Valuation of a New Class of Commodity-Linked Bonds with Partial Indexation Adjustments

Valuation of a New Class of Commodity-Linked Bonds with Partial Indexation Adjustments Thomas H. Kirschenmann Institute for Computational Engineering and Sciences University of Texas at Austin and Ehud I. Ronn Department of Finance University of Texas at Austin December 2008 Revised: May 28, 2009 Communications Author: Address: Ehud I. Ronn Department of Finance McCombs School of Business 1 University Station, B6600 University of Texas at Austin Austin, TX. 78712-1179 Tel.: (512) 471-5853 FAX: (512) 471-5073 E-mail: eronn@mail.utexas.edu

Abstract The issuance of debt instruments with indexation of coupon payments to consumer or commodity prices has occurred with increasing frequency in both the developed and developing world. Such issuance has frequently been made by sovereign or near-sovereign issuers, and occasionally replicated through derivative structures provided by investment bankers. In this paper we provide a valuation for the recent issuance of a new type of commodity-linked bond, wherein the indexation is partial, reflecting the growth if any in the underlying price index since the time of the last price adjustment. We explore the circumstances under which the bond s rate of return dominates that of the traditional indexed instrument. We demonstrate such a bond fails to provide the constant, fixed real rate of return typically guaranteed by a CPI-linked instrument such as TIPS. Key Words: Commodity-linked bonds, derivative structures, indexation, real rates of return i

Valuation of a New Class of Commodity-Linked Bonds with Partial Indexation Adjustments Example of a New Commodity-Linked Bond: At redemption, holder receives par. In addition, holder receives semiannual coupon. Those payments are.82 (percentage gain in the NYMEX WTI). Say the NYMEX WTI goes from $50/bbl to $55/bbl, coupon payment on a $1000 par bond would be.82 (.1) (1000) or $82. Next coupon payment would be determined off a new base price of $55. 1 Introduction From 2007 Offer Sheet of a Major Commodity Dealer The issuance of debt instruments with indexation of coupon payments to consumer or commodity prices has occurred with increasing frequency in the developed and developing world. Dating back to the 1970 s, issuers of such securities typically bore sovereign or near-sovereign status. With respect to issues of sovereign entities, CPI-linked securities have been issued by the governments of the United Kingdom, Norway, Brazil and Israel for several decades. A more recent example is Treasury Inflation-Protected Securities, or TIPS, issued by the U. S. Treasury since 1997. An early example of commodity-linked bonds is the issue of the United Mexican States Government back in 1979. Using the notion of dynamic replication, for some time now investment bankers have marketed to retail investors structured products whose payoffs are determined in whole or in part by changes in the underlying commodity In this paper we provide a valuation for the recent issuance of a new type of commodity-linked bond, wherein the indexation is partial, reflecting only the growth if any in the underlying price index since the time of the last price adjustment. We explore the circumstances under which the bond s rate of return dominates that of the traditional indexed instrument. We demonstrate such a bond fails to provide the constant, fixed real rate of return typically guaranteed by a standard CPI-linked instrument. Over the past several years, investors have increasingly focused on commodities as an investmentasset class. Extraordinary returns in recent years have attracted investors seeking exposure to commodity price movements through commodity-linked derivatives, which provide exposure to commodity markets without the purchase of standard futures contracts. Commodity-linked financial 1

