6

Debt Instruments and Their Valuation

Traditionally, debt has been identified as a long-term source of fund for a corporate enterprise. Bonds, usually, are expected to offer a fixed rate of interest and bondholders have a claim on the issuer of the security for the payment of the principal amount at the time of repayment of the debt. Though debt has been viewed over a period of time as long-term debt, there are many instruments that have been designed to meet the short-term requirements of the market participants.

The terminology used for debt instruments differs among countries and among instruments. For instance, in Germany, debt instruments are known colloquially as “bunds”. The classifications are “Bundesschatzanweisungen” (2–4 years maturity); “Bundesobligationen” (5 years maturity); and “Bundesanleihen” (10–30 years maturity). In France, government debt instruments are called “Obligations Assimilable de Tresor” (OATS). In UK, government debt instruments are categorised as “Gilt Edged Securities”. In Italy, they are called “Buoni del Tesoro Poliennali”. In the US they are called “Treasury Notes”. In India, debt instruments are called “Bonds”.

In most countries, private sector debt instruments are called “corporate bonds”. However, within this category, there are a number of instruments differentiated mainly by their claim on the enterprise’s assets. In this context, debentures are debt instruments that are secured on the assets of the enterprise issuing the instrument. The security may be by a fixed charge (on specific assets) or a floating charge (on the assets in general). Private enterprise debt instruments that do not have any fixed or floating charge on the assets of the company are known as “unsecured bonds”. Some debt instruments could be guaranteed by a third party, usually the parent company or a group to which the enterprise belongs; such debt instruments are known as “guaranteed bonds”.

Corporate debt instruments can also called “convertibles”. These instruments, in addition to the normal characteristics of a bond, carry an option at future time duration to convert to other types of financial instruments say equity shares. Other corporate debt instruments with options are termed “callable bonds” or “puttable bonds”. The redemption (repayment) date of these bonds can be decided later, at the discretion of the issuer (callable) or the holder (puttable). Some corporate bonds are double dated, implying that their redemption can take place at any time between two dates, usually agreed upