The Indian Bond Market

Transcription

1 The Indian Bond Market SUCHISMITA BOSE DIPANKAR COONDOO SUMON KUMAR BHAUMIK I C R A B U L L E T I N Abstract It would perhaps be an understatement to say that the Indian bond market is underdeveloped. The major problem is not even that the pricing algorithms and instruments are not sophisticated; sustained competitive trading itself is at a premium. Indeed, market insiders agree that once a bond is introduced in the market, the process of price-search continues for a few months, and thereafter the lion s share of the bonds disappear from the secondary market. Typically, they are held to maturity by banks, insurance companies and mutual funds. As a consequence, trading is thin in the secondary market, especially for bonds with longer maturities, and the situation is aggravated by the fact that bulk of the trading takes place among a handful of large traders. Hence, the secondary market is marked by significant differences between prices-ytms of bonds with similar characteristics, and of similar maturity. This indicates that the pricing process is divorced from interest rate expectations which, at least in principle, should be the only determinant of bond prices. The paper examines the extent of intra-month variations in prices and YTMs of government securities over time, and concludes that price formation in the secondary market for bonds is indeed perverse and inconsistent with the postulates of finance theory. Introduction The Indian financial market has seen steady liberalisation since the early 1990 s. All segments of the market have witnessed changes, but it is the banking sector and the equity markets that have remained at the heart of coffee-table discussions and policy debates alike. The bond market has remained relatively neglected, even though, in the recent past, bonds of various kinds have helped raise more than half the capital raised by the financial and non-financial companies. The government too has used the bond market extensively to bridge its fiscal deficit. In other words, it is perhaps time that the trends in the Indian bond market are analysed methodically, and the evolution of the market traced over time. The importance of the bond market does not lie merely in the fact that bonds are a way to raise money for corporate and government spending. The prices of bonds in the secondary market reflect the interest rate expectations of the market participants. These expectations, in turn, give us the so-called yield curve. The yield curve is perhaps one of the most important tools in the hands of financial analysts and economic policy makers. On the one hand, it is essential for pricing new bonds and derivatives products. On the other hand, it highlights the impact of changes in monetary and other policies on the interest rate expectations of the market participants. The paper examines the extent of intramonth variations in prices and YTMs of government securities over time, and concludes that price formation in the secondary market for bonds is indeed perverse and inconsistent with the postulates of finance theory. 45

2 I C R A B U L L E T I N thinness of the Indian bond market results in perverse pricing.... we should verify whether, over time, the market has become more efficient,... Since government bonds constitute the bulk of trading in the secondary market, the data for this exercise has been restricted to those from secondary market transactions in government securities. In an earlier issue of, we had shown that the thinness of the Indian bond market results in perverse pricing. The consequence of this phenomenon is easily understood. Suppose that only 2 government securities with 3 years to maturity are traded during a month. If, now, the weighted average yield to maturity (YTM) of one of the securities is 10.5 while that of the other is 11.5, then we face a unique dilemma. 1 It is obvious that the yield difference between the 2 bonds of similar riskiness cannot be 1 percentage point. But which of these YTMs can we accept as the accurate one, given that both YTMs are market determined? If, therefore, some average of these rates is used to construct the yield curve, the curve would be largely meaningless, at least in so far as the yield for the 3 year horizon is concerned. It is evident from the above discussion that the extent of the pricing anomaly in the Indian bond market should come under scrutiny. Specifically, we should verify whether, over time, the market has become more efficient, i.e., whether the variance in the prices/yields of comparable securities measured within a reasonably short span of time, have declined over time. If they have, the market can be said to have become more efficient. If not, we would have to speculate about possible changes in infrastructure and trading practices that can yield the desired trend. This forms the crux of the current exercise. Data and Methodology Data Since government bonds constitute the bulk of trading in the secondary market, the data for this exercise has been restricted to those from secondary market transactions in government securities. The data consists of three major parts. The first of these has been obtained from the SGL accounts of the Reserve Bank of India (RBI). This data spans 17 months, from September 1995 to January It consists of information about volumes, prices, and settlement dates of all repo and non-repo transactions. The second part of the data has been culled out of business dailies. This data spans 12 months of the calendar year It consists of information about the volume adjusted YTMs associated with non-repo trades in government securities, and their dates of settlement at the wholesale debt segment (WDM) of the National Stock Exchange (NSE). Finally, third and the smallest part of the data has been obtained from the Weekly Supplement to the Monthly Report of the RBI, and the NSE s web site. The Weekly Supplement has provided data for ranges of prices of government securities with similar maturities for November-December The NSE web site, on the other hand, has provided the data about the trades (in government securities) recorded at the NSE during November For example, during January 1998, 7 government coupon bonds with 4 years or less maturity were traded. Of these, the average YTM for the 13.31% security maturing in November 2001 was 12.95, and that for the 13.55% security maturing in the same month was The average YTMs of the other bonds, with maturities between March and July, were in the range It is evident that the determination of the appropriate average YTMs for the 3-4 year horizons would be extremely difficult.

3 Methodology The research exercise proceeds in some discreet steps. At the outset, the exercise seeks to verify whether the depth and the width of the gilts market have increased over time. To this end, frequency distribution tables have been drawn up for the shorter (time-to-maturity less than equal to 3 years) and longer (greater than 3 years) bonds with respect to the number and average frequency of transaction, for a 17 month period (September 95 to January 97). The relation between the number of bonds and the width of the market is easily understood: the greater the number of bonds traded in the market, the more is the so-called width. But the choice of (average) frequency of transaction per bond as the proxy for the depth of the market needs some clarifications. The usual proxy for the market depth is the (rupee) volume of transactions. However, at least in the Indian context, this measure can be notoriously unrevealing. For example, if a gilt is transacted to the tune of Rs. 220 crore in a day, the market price of the bond being Rs. 110, it is not evident whether or not Rs. 110 is the market clearing price. The entire transaction of Rs. 220 crore could have been concluded between two traders in a telephone market such that the market price is an aberration. If, on the other hand, the same volume is traded in (say) ten equal transactions of Rs. 22 crore each, involving 10 pairs of traders, then Rs.110 in our opinion, is much more acceptable as a market clearing price. Even in a telephone market, imperfect as it is, as the information about each trade gets recorded, potential traders update their information set and revise their expectations. At the very least, a number of transactions between independent and rational traders helps mimic a quasi efficient market. In other words, if the market is dominated by a few large players, thereby leading to low frequency of transactions, the market is likely to remain inefficient. Next, the exercise traces the prices of specific government securities over time and verifies whether (i) the prices converge to the sum of face value plus coupon payment as the bonds near maturity, and (ii) the variance of the prices of these bonds are significant for intra month trading, and whether there is any pattern in the movement of these variances across the months. The rationale underlying hypothesis (i) has its roots in the basic premise about bond pricing [see Appendix]. The logic for the first part of hypothesis (ii) too is easily understood; in an efficient market, the variance should be low within a short period like a month, so as not to offer arbitrage opportunities in the absence of significant changes in expectations about the term structure of interest rates. But why can we expect a trend or pattern in the movement of the variance over time? Suppose that, for a representative gilt, frequency of trade increases with a decrease in its time to maturity. If, as argued above, greater frequency of trades implies greater market efficiency, the variance in the traded prices should decline over time, as the gilt nears maturity. There might be some deviations from this expected pattern because bond owners might hold rather than sell immediately prior to the maturity of the bonds, thereby reducing the frequency of trading sharply. I C R A B U L L E T I N The research exercise proceeds in some discreet steps. At the outset, the exercise seeks to verify whether the depth and the width of the gilts market have increased over time. Next, the exercise traces the prices of specific government securities over time and verifies whether (i) the prices converge to the sum of face value plus coupon payment as the bonds near maturity, and (ii) the variance of the prices of these bonds are significant for intra month trading,... 47

4 I C R A B U L L E T I N Analysis of the data suggests that a necessary though not sufficiently strong manifestation of market efficiency can be found in the secondary market for government securities. Over time, as a bond nears its maturity, its price approaches the sum of its face value and coupon payable Indeed the issue of efficiency is so central to any analysis about any market that, as mentioned in the previous section, this has been the focus of the exercise. The results obtained with the SGL data, therefore, have been cross-verified with the data obtained from dailies and RBI s Weekly Supplement. The results have been discussed in the following section. Results Analysis of the data suggests that a necessary though not sufficiently strong manifestation of market efficiency can be found in the secondary market for government securities. Over time, as a bond nears its maturity, its price approaches the sum of its face value and coupon payable [see Table 1] 2. As mentioned earlier, such price behaviour is consistent with the theoretical basis for estimation of bond prices. TABLE 1 ZCB 1999 maturing on 18/01/99 Month f P a s P min P max Sep % Oct % Nov-95 0 Dec % Jan % Feb % Mar % Apr % May % Jun % Jul % Aug % Sep % Oct % Nov % Dec % Jan % Note : f =frequency of trade Pa = average price s: standard deviation of prices P min = minimum price P max = maximum price 48 2 A zero coupon bond was chosen for the illustration because prices of coupon bonds are affected by the magnitude of accrued coupon payments and their tax treatment. However, the exercise was repeated with several other coupon and zero coupon bonds, and the pattern highlighted in Table 1 was valid for all of them.

5 However, both price and YTM data suggest that (i) during many a month the variations have been significant, and (ii) there is no pattern or trend so far as the movements of the variance in price/ytm are concerned, neither over time nor cross-sectionally across bonds of different maturities [see Tables 2 and 3]. This is somewhat inconsistent with the observation that bonds that are closer to maturity are on average traded more frequently than the bonds that are set to mature well into the future [see Figure 2] 3. Given the aforementioned hypothesis linking frequency of transactions with price discovery and low price/ytm variance, it would be reasonable to expect that the variance of the prices/ytms for near maturity bonds would by and large be lower than those for longer maturity bonds. The data clearly does not bear out this expected relationship. I C R A B U L L E T I N TABLE 2 S.Ds of prices of bonds of different maturities over the months Month 12.75% 13.5% 13.5% 13.65% ZCB 12% 12.5% Sep % 1.6% 18.9% 6.0% 180.5% 10.1% Oct % 7.2% 10.5% 0.5% 3.4% 0.0% Nov % 26.0% 26.0% 6.3% 45.0% Dec % 26.7% 0.0% 2.8% Jan % 52.1% 17.6% 0.0% 41.6% 31.0% Feb % 6.6% 58.7% 11.7% 22.1% 366.0% Mar % 17.6% 47.7% 45.5% 19.8% Apr % 6.1% 11.8% 16.0% 80.7% 16.1% May % 8.0% 0.0% 20.0% 16.4% 21.8% Jun % 4.3% 20.3% 0.0% 12.4% Jul % 18.7% 30.4% 18.3% 14.5% 13.6% Aug % 13.1% 4.4% 11.8% 8.4% 16.9% Sep % 16.0% 6.4% 23.6% 9.0% 64.4% Oct % 57.2% 34.5% 80.2% 41.9% 64.7% Nov % 36.3% 0.0% 31.8% 28.8% 18.6% Dec % 37.8% 0.0% 27.7% 28.9% 10.3% Jan % 21.0% 6.4% 47.5% 17.6% 14.8% However, both price and YTM data suggest that (i) during many a month the variations have been significant, and (ii) there is no pattern or trend so far as the movements of the variance in price/ YTM are concerned, neither over time nor cross-sectionally across bonds of different maturities... 3 Figure 1 indicates that longer term bonds are traded in larger numbers. Although this is per se not inconsistent with the observation that shorter term bonds are traded more frequently, it demands a closer examination of the data. Information available from RBI s publications and the NSE s web site indicates that more longer term bonds are traded simply because there are more longer term bonds in existence. However, a much higher fraction of the outstanding shorter term bonds are traded than the outstanding longer term bonds. For example, during November 1998, about 78% of the outstanding bonds with less than one year to maturity were traded, compared to about 45% of the outstanding bonds with greater than four years to maturity. 49

6 I C R A B U L L E T I N TABLE 3 S.Ds of YTMs of bonds of different maturities over the months Month 13.5% 13.65% ZCB 12% ZCB 11.75% 12.5% Jan % 533.5% 389.7% 142.8% 95.9% 157.0% Feb % 103.0% 30.2% 356.4% 11.3% 11.3% Mar % 70.9% 20.0% 70.4% 44.2% 47.7% 26.6% Apr % 36.0% 68.3% 60.4% 25.6% 13.9% 11.5% May % 132.4% 106.9% 45.2% 32.3% 6.5% Jun % 19.8% 20.4% 11.2% 6.6% 7.2% Jul % 37.1% 20.5% 11.7% 3.9% 7.5% Aug % 16.8% 59.4% 22.0% 16.5% 7.8% Sep % 21.9% 11.5% 3.2% 5.0% Oct % 19.5% 21.7% 3.1% 10.5% Nov % 47.1% 50.4% 3.1% 5.7% Dec % 35.6% 44.3% 2.2% 4.2% FIGURE 1 FIGURE Bond Trades per Month Bond Trades Per Month 3 yrs or less to maturity > 3 yrs to maturity 3 yrs or less to maturity > 3 yrs to maturity 50 The latest scenario, involving the last six trading weeks of 1998, with respect to variance in gilt prices too has been examined, albeit with recourse to looser methodology. Data available from RBI s Weekly Supplement suggest that there still exists significant dispersion among prices/ytms of gilts with similar time to maturity [see Table 4]. The lower spreads among prices of longer term securities possibly reflect the relative flatness of the higher end of the yield curve. The exercise has highlighted the fact that while the bond traders in India are clearly rational in the economic sense of the word, the market as yet is far from being efficient. The rationale for this phenomenon perhaps

7 TABLE 4 Range of YTMs for bonds of different maturities over weeks Week ended Time of Maturity 27-Nov-98 4-Dec Dec Dec Dec-98 1-Jan % 68.0% 55.0% 112.0% 80.0% 172.0% % 95.0% 105.0% 141.0% 151.0% 121.0% % 37.0% 39.0% 46.0% 32.0% 19.0% % 125.0% 17.0% 6.0% 6.0% 4.0% % 2.0% 17.0% 3.0% 1.0% 4.0% % 72.0% 22.0% 20.0% 22.0% 22.0% % 27.0% 71.0% 30.0% 27.0% 19.0% % 9.0% 12.0% 14.0% 11.0% 13.0% % 23.0% 28.0% 95.0% 9.0% 9.0% lies in the nature of the market itself. Trading in government securities is largely restricted to a telephone market, and on-line two way quotes are available from very few traders and for very few securities. The trades are subsequently reported to the WDM segment of the NSE, but the reporting itself need not be real time. Finally, the trades are recorded in the SGL account of the RBI, typically with a further lag. Hence, on the one hand, market making is limited and, on the other hand, informational asymmetry continues to be a dominant feature of the market. While the extent of informational asymmetry is debatable, and has been questioned by some market participants, there is no doubt about the fact that informational transparency is not a hallmark of the Indian bond market, where on-line quotes and interest rate movements are a very recent phenomenon, thanks to information vendors like Reuters and Bloomberg. As mentioned earlier, this transparency too has de facto been limited to a handful of securities, but at least some progress has been made on this front. However, two-way quotes alone are unlikely to act as a panacea in a market that is dominated by a few large traders. So long as the market remains limited to these participants, trades will continue to be settled on a bilateral basis rather than through a competitive bidding process. The need of the hour therefore, is a manifold increase in the number of active portfolio managers who would roll over their bond portfolios often, and who would collectively provide two way quotes for a wide array of securities. Given that a two way quote is a proxy for a trade, albeit imperfect, the depth of the bond market can then increase sufficiently, thereby increasing the market s efficiency. The government s attempt to increase the depth of the market by allowing foreign institutional investors (FIIs) to trade in both T-bills and dated government securities has come to nought. The exposure of the FIIs to Indian debt instruments is marginal at best. The salvation perhaps lies in the opening up of the insurance and pension funds sectors. I C R A B U L L E T I N While the extent of informational asymmetry is debatable, and has been questioned by some market participants, there is no doubt about the fact that informational transparency is not a hallmark of the Indian bond market, where on-line quotes and interest rate movements are a very recent phenomenon,... 51