instruments provide a means for both commercial and financial investors to satisfy their investment objectives. Interest in the two types of commodity-linked structured products we consider here i.e., both the traditional as well as the partial-indexation structures is driven by the desire to gain commodity-price exposure. As we show below, the partial-indexation structure is particularly appropriate for investors with especially strong views regarding a continuous upward movement in commodity prices. Commercial investors, such as corporations, utilize commodity-linked instruments to reduce their overall risk. Whereas commercial users are primarily concerned with hedging their exposure to commodity prices in a view-consistent, cost-efficient manner, financial users seek out such exposure for its return and diversification properties. Financial users seeking diversification note that commodity returns are historically weakly correlated with equity and fixed income products, so that commodity-linked instruments can be viewed as a separate asset class. Commodity prices can provide protection against inflation due to economic growth, and are thus a desirable means of diversification through which to preserve purchasing power. Commodity prices are correlated with non-economic drivers, such as weather, environmental issues, supply constraints, etc. Indeed an investor may address specific risk exposures in an efficient manner through commodity-linked derivatives, making these products an essential tool for risk management strategies. Consider a commodity-linked bond which pays the holder a fraction of par at the time of redemption as well as semiannual coupons in an amount dependent on a fixed multiple of par and the percentage increase in the commodity futures price. The valuation of this type of instrument was explored by Schwartz (1982). It is important to note the coupon is paid only in the case where the price of the futures contract increases. Clearly the value of a commodity-linked bond is equal to the face value of the bond plus an option to buy the reference commodity bundle at a specified exercise price. Schwartz utilizes Black-Scholes option pricing techniques to formulate a pricing algorithm for the commodity-linked bond. Additionally, commodity-linked bonds may be structured to include linkage to underlying corporate assets, although our examples do not include this variation. This paper is now organized as follows. Section 2 introduces the new type of partial indexation security, offered no later than mid-2007 by a large investment brokerage house active in the crudeoil commodity markets. Section 3 weaves the connection between this new security and traditional 2

TIPS-type securities which deliver a known real rate of interest to their holders. Section 4 provides two valuation approaches an analytical approach based on the Stulz (1982) model, and a Monte Carlo verification approach. Section 5 reports comparative statics across different values of volatility as well as path-dependent evolutions of the underlying futures contracts. Section 6 concludes. 2 A New Type of Commodity-Linked Bond with Partial Indexation Adjustment 2.1 Traditional Commodity-linked Bond We first consider the valuation of a standard commodity-linked bond with coupon payments made at times T/2 and T. For semi-annual payments, T = 1. Each coupon payment has payoff equal to { } FT max 1, 0 = 1 max {F T F 0, 0}, (1) F 0 F 0 where F t is the date-t value of a futures contract. The coupon payment at time T/2 is max { F T/2 /F 0 1, 0 }, and the coupon payment at time T is max {F T /F 0 1, 0}. It is important to note the basis for the indexation of the coupon is fixed at the value F 0. At redemption the holder is paid par if the price of the futures contract has fallen, but if the price has increased, then the holder receives par plus the additional contribution from the coupon. In the case of a commodity-linked bond, for contractually determined values of α and β, the payoff for the bond at time T is { } FT Payoff T = α Par + β Par max 1, 0, (2) F 0 where α is the percentage of par that is guaranteed to be received by the holder, and β is a fixed multiplier of par and the percentage gain in the commodity price. 1 α and β are typically set so as to give the time 0 value of the commodity-linked bond a value (close to) par. 2.2 A New Class of Commodity-linked Bond Now consider a new class of commodity-linked bonds, for which the base is reset upwards if F T/2 > F 0. The first coupon payment, occurring at time T/2, is identical to that of the traditional commodity-linked bond. However, the second coupon payment, which occurs at time T, is 1 The principal payments may also be indexed for price movements. 3

proportional to Payment T = F T max max } 1, 0 F T/2, F 0 = { { } } FT F T max min, 1, 0 F 0 F T/2 (3) (4) The upward ratcheting effect on the basis is clearly seen in (3). When F T/2 > F 0 the basis will be reset, and the payment at date T will be based on the higher price F T/2 rather than F 0. As a result, any coupon payment at date T will be weakly lower than that payable on a traditional commodity-linked bond wherein the basis remains fixed at F 0 : { { } } { } FT F T FT max min, 1, 0 max 1, 0. F 0 F 0 F T/2 Accordingly, the analogue to (2) is Payoff T = α Par + β Par max { min { FT F 0, F T F T/2 } 1, 0 } (5) 3 TIPS-Type CPI-Linked Bonds and the Real Rate of Interest The properties of the new form of commodity-linked bond with respect to inflation can be contrasted with Treasury Inflation-Protected Securities (TIPS), where the base of the bond is fixed. If inflation is positive and has increased the principal value of the underlying security during its term, the Treasury will pay the owner the higher inflation-adjusted coupon and principal. Roll (2004) provides a discussion of the valuation of TIPS and their usage for estimating the structure of anticipated inflation. Thus TIPS provide an effective means of protection against inflation, whereas a commodity-linked bond with a base that is reset to a higher value does not provide full inflation protection. To see the lack of inflation protection precisely, consider the discounted cash-flows of a TIPS bond in an environment with annual inflation rate π and coupon payment C, the bond with maturity T has value V given by V = T t=1 C (1 + π) t Par (1 + π)t t + [(1 + r) (1 + π)] [(1 + r) (1 + π)] T, (6) where 1 + R (1 + r) (1 + π) is one plus the nominal rate of interest (i.e., yield-to-maturity) used to discount the nominal cash flows C (1 + π) t and Par (1 + π) T, and where the real rate of interest 4

