issuers also are likely to find more benefit the greater their leverage. Generally, more levered issuers value the opportunity to restructure their debt more than those that are less levered. For example, at issuance, more highly levered issuers may have to agree to restrictive covenants Restrictive covenants Provisions that place constraints on the operations of borrowers, such as restrictions on working capital, fixed assets, future borrowing, and payment of dividends. which may be distasteful to the issuer, but making the issue callable presents greater opportunity to eliminate them sooner than otherwise. As a general rule of thumb, an issuer will likely have to pay less of a premium to include a call when interest rates are stable and the term structure is comparatively flat. In such a case, the market will not demand a high premium with respect to the pure option value effects described below and the issuer will have the right to call for reasons of restructuring restructuring - The transformation from one representation form to another at the same relative abstraction level, while preserving the subject system's external behaviour (functionality and semantics). and improved credit quality without paying a high price in terms of greater yield spreads over noncallables. Effect of Options on Yield Perhaps the best way to understand the impact of including a call option on a municipal bond is to look at the value of a callable bond as if it were the value of a noncallable bond minus the value of the embedded Inserted into. See embedded system. call. That is, CB = NCB (Network Control Block) A packet structure used by the NetBIOS communications protocol. - OVB OVB Orszagos Valasztasi Bizottsag (Hungarian: National Election Commitee) OVB Novosibirsk, Russia - Tolmachevo (Airport Code) OVB Outer Vascular Bundle OVB Overflow Flag for Accumulator B in STO where CB is the value of a callable bond, NCB is the value of a noncallable bond, and OVB is the option value of the embedded call. A callable bond is worth less to the investor than a noncallable because the issuer holds the option to call the bond when it is to their own advantage (investor's disadvantage). In other words Adv. 1. in other words - otherwise stated; "in other words, we are broke" put differently , the bondholder Bondholder A firm often has stockholders and bondholders. In a liquidation, the bondholders have first priority. bondholder An individual or institution that owns bonds in a corporation or other organization. is short the call option. The greater the value of the call option, the greater the difference in value between callable and noncallable bonds and the greater the spread between callable and noncallable bonds. Valuing a call option is quite complex and before the 1990s, attempts to do so were quite crude. However, in the 1990s, a number of finance theorists developed rigorous and accurate models of debt option value wherein where·in adv. In what way; how: Wherein have we sinned? conj. 1. In which location; where: the country wherein those people live. 2. the models are made to fit the existing term structure to prevent arbitrage arbitrage: see foreign exchange. arbitrage Business operation involving the purchase of foreign currency, gold, financial securities, or commodities in one market and their almost simultaneous sale in another market, in order to profit from price profits (arbitrage free). The calculations for this research will use the Hull and White model (1993) which is based on large "trees" that represent potential movements in interest rates. For simplicity, consider a two-year bond shown in Exhibit 1. The objective is to find the call option value (OVB) and coupon rate Coupon rate In bonds, notes, or other fixed income securities, the stated percentage rate of interest, usually paid twice a year. (icb) which results in the callable bond (CB) selling at par (100) at issuance (time zero). Assume the bond can be called at par. To find these values, 100 + c (c for dollars of coupon payment) is assumed received at time 2 (maturity). Here c is 100 times icb as 100 is par value. The noncallable bond value at issuance ([NCB.sub.0]) is the expected value Expected value The weighted average of a probability distribution. Also known as the mean value. of the bond (based on the probabilities of moving to the different levels of interest rates represented in the tree) and the first coupon (c) properly discounted. The different levels of interest rates are represented by the different nodes. That is, there is some likelihood of moving to the upper node ([NCB.sub.1,u]) where interest rates move upward, some likelihood of moving to the middle node ([NCB.sub.1,m]) where interest rates remain unchanged, and some likelihood of moving to the lower node ([NCB.sub.1,d]) where interest rate s move downward. The character "j" is used to generally represent the upper, middle, and lower nodes below. The call option value at issuance ([OVB.sub.0]) is the maximum of the value of immediate exercise and the present value of waiting to exercise. That is, an issuer will not necessarily exercise even if it is beneficial because the expected value of waiting to exercise later may be greater (here we assume the issuer does not exercise immediately). The value of the callable bond at issuance ([CB.sub.0]) is, again, the