8 I C R A B U L L E T I N 52 Two further areas of study are: i) there should be a link between the yield structure and expectations about financial conditions of the governments and the PSUs in the future. The strength and nature of this link can be explored in an exercise, ii) to verify the extent to which the expectations about the spot rates for period (t+i) estimated during period t, and the actual spot rate for that period have differed in the recent past. Summing Up It was felt that this preliminary survey of the Indian bond market should restrict itself to raising certain issues and highlighting some basic characteristics of the market. Hence, as mentioned at the outset, this exercise limited itself to an analysis of the extent of pricing anomaly (i.e. imperfection) in the bond market. However, there are two other issues that beg discussion, and that should be discussed in the future. First we know that bond prices are determined by expectations about the future spot rates. These spot rates are affected by the extent of liquidity in the market, and in a country like India liquidity significantly depends on the earning and expenditure patterns of the governments, and the actions of the public sector undertakings (PSUs). Their bonds together comprise of a very high proportion of the money raised through the bond market. In other words, there should be a link between the yield structure and expectations about financial conditions of the governments and the PSUs in the future. The strength and nature of this link can be explored in an exercise, subject to availability of appropriate data. Second, we have already observed that near maturity bonds are traded much more often than longer maturity bonds. If traders are rational, this phenomenon should be observed only if the uncertainty about longer term interest rates are very high, and if this is common knowledge. An interesting exercise, therefore, would be to verify the extent to which the expectations about the spot rates for period (t+i) estimated during period t, and the actual spot rate for that period have differed in the recent past. This exercise is feasible, but a longer time series would be required for it than what we have at our disposal. The bond market is a fascinating area of research. Given the functional relationship between the bond prices and expected interest rates, it allows us to test interesting hypotheses that cannot be analysed within the paradigm of the equity market. For example, whether or not an equity is overpriced is anybody s guess. But it is possible to verify whether or not a bond is rich. The nature of bonds, particularly the array of possible structures that can be developed with embedded options, and the impact of perceived creditworthiness of the issuer on a bond s price make bond markets a fascinating study even at the micro level. An analysis of the Indian bond market, therefore, would be a recurring theme in future issues of.

9 APPENDICES 1: Pricing Bonds Price as a discounted cash flow It is well known that the price of a bond (P) is given by the discounted value of the income stream that is expected to be generated from it in the future. In other words, if a bond promises (say) annual coupon payments at the rate of x% of the face value, the amount available upon maturity is B, the discount rate is r, and T is the number of years to maturity, then the price of the bond is given by: 4 I C R A B U L L E T I N However, a price per se does not tell us whether a bond is worth buying. A price might seem to be low but, given that economic agents make their decisions in accordance with the opportunity costs of their actions, one has to know whether the price is low enough such that the implicit return from the bond is equal to the opportunity cost of the investment. Hence, bonds are often traded on the basis of their yield to maturity (YTM). The YTM of a bond is the value of r for which equation (1) holds, given P, x, B and T. If the YTM of a bond is equal to or, on occasions, greater than the YTM of comparable bonds, then it is deemed to be a good buy. Economic theory suggests that the YTM of a bond varies inversely with the credit worthiness of the bond issuer, and directly with the extent to which the interest income from the bonds are taxed. These hypotheses are supported by available market data. Yields to maturity and call While the YTM of a bond is an useful indicator of its purchasability, it is not always an accurate measure of the richness of the bond. For example, today an increasing number of bonds are bundled with call and put options. In other words, some of these bonds might be redeemed prior to their maturity. Further, coupon payments obtained from a bond can be reinvested, and the returns from reinvestment can differ significantly. Therefore, the difference between the overall returns from two comparable bonds can differ significantly from the difference between their YTMs. It is difficult to objectively compare the richness of two bonds based on their overall returns, because the rates of return from reinvestment will vary according to the preferences, risk appetite, and expectations of the investors. Hence, market observers usually shy away from the concept of overall or total return. But they try to deal with embedded options using measures of return like the yield to call (YTC). The YTC of a bond can be computed from equation (1) by substituting the exercise value of the call option (C) for B, and the strike period for the maturity period, T. 4 Note that if the discount rate is a function of the opportunity cost of funds, it is unlikely to be constant throughout the life of a bond. In that case, the discount rates shall be given by r i for the i-th period, when r I is the forward rate for that period. However, the current spot rate and the arbitrage-free forward rates are related such that discounting cash flows at the arbitrage-free forward. rates is equivalent to discounting at the current spot rates (Kalotay et al., 1993). 53