on the bond r is implicitly given by (6). If V = Par, the real rate of interest r is equal to the statutory coupon rate C. When T = 1 for a semi annual-pay coupon bond, (6) may be rewritten as: V = = 1 t=0.5 1 t=0.5 (C/2) (1 + π) t Par (1 + π)t t + [(1 + r) (1 + π)] [(1 + r) (1 + π)] T (C/2) (F t /F 0 ) (1 + r) t (F t /F 0 ) + Par (F T /F 0 ) (1 + r) T (F T /F 0 ) (7) where we have made the substitution (1 + π) t F t /F 0. With r now defined as the real rate of interest with respect to the price of the given commodity F t, the condition V = Par implies the real rate of interest r is equal to the statutory coupon rate C. 2 However, when the base may be reset between coupon payments, and even assuming the Par amount is afforded full commodity-price indexation, the inflation-adjusted numerator and denominator no longer cancel: V = C (F 0.5 /F 0 ) (1 + r) 0.5 (F 0.5 /F 0 ) + C min {(F 1/F 0 ), (F 1 /F 0.5 )} + Par (F 1/F 0 ) (1 + r) (F 1 /F 0 ) (1 + r) (F 1 /F 0 ) (8) Since min {(F 1 /F 0 ), (F 1 /F 0.5 )} F 1 /F 0, the second term on the RHS destroys the inflationprotection investors may be expecting from commodity-linked bonds. Investors should thus be aware that a portfolio holding these securities is at least partially exposed to inflationary risk. This result obtains a fortiori if the Par amount, i.e., the third term on the RHS of eq. (8), is also impacted by the same min operator min {(F 1 /F 0 ), (F 1 /F 0.5 )}. 3 2 Technically, of course, it is the semi-annually compounded rate r that is equal to C : V = C 2 = C 2 1 t=0.5 1 t=0.5 = r = C 1 (1 + r) t + Par (1 + r) T 1 (1 + r /2) 2t + Par (1 + r /2) 2T 3 In practice, TIPS provisions preclude the principal value from falling below par if deflation prevails on average over the life of the bond. We abstract from this floor issue in the current analysis. 5

4 Model We now seek to obtain the valuation of bonds represented by the payoffs in eqs. (2) and (5). In particular, we are interested in the values of the two embedded options whose payoffs are (1) and (4): { } FT max 1, 0 F 0 vs. max { min { FT F 0, F T F T/2 } 1, 0 To evaluate these expressions, we make the traditional assumption of geometric Brownian motion }. df F = σ dz, (9) where the risk-neutral drift for F is zero. where If we define V ( ) as the date 0 valuation operator, the valuation of max {F T /F 0 1, 0} is V ( 1 F0 ) max {F T F 0, 0} r = is the risk free rate of interest to maturity T = time to payment σ = volatility = 1 F 0 V (max {F T F 0, 0}) = 1 F 0 c (F 0, K = F 0, r, T, σ) and c (F 0, K = F 0, r, T, σ) is the value of a T -period European-style call option with parameters {F 0, K = F 0, r, T, σ}. Assuming a zero probability of issuer default, the value of the traditional commodity-linked bond at date T is V (Payoff T ) = α Par (1 + r T ) T + β 1 F 0 c (F 0, K = F 0, r, T, σ). Our analysis compares the valuation of both the traditional and the new form of commoditylinked bonds in which the base is allowed to reset at the time of a coupon payment. We demonstrate quantitatively that an upward ratcheting effect on the base in the new class of bonds reduces their value relative to the traditional commodity-linked bond. 6