value of a noncallable less the call option. The coupon, c, and related interest rate, icb, which makes the callable bond sell at par, is solved by trial and error. Values for NCB, OVB, and CB are solved for in this way (trial and error) except for the period before maturity (time 1 in this special case), where [OVB.sub.1,j] = Max [[NCB.sub.1,j] -100,0]. That is, the option value is the greater (maximum) of either 1) the noncallable value less par or 2) zero. Here [NCB.sub.1,j] is just par plus the coupon properly discounted and the value of the callable bond is, as always, noncallable price less the option value. If, for example, the bond were a 30-year maturity, and year 29 was the last chance for exercising the call, then the value of the callable would be expressed as [CB.sub.29,j] = [NCB.sub.29,j] - [OVB.sub.29,j] Of course, representation of such a more realistic case would require a tree taking many pages. This research also calculates noncallable par coupon rates (incb) to compare to the callable par rates. To do this, a similar but simpler tree is calculated where only the coupon resulting in [NCB.sub.0] selling at 100 is found. The spread in this research is icb - incb, the extra cost an issuer has to pay if the bond is callable. The greater the option value, the greater the spread. [2] Factors Affecting Call Option Value Two market conditions clearly affect the value of the option and the resulting spread between callables and noncallables. The first is the volatility of short-term interest rates Short-term interest rates Interest rates on loan contracts-or debt instruments such as Treasury bills, bank certificates of deposit or commerical paper-having maturities of less than one year. Often called money market rates. , represented as sigma ([sigma]). Standard option pricing theory states that the value of any option increases with volatility. The basic idea is that greater volatility means there is a greater chance that interest rates will decline by a large margin, allowing the issuer to profitably replace the higher coupon debt with considerably lower cost debt. The likelihood that interest rates could rise is also greater with higher volatility, but there is no incremental Additional or increased growth, bulk, quantity, number, or value; enlarged. Incremental cost is additional or increased cost of an item or service apart from its actual cost. loss involved when rates move higher because the issuer can simply choose not to exercise the option. In other words, as volatility increases, the potential loss to the issuer is limited, but the potential gain is not. Thus, the value of the call option increases when interest rates are volatile. The second market condition that affects the price of the call option is the shape of the term structure of interest rates Term Structure of Interest Rates A yield curve displaying the relationship between spot rates of zero-coupon securities and their term to maturity. . A rising term structure (increasing interest rates for later maturities) implies that interest rates are expected to increase. The more steeply sloping the term structure, the stronger the suggestion that interest rates will rise. A flat term structure implies no expected change in interest rates, and a declining term structure (rare for municipals) implies an expectation of declining interest rates. The Hull and White model, like others, computes a call option value that is lower the greater the slope of the term structure. This is because the likelihood of a profitable decline in interest rates is more remote the more positive the term structure. A nearly flat or negative term structure will result in a larger option value. For the purpose of tables to illustrate the callable/noncallable spread, a term structure must be assumed. Thus the following describes the shape of the term structure for spot rates, R (n), where a positive beta [(beta)] means a positive term structure slope. Alpha [(alpha)] is the vertical intercept intercept in mathematical terms the points at which a curve cuts the two axes of a graph. , the one-year adj. 1. completing its life cycle within a year. Adj. 1. one-year - completing its life cycle within a year; "a border of annual flowering plants" annual phytology, botany - the branch of biology that studies plants rate, and In is the natural logarithm Natural logarithm Logarithm to the base e (approximately 2.7183). of the maturity. R(n) = [alpha]+[beta] [ln (n)] / 100 Two aspects of the call feature chosen by the issuer clearly affect option value. The first is the call protection period. This refers to the period of time which must pass before the issuer can call the bonds for the purpose of replacing them with lower cost debt. For example, a callable bond may have a 30-year maturity but not be callable for 10 years. Thus, even if interest rates decline, say, five years after issuance, the bond cannot be retired. The longer the call protection period, the lesser the call option value and the lesser the spread of callables over noncallables (icb - inch). The call (exercise) price is a second aspect of the call feature that affects value of the call option. Some bonds are callable at par, but many are not. For example, Broward Broward could refer to:
Florida (flôr`ĭdə, flŏr`–), state in the extreme SE United States. A long, low peninsula between the Atlantic Ocean (E) and the Gulf of Mexico (W), Florida is bordered by Georgia and ) sold $22.6 million of bonds in June June: see month. 1999 maturing in 2034 first callable in 2009 at 102 percent of par However, the call price declines to par in 2011. A higher call price of course reduces the value of the call option as the issuer must pay more to exercise the option. Additionally, the maturity of the bond, although it is not an explicit call option feature, can have an important impact on the value of the call option. Generally (keeping the call protection period constant), the longer the maturity, the greater the option value as there is more opportunity for interest rates to decline to permit profitable exercise. The next section will show how variations in market conditions and call features can affect option value and the spread. In some cases, it will be shown that the spread is small and in others it is quite significant. Also, in some cases a variation in a specific factor can have a small impact on option value, but in other cases, a variation can have a quite significant impact. How the Call Feature Affects Yield Exhibits 2 and 3 are tables depicting the impact of a call feature on municipal bond yields as dependent upon the myriad Myriad is a classical Greek name for the number 104 = 10 000. In modern English the word refers to an unspecified large quantity. The term myriad is a progression in the commonly used system of describing numbers using tens and hundreds. of factors determining the value of a call. The headings of the exhibits contain assumptions needed for calculation. In both exhibits, the call price is assumed to be par. Fixed refunding costs to replace the existing bond are 2 percent of bond value where this effectively increases the exercise price. It is assumed that one year rates are 2 percent, which is the lower anchor of the term structure, and that the mean reversion Mean Reversion A strategy that involves purchasing an underperforming stock or another type of security and holding the position until the market rebounds. Notes: factor (from Hull and White model) for interest rates is 0.05. Panel a of each table is for a low volatility of interest rates ([sigma]=0.005) and Panel b is for a considerably greater volatility ([sigma]=0.015). Bond maturities of 20 and 30 years are given in each panel. Exhibit 2 is for a call protection of 10 years while Exhibit 3 represents a call protection of five years. Yields for both callables and noncallables are given, as is the spread or difference. Also, the embedded option Embedded Option An option that is an inseparable part of another instrument. Compare this to a normal (or bare) option, which trades separately from the underlying security. Notes: A common embedded option is the call provision in most corporate bonds. value is given although one obviously cannot observe A type of fire control which indicates that the observer or spotter will be unable to adjust fire, but believes a target exists at the given location and is of sufficient importance to justify firing upon it without adjustment or observation. this. As the embedded option increases, the spread increases. In each exhibit, the term structure assumption, represented by beta ([beta]), is given in the last column. These exhibits show that the impact of the call theoretically can vary tremendously in that the range of spreads is from one to 113 basis points. The spread is only one basis point if volatility is low, call protection is ten years, and the term structure is steep ([beta] = 1.25) in Panel a of Exhibit 2. However, if volatility is high, call protection is only five years, and the term structure not very steep, the spread is 113 basis points in Table 3, Panel b. Increasing [beta] always reduces the spread but the size of the reduction varies considerably dependent upon other things. A greater volatility in both exhibits makes the spread much more sensitive to [beta], as can been seen by comparing Panels a and b of both exhibits. Comparing Panels a and b in both exhibits illustrates the strong impact volatility can have. Panel b results, which represent greater volatility, always show a much greater spread than for otherwise equivalent cases in Panel a. In Exhibit 2, increasing volatility for a 20-year maturity when [beta] is one makes the spread 41 basis points instead of three. Such a pattern is stronger for cases with less steep term structures and lesser call protection (Exhibit 3). Comparing Exhibits 2 and 3 illustrates the impact that varying call protection from 10 to five years can have. Each of the spreads in Exhibit 3 (call protection of five years) are greater than otherwise equivalent Exhibit 2 (call protection of 10 years) cases. Increasing maturity from 20 to 30 increases the spread except in cases of a steep term structure and low volatility where the impact is minimal. Summary A municipal issuer's decision to include a call should be based on the following factors. * Interest Rates. If interest rates are high compared to historical standards, a call is advisable ad·vis·a·ble adj. Worthy of being recommended or suggested; prudent. ad·vis  a·bil . However, the issuer should expect to pay
more (in terms of greater interest costs) when the call is included
under such circumstances.