10 I C R A B U L L E T I N Pricing bonds using binomial interest tree More sophisticated pricing models for bonds, which take into consideration the possibility of interest rate volatility, use the binomial interest rate tree based pricing model. As suggested by Kalotay et al. (1993), [t]his tree is nothing more than a discrete representation of the possible evolution over time of the one-period rate based on some assumption about interest rate volatility. How do we construct such a tree? Let us assume that the spot interest rate for the present (or the base) period is r 0, and that the volatility of the interest rate during all future time periods is given by σ. It is further assumed that, given any interest rate in period t, there are two possible rates of interest in period t+1: a high rate and a low rate, and that each of these rates occur with equal probability. In other words, it is assumed that the forward rates follow a lognormal random walk, albeit with a known and certain volatility (Fabozzi, 1996). These assumptions ensure that the relationship between the low rate and the high rate, one year into the future, is as follows: 5 r 1,H = r 1,L (e 2σ ) (2) Similarly, the interest rates two years into the future would be given by the following equations: r 2,HH = r 2,LL (e 4σ ) (3a) r 2,HL = r 2,LH = r 2,LL (e 2σ ) The high and low rates for the i-th period can be similarly obtained. Therefore, the binomial interest tree for a three year period is given by: Figure 1 (3b) r 3,LL (e 6 s) r 2,LL (e 4 s) r 1,L (e 2 s) r 3,LL (e 4 s) r 0 r 2,LL (e 2 s) r 1,L r 3,LL (e 2 s) r 2,LL r 3,LL In figure (1), the volatility of the forward interest rates at all nodes has been ascribed the symbol σ. This, however, is a simplification. In 54 5 In effect, an investor would require to know only the value of r 1,L. This rate, known as the one-year forward rate (for period 0), can be estimated using the spot t-year interest rates obtained from an yield curve, and the principle that interest rate arbitrage is not possible in an efficient market. For details, see Kolatay et al. (1993).