4.1 Pricing a Call Option on the Minimum Recognizing the valuation of the coupon payment as equivalent to a call option on the minimum of two assets, we apply the results of Stulz (1982) for pricing a call option on the minimum of two risky assets. Stulz shows that a call option on two assets S 1 and S 2 with a payoff at time T of max {min {S 1, S 2 } X, 0} has a present value given by: c min (S 1, S 2, X, T ) = S 1 e (b1 r)t N 2 (y 1, d; ρ 1 ) +S 2 e (b2 r)t N 2 (y 2, d Σ T ; ρ 2 ) Xe rt N 2 (y 1 σ 1 T, y2 σ 2 T ; ρ) (10) where d = ln (S 1/S 2 ) + (b 1 b 2 + Σ 2 /2) T Σ T y 1 = ln (S 1/X) + (b 1 + σ 2 1/2) T σ 1 T y 2 = ln (S 2/X) + (b 2 + σ 2 2/2) T σ 2 T Σ = σ 2 1 + σ 2 2 2ρσ 1 σ 2 ρ 1 = σ 1 ρσ 2 Σ N 2 ( ) is the cumulative bivariate normal distribution ρ 2 = σ 2 ρσ 1 Σ σ 1 and σ 2 are the volatilities of S 1 and S 2 respectively ρ is the correlation of S 1 and S 2 r is the risk-free rate of interest to maturity T b 1 and b 2 are the dividend yield of S 1 and S 2 respectively We now apply this valuation to the coupon of the base-reset class of bonds by defining f 1 F T /F 0 and f 2 F T /F T/2 and setting both b 1 and b 2 equal to zero as the futures contract does not provide a dividend. The volatility of f 1 is σ 1 = σ, which is the volatility of the underlying futures contract through time T. The volatility of f 2 is σ 2 = σ 2/2, since this is the volatility of the return over the period beginning at time T/2 and ending at T. Substituting these values into the above equation shows the value of the coupon payment when viewed as a call option is then given by V (max {min {f 1, f 2 } 1, 0}) = c (f 1, f 2, r, σ, T ) ( = f 1 e rt N 2 (y 1, d, ρ 1 ) + f 2 e rt N 2 y2, d Σ ) T, ρ 2 ( ) e rt N 2 y1 σ 1 T, y2 σ 2 T, ρ (11) where 7

d = σ T 2 2, Σ = σ 2, y1 = σ T, σ1 = σ 2 2 σ 2 = σ 2, y2 = σ T 2 2 2 2, ρ 1 = ρ = 2, ρ 2 = 0 N 2 ( ) is the cumulative bivariate normal distribution σ is the volatility of the underlying futures contract ρ is the correlation of f 1 and f 2 r is the risk-free rate of interest to maturity T The value of the base-reset class of commodity-linked bonds can now be calculated analytically, since the value of the coupon payment is given by (11). This solution is independently verified via Monte-Carlo simulation on the coupon payments in the next section, and the close agreement between the two methods provides confidence the analytical solution can be used by investors to accurately value commodity-linked bonds with the base-reset feature. 4.2 Pricing a Put Option on the Minimum With commodity prices declining in the latter part of 2008, we can also structure a put option variant of the commodity-linked bond for investors who wish to hedge or gain exposure to decreasing commodity prices. The traditional fixed base commodity-linked bond would have a coupon payment given by { } { F0 F T Payment T = max, 0 = max 1 F } T, 0 F 0 F 0 To create the analogous base-reset coupon, we reset the base at time T/2 if the price has fallen, so that the coupon payment at time T is proportional to Payment T = max 1 F T min { }, 0 F 0, F T/2 { { } } FT F T = max 1 max,, 0 F 0 F T/2 { { = max min 1 F T, 1 F } } T, 0 F 0 F T/2 Clearly, { { max min 1 F T, 1 F } } T, 0 F 0 F T/2 { max 1 F } T, 0, F 0 8

so that the put-option variant of the base-reset class of commodity-linked bonds is less valuable than the analogous traditional commodity-linked bond lacking the base-reset feature. The valuation of the base-reset put variants of the commodity-linked bond can be done in analogous manner to the previous subsection s use of Stulz s valuation of options on the minimum of two risky assets. The base-reset examples in the remainder of this paper are in reference to the upward-reset call option style outlined in Section 4.1. 4.3 Validation Through Monte Carlo Simulation We perform a Monte Carlo simulation directly on the risk-neutral process for the futures contract F, providing a direct and independent validation of the valuations derived above. The Monte Carlo simulation is also used to validate the closed form solution, and shows that it does indeed provide an accurate solution for a wide range of volatility. The Monte Carlo simulation assumes geometric Brownian motion of the futures contract and is implemented by two coupled stochastic equations ln F T/2 = ln F 0 1 4 σ2 + σ 2 ɛ 1 ln F T = ln F T/2 1 4 σ2 + σ ɛ 2 2 where ɛ 1 and ɛ 2 are independent samples from the standard Normal distribution N (0, 1). It is easily seen that each path is a two-step process, producing futures prices at dates T/2 and T. We use these prices to directly compute the value of the coupon payments of both types of commodity-linked bonds. Repeating the simulation while varying the volatility, we compare the valuations of the two classes of bonds. We find excellent convergence to the analytical solution after sampling only a few thousand paths, so that the run-time of such a valuation is not a significant factor to risk managers. 5 Numerical Examples 5.1 Monte Carlo Results Validate the Analytical Solution The following simulation data is computed for a futures contract with a one-year maturity, an initial price of $100, and a risk-free interest rate of 5% per annum. We denote the Monte Carlo based valuations of the traditional commodity-linked bond as Valuation 1 and that of the new class of 9

bonds as Valuation 2. The following graph shows the close agreement of the closed-form solution (11) derived above and the Monte Carlo-based Valuation 2. Figure 1: Comparison of Closed Form Valuation and Monte Carlo based Valuation 2 for volatilities up to 200% We note that the accuracy is particularly good in the region for which the volatility is less than 100%, which is useful for modeling most commodities. As a check to confirm that the coupon payments of the traditional commodity-linked bonds are being accurately valued, we look for close agreement between Valuation 1 and the well-known Black (1976) valuation of a one-year European-style call option. Now confident in the accuracy of the valuations, we compare the two valuations directly in Figure 2 and recognize the value of the new class of commodity-linked bonds is consistently less than the traditional form, no matter the size of the underlying volatility. The reduced value of the base-reset style commodity-linked bond becomes even more significant when the volatility of the underlying asset is high. Intuitively, the base may have been ratcheted upwards due to the larger volatility, and the magnitude of the subsequent coupon payment is lower, thereby reducing the value of the bond. Similarly, we see the reduced value of the closed form solution with respect to Valuation 1. 10

Figure 2: Comparison of the Closed Form Solution Valuation 2 and Black (1976) for Volatilities up to 200% Noting that all four valuations approach zero as the volatility is reduced to zero, we seek to quantify the convergence of the ratio Closed-Form / Black (1976) as the volatility is reduced. Figure 3 displays the ratio Closed-Form / Black (1976) as a function of volatility. Figure 3: Comparison of the Ratio of Closed-Form / Black (1976) Valuations for Volatilities up to 200% 11

It is important to note that as the volatility tends to zero, the Closed-Form / Black (1976) monotonically increases to 0.5. As volatility becomes negligibly small, the value of the base-reset option is half that of the traditional fixed-base bond. A detailed comparison of rates of return on the two bonds over all sets of possible price paths is provided in the following section. 5.2 Analysis of Commodity-Linked Bonds Rates of Return for Positive and Negative Returns Over the Two Periods For investors who seek exposure to upward commodity price movements as part of a risk management strategy, we consider the effectiveness of the base-reset style commodity-linked bond. 4 In an environment of continuous upward price movements, the base is always reset to the futures price at the time of the most recent coupon payment. We find that when the return over the second period is greater than the return on the first, the base-reset bond renders a higher return than the traditional commodity-linked bond. Specifically, we seek the range of returns on the futures price in which the new class of base-reset commodity-linked bonds provides a higher rate of return than the traditional commodity-linked bond. We assume an underlying initial commodity price of 100 and then define: where Ratio 1 max {F 2 100, 0} Ratio 2 max {F 2 100 (1 + max {r 1, 0}), 0} C 2 (σ) r 1 = (F 1 100) /100, i.e., the rate of return over the first period r 2 = (F 2 F 1 ) /F 1, i.e., the rate of return over the second period 1 (12) 1 (13) is the European call option used to value the coupon payment of the traditional commodity-linked bond C 2 (σ) is the call option valuation of the coupon for the base-reset style bond as outlined in Section 2.2 4 Investors recognize that under the risk-neutral measure, E (Ratio2 Ratio1) = 0, so that those who prefer the base-reset option must believe that under their own subjective measure, E (Ratio2 Ratio1) > 0. 12