* Credit Prospects. An issuer who is confident that the creditworthiness Creditworthiness The condition in which the risk of default on a debt obligation by that entity is deemed low. Creditworthiness Eligibility of an individual or firm to borrow money. of his/her governmental unit will likely improve should seriously consider a call because improved credit allows the government to replace the debt at a better rate in the future. * Issue Size. Larger issuers should always consider a call because, in their case, future savings will likely offset fixed costs fixed costs, n.pl the costs that do not change to meet fluctuations in enrollment or in use of services (e.g., salaries, rent, business license fees, and depreciation). of a new issue. * Length of Maturity. It is advisable for longer maturity bonds to include a call since there is a stronger likelihood of wanting to rearrange re·ar·range tr.v. re·ar·ranged, re·ar·rang·ing, re·ar·rang·es To change the arrangement of. re the debt structure before maturity. * Leverage. Issuers who are more heavily levered and subject to overly restrictive covenants may wish to make debt callable. The extra yield that investors will demand when a call option is included depends on the interaction of the following market conditions and features of the call: * Market conditions include the shape of the term structure and volatility of interest rates. The greater the slope of the term structure, the lower the value of the call option. By contrast, the value of the call option increases when interest rates are volatile. * Features of the call include the call price and the call protection period. A higher call price (greater than par) reduces the value of the call option because the issuer must pay more to exercise the option. Likewise, the longer the call protection period (the minimum time before the bonds can be called), the lesser the call option value. If the call protection is short, interest rates are very volatile, and the term structure is comparatively flat, issuers can expect to pay a sizeable premium for the privilege to call. DUANE Duane is a given name for a male. In the United States this spelling is typically associated with white Southerners. It seems to have a convergent etymology from several sources. One of its origins in the Norman "D'Wain", meaning "of Wagons". R. STOCK, PH.D., is the Price Investments Professor at the Price College of Business at the University of Oklahoma University of Oklahoma, abbreviated OU, is a coeducational public research university located in the U.S. state of Oklahoma. Founded in 1890, it existed in Oklahoma Territory near Indian Territory 17 years before the two became the state of Oklahoma. . He has published numerous articles concerning the municipal bond market. NOTES (1.) See Spivey, Michael Michael, archangel Michael (mī`kəl) [Heb.,=who is like God?], archangel prominent in Christian, Jewish, and Muslim traditions. In the Bible and early Jewish literature, Michael is one of the angels of God's presence. , "The Cost of Including a Call Provision in Municipal Debt Contracts," Journal of Financial Research, Volume 12, Fall 1989, pages 203-216 for more details on the details to include a call. (2.) See Hull, John Hull, John (1624–83) merchant, goldsmith; born in England. He came to Boston in 1635. He became the mintmaster for Massachusetts (1652), coined the colony's pine tree emblem shillings (1662) and served as the colony's treasurer (1676–80). and White, Alan, "One-Factor Interest Rates Models and the Valuation of Interest Rate Derivative An interest rate derivative is a derivative where the underlying asset is the right to pay or receive a (usually notional) amount of money at a given interest rate. The interest rate derivatives market is the largest derivatives market in the world. Securities," in Journal of Financial and Quantitative Analysis Quantitative Analysis A security analysis that uses financial information derived from company annual reports and income statements to evaluate an investment decision. Notes: , (June 1993) for details on their model. Also, see Stanhouse, Bryan and Stock, Duane, "The Impact of Volatility on Duration of Amortizing Debt with Embedded Call Options," Journal of Fixed Income, Volume 8, September 1998, pages 87-94 for more information on modeling option values.