11 reality, the volatility of a one-year forward rate is given by r 0 σ, when r 0 is the spot rate for the current period, and s is the current period volatility. The volatility of interest rate for all i periods can thus be obtained. How can the binomial interest tree enable us to compute prices of bonds with embedded options? Let us go back to the example of the 3-year bond. For the moment, let us assume that it is a straight bond. The price of the bond in the current period, i.e., when t=0, should equal the discounted value of the cash flows accruing in periods 1, 2 and 3, and the discount rates would be given by the forward rates highlighted in figure (1). Suppose, that the face value of the bond is B, and that the (annual) fixed coupon rate is x. Then the cash flow expected to accrue at the end of period 3 is (1+x)B. The price of the bond at any node of the second period is then given by: I C R A B U L L E T I N when r 2,j is the one-year rate at the j-th node of period 2, and ½ indicates that the high and low values of r are equally likely in period 3. 6 These discounted values of cash flow from period 3 can be further discounted using forward rates of period 1 to obtain the values of the bond at the two nodes of period 1. One final round of discounting using the current rate yields the current price of the bond, i.e., the discounted value of cash flows as expected in period 0. The pricing process can be described with the help of the following binomial tree: Figure 2 V 3,HHH = (1+x)B V 2,HH = ½(V 3,HHH + V 3,HHL )/(1+r 2,HH ) V 1,H = ½(V 2,HH + V 2,HL )/(1+r 1,H ) V 3,HHL = (1+x)B V 0 = ½(V 1,H + V 1,L )/(1+r 0 ) V 2,HL = ½(V 3,HHL + V 3,HLL )/(1+r 2,HL ) V 1,L = ½(V 2,HL + V 2,LL )/(1+r 1,H ) V 3,HLL = (1+x)B V 2,LL = ½(V 3,HLL + V 3,LLL )/(1+r 2,LL ) V 3,LLL = (1+x)B 6 Clearly, this method also allows investors to value floating rate bonds. In that event, the coupons for the bond will be x high and x low for the high and low interest rate regimes respectively. Hence, equation (4) has to be modified only modestly to take into account the floating-rate nature of the bond. 55

12 I C R A B U L L E T I N Pricing bonds with embedded options Suppose that the bond has an embedded call option that can be exercised at the end of the first period at the discretion of the issuer. The bond will be called if the one-year forward rate at the end of period 1 is lower than the expected level of interest rate at that point in time. If so, the discounted value of future cash flows at one or more nodes of the first period will be greater than the face value of the bond, B. Let us assume that this occurs at only one of the nodes, and let this discounted value be called B. In the binomial tree, therefore, the value B at the appropriate node will be replaced by the value B, and the process of discounting will yield an optionadjusted price for the bond. It is obvious that if a higher cash flow B is replaced by a lower cash flow B, then the price of the bond in the current period will be lower than the price of the option-free bond described in the previous section. The difference between the prices of the straight bond, and that with an embedded option is the (implicit) price of the call option that benefits the issuer. It can easily be verified that if the option is a put which benefits the buyer, then the price of the bond with the option will be higher than the price of the option-free bond. Option adjusted spread As mentioned earlier, even though an investor de facto wants to purchase a bond whose market price is lower than its option-adjusted value, (s)he prefers to think in terms of yields on his/her investment, as opposed to the prices themselves. If a bond is option-free then the investor can simply compare its YTM with that of a comparable on-the-run bond. 7 However, since YTM is not a meaningful measure of the yield in the event that a bond has an embedded option, an investor has to take into consideration the option adjusted spreads of such bonds. 8 What is an option adjusted spread (OAS)? Suppose that the theoretical price of a bond obtained from the binomial tree (P) is not equal to the market or actual price of the bond (P ). Then the OAS of the bond is the constant spread which, when added to all the forward rates of the tree, will ensure that P equals P. It is obvious that the spread will be lower for a rich bond and higher for a cheap bond. If the theoretical price of the bond equals its market price then the OAS will be zero An on-the-run yield curve highlights [t]he relationship between the yield-tomaturity and maturity for bonds of similar credit quality trading at par (Kolatay et al., 1993). 8 Further, YTM is an average measure of yield, and does not take into consideration the shape of the underlying yield curve. It is evident from the above discussion that the shape of the yield curve, which will determine the forward rates, and hence the discount rates for the different time periods, will have a significant impact on the theoretical (option-adjusted) value of a bond.