1. We first note that when (1 + r 1 )(1 + r 2 ) 1, both the traditional option and the base-reset option expire out of the money, so we focus our attention on the case when (1+r 1 )(1+r 2 ) > 1, where r 1 = (F 1 100) /100 and r 2 = (F 2 F 1 ) /F 1. The rates of return are calculated and then summarized in the table at the end of the section. Ratio2 Ratio1 = max {F 2 100 (1 + max {r 1, 0}), 0} C 2 (σ) max {F 2 100, 0} Substituting F 2 = 100(1 + r 1 )(1 + r 2 ) into the above equation and simplifying terms gives Ratio2 Ratio1 = 100 [ max {(1 + r1 )(1 + r 2 ) 1 max {r 1, 0}, 0} C 2 (σ) max {(1 + r ] 1)(1 + r 2 ) 1, 0} which reduces to [ max {(1 + r1 )(1 + r 2 ) 1 max {r 1, 0}, 0} Ratio2 Ratio1 = 100 (1 + r ] 1)(1 + r 2 ) 1 C 2 (σ) (14) 2. There are now two cases to consider. The first is when the futures contract has decreased at T = 1/2, i.e., r 1 0, and the second case is when r 1 > 0: (a) If the price declined, r 1 0, and the rate of return of the base-reset option always exceeds that of the traditional one. [ (1 + r1 )(1 + r 2 ) 1 Ratio2 Ratio1 = 100 (1 + r ] 1)(1 + r 2 ) 1 C 2 (σ) [ 1 = 100 [(1 + r 1 ) (1 + r 2 ) 1] C 2 (σ) 1 ] > 0 since C 2 (σ) < (b) We now consider the case when the price has increased at T = 1/2, i.e., r 1 > 0, so that eq. (14) becomes [ max {r2 (1 + r 1 ), 0} Ratio2 Ratio1 = 100 (1 + r ] 1) (1 + r 2 ) 1 C 2 (σ) = Ratio1 = 100 (1 + r 1) (1 + r 2 ) 1 < 0 3. If r 2 0, the new option expires worthless, Ratio2 Ratio1 = Ratio1 < 0, and the rate of return on the traditional option exceeds that of the base-reset option. 13