VARIATION IN SPREAD OVER NONCALLABLE PAR COUPONS DUE TO TERM STRUCTURE
SLOPE, VOLATILITY, AND MATURITY
Call protection = 10 years, [alpha] = 0.02, mean reversion = 0.05,
refinancing cost = 2% of par
Panel a
Volatility = 0.005
Maturity of Twenty Years
Callable Par Coupon Noncallable Par Embedded Option Coupon Basis
(icb) Coupon (incb) Value Point Difference
0.0350 0.0340 1.44 10
0.0477 0.0474 0.481 3
0.0540 0.0539 0.296 1
Maturity of Thirty Years
0.0370 0.0355 2.758 15
0.0502 0.0499 0.780 3
0.0567 0.0566 0.415 1
Panel b
Volatility = 0.015
Maturity of Twenty Years
0.0393 0.0337 8.29 56
0.0512 0.0471 5.41 41
0.0570 0.0536 4.24 34
Maturity of Thirty Years
0.0430 0.0352 14.73 78
0.0548 0.0497 8.07 51
0.0606 0.0565 6.10 41
Panel a
Volatility = 0.005
Maturity of Twenty Years
Callable Par Coupon Beta ([beta])
(icb)
0.0350 0.5
0.0477 1.0
0.0540 1.25
Maturity of Thirty Years
0.0370 0.5
0.0502 1
0.0567 1.25
Panel b
Volatility = 0.015
Maturity of Twenty Years
0.0393 0.5
0.0512 1.0
0.0570 1.25
Maturity of Thirty Years
0.0430 0.5
0.0548 1
0.0606 1.25
VARIATION IN SPREAD OVER NONCALLABLE PAR COUPONS DUE TO TERM STRUCTURE
SLOPE, VOLATILITY, AND MATURITY
Call Protection = 5 years, [alpha] = 0.02, mean reversion = 0.05,
refinancing cost = 2% of par
Panel a
Volatility = 0.005
Maturity of Twenty Years
Callable Par Coupon Noncallable Par Embedded Option Coupon Basis
(icp) Coupon (incb) Value Point Difference
0.0358 0.0341 2.56 17
0.0481 0.0474 0.94 7
0.0543 0.0536 0.61 7
Maturity of Thirty Years
0.0377 0.0355 4.02 22
0.0507 0.0499 1.25 8
0.0571 0.0564 0.71 7
Panel b
Volatility = 0.015
Maturity of Twenty Years
0.0434 0.0337 14.23 97
0.0545 0.0471 9.63 74
0.0598 0.0536 7.78 62
Maturity of Thirty Years
0.0465 0.0352 21.13 113
0.0575 0.0497 12.27 78
0.0631 0.0565 9.66 66
Panel a
Volatility = 0.005
Maturity of Twenty Years
Callable Par Coupon Beta ([beta])
(icp)
0.0358 0.5
0.0481 1.0
0.0543 1.25
Maturity of Thirty Years
0.0377 0.5
0.0507 1
0.0571 1.25
Panel b
Volatility = 0.015
Maturity of Twenty Years
0.0434 0.5
0.0545 1.0
0.0598 1.25
Maturity of Thirty Years
0.0465 0.5
0.0575 1
0.0631 1.25
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