13 2. Duration and Convexity Duration Apart from ensuring that the bonds in their portfolios are liquid and have acceptable levels of counter-party risk, fixed-income portfolio managers have to take into consideration the nature of the expected cash flow from these securities. For example, a pension fund manager has to ensure that the monthly returns from its portfolio is enough to meet the fund s monthly pension related obligations. Hence, (s)he would be much more interested in a coupon bond with (say) quarterly coupon payments, than in a zero-coupon bond with the same maturity as the coupon bond. Moreover, a fund manager might have to take into account the possibility that (s)he might face a short-term liquidity problem which will force him/her to liquidate a significant part of the portfolio. 9 In that case, (s)he would also be interested in the extent to which bond prices change with changes in the interest rate. This price-interest rate relationship is captured by two measures known as the duration and the convexity of bonds. The duration of a bond is commonly defined as the weighted average of the maturities of the bond s coupon and principal repayment cash flows, where the weights are the fractions of the bond s price that the cash flows in each time period represent (Edwards and Ma, 1993, p. 318). The mathematical relationship between duration and the cash flows is given by: I C R A B U L L E T I N when D is the duration of the bond, P is its market price, r is the YTM of the bond divided by the number of coupon payments in a year, m is the number of payment periods, and X t is the cash flow from the bond in period t. Note that the duration of a bond is expressed in terms of number of payment periods, as opposed to number of years. However, the yeardenominated duration of a bond can be obtained by multiplying the perioddenominated value by 1/f when f is the frequency of coupon payments per year. Finally, the duration of a portfolio of bonds equals the weighted average of the duration of the bonds in the portfolio. Modified Duration As evident from the above example involving a zero-coupon and a coupon bond, in order to avoid a short-term liquidity problem, a fund manager has to ensure that the duration of the firm s assets is similar to the duration of its liabilities. 10 However, in order to realise the usefulness of 9 For example, suppose that a life insurance company has agreed to provide a fixed rate loan to its policyholders. If, therefore, the market rate of interest exceeds this agreed upon fixed rate, a significant proportion of the policyholders might take a loan, thereby precipitating a liquidity crisis. In such an event, the fund manager of the life insurance company might have to liquidate part of his/her portfolio to meet the company s contractual obligations. 10 It can easily be verified that the duration of a zero coupon bond equals its time to maturity. For coupon bonds, the duration is typically lower than their time to maturity. 57

14 I C R A B U L L E T I N this measure in the context of marking to the market a bond portfolio, given a change in the interest rate(s), one has to introduce a related measure known as modified duration. Algebrically, the relationship between duration and modified duration is given by: when R is the annualised YTM, and f is as defined in the previous paragraph. More importantly, however, the modified duration of a bond equals the ratio of the percentage change in its price to the percentage change in its YTM. Hence, given any change in the interest rates, which would also alter the YTM of a bond, the change in the bond s price is given by the following relationship: percentage change in P = -D mod x percentage change in (1+R) (7) Convexity Clearly, modified duration is the slope of the price-ytm curve. However, while this curve is generally convex in nature, modified duration is essentially the slope of a straight line approximation of this convex curve. Hence, modified duration overestimates the price change if the yield increases, and vice versa. Therefore, in order to obtain an accurate value of the price change, given a change in the YTM, a portfolio manager will also have to take into account the extent of the curvature of the price-ytm curve. 11 The resultant measure for a bond is its so-called convexity. Mathematically, convexity is defined as: when the symbols have the same interpretations as in the case of duration. The relationsip between a bond s C and its price is given by: percentage change in price = ½ x C x (change in yield) 11 (9) As before, the year-denominated value of C can be obtained by dividing the period-denominated value of C by the square of f. Effective duration and convexity The estimates of modified duration and convexity of a bond, as described above, are obtained under the assumption that the future cash from the instrument will not vary with changes in future interest rates. However, this assumption is unlikely to hold if the bond has embedded options. For example, the modified duration of a bond that is callable at the An accurate measure of the extent of the change in a bond s price can be obtained by summing the estimates of the changes obtained using modified duration and convexity.

15 discretion of the issuer is likely to be lower than that of a straight coupon bond. Hence, portfolio managers are increasingly using effective duration and effective convexity, in lieu of modified duration and convexity. These measures can be approximated as follows: I C R A B U L L E T I N where P 0 is the initial price of the bond, and P - and P + are the prices that can be obtained by shifting the aforementioned binomial tree by AR basis points. 59

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