4. If r 2 > 0, then Ratio2 Ratio1 = 100 The above equation is positive if and only if [ r2 (1 + r 1 ) C 2 (σ) r 2 (1 + r 1 ) (1 + r 1 )(1 + r 2 ) 1 > C 2(σ). (1 + r ] 1)(1 + r 2 ) 1. Since C 2 (σ)/ < 1/2 for σ [0, 100%], Ratio2 Ratio1 < 0 when r 2 > r 1 1 + r 1. The case when r 1 > 0 can be succinctly summarized: The rate of return of the base-reset option exceeds that of the traditional option when r 2 > r 1 / (1 + r 1 ). The difference in the return rates, Ratio2 Ratio1, is summarized in the table below. Case Ratio2 Ratio1 Conditions on r 1, r 2 Verbal Explanation 1. 0 (1 + r 1 )(1 + r 2 ) 1 If the forward price suffers a price decline [ 2. [(1 + r 1 ) (1 + r 2 ) 1] 1 C 2 (σ) 1 over the full period [0, T ], both options expire out-of-the-money. ] > 0 r 1 0, (1 + r 1 )(1 + r 2 ) > 1 If the price declines in the [0, T/2] period but then ends up above the time-0 price, the two options have identical payoffs, and the new consequently outperforms due to its lower price. 3. (1 + r 1) (1 + r 2 ) 1 [ r2 (1 + r 1 ) 4. 100 C 2 (σ) [ r2 (1 + r 1 ) 5. 100 C 2 (σ) < 0 r 1 > 0, r 2 0 If the price increases in the [0, T/2] period but subsequently declines, the new option expires worthless. (1 + r ] 1)(1 + r 2 ) 1 < 0 r 1 > 0, r 2 < r 1 If the price increases in the first period 1 + r 1 and increases in the second period by an amount less than r 1 / (1 + r 1 ), the standard option will dominate (1 + r ] 1)(1 + r 2 ) 1 > 0 r 1 > 0, r 2 > r 1 If the price increases in the first period 1 + r 1 and increases in the second period by an amount greater than r 1 / (1 + r 1 ), the new option will dominate In conclusion, the new option will dominate if: r 1 when r 1 > 0 1 + r 1 r 2 > 1 1 when r 1 < 0 1 + r 1 (15) 14

5.3 Valuation for Arbitrary Base-Reset Time Noting that the ratio of the two valuations described in Figure 2 approaches 0.5 as the volatility is reduced, we seek to determine how this limit varies when the base-reset time is altered. By directly modeling these contracts with a Monte Carlo simulation analogous to the one described in Section 4.3, we determine the ratio of the two valuation times for arbitrary base-reset times. The figure below displays the ratio of the two valuations computed at different base-reset times at a fixed 2% volatility. Figure 4: Ratio of Valuation 2 / Valuation 1 while varying the base-reset time as a percentage of the expiration of the traditional option The Monte Carlo simulation demonstrates that a later reset time decreases the ratio of the basereset valuation relative to the traditional valuation. (Note that for the initial semi-annual coupon bond when T reset /T = 0.5, the ratio Valuation 2 / Valuation 1 = 0.5.) Intuitively, the relative value reduction is due to the reduced amount of time that the base-reset style bond has to recover from the ratcheting effect of higher base for coupon payments. 15

6 Conclusions Our paper explores the valuation, both analytical and numerical, of the base-reset style commoditylinked bond and shows how the value is reduced relative to the traditional bond whose base is fixed. We provide a closed form analytical solution for the new class of commodity-linked bonds and verify the solution with an independent Monte Carlo simulation. Investors who are particularly concerned about inflationary protection should be aware of the incomplete inflation protection provided by the base-reset style bonds, relative to TIPS-style full indexation provisions. With the increased attention to commodities in recent times, it is important for both financial and commercial investors to understand the relative performance of the new class of base-reset bonds with respect to the traditional commodity-linked bond. As an important separate asset class, the correct valuation is vital for the accurate implementation of many risk management strategies. 7 References 1. Black, F. and M. Scholes., 1973. The Pricing of Options and Corporate Liabilities. The Journal of Political Economy, 81, 637-654. 2. Domanski, D. and Heath, A., 2007. Financial Investors and Commodity Markets. BIS Quarterly Review, 5, 53-67. 3. Lessard, D., 1989. Financial Risk Management Needs of Developing Countries: Discussion. American Journal of Agricultural Economics, 71, No. 2, 534-535. 4. Myers, R., 1992. Incomplete Markets and Commodity-Linked Finance in Developing Countries The World Bank Research Observer, 7, No. 1, 79-94. 5. Roll, R., 2004. Empirical TIPS. Financial Analysts Journal, 60, No. 1, 31-53. 6. Schwartz, E., 1982. The Pricing of Commodity-Linked Bonds The Journal of Finance, 37, No. 2, 525-539. 7. Stulz, R., 1982. Options on the Minimum or the Maximum of Two Risky Assets: Analysis and Applications Journal of Financial Economics, 10, No. 2, 161-185. 16

8. Wright, B. and Newbery, D., 1989. Financial Instruments for Consumption Smoothing by Commodity-Dependent Exporters. American Journal of Agricultural Economics, 71, No. 2, 511-516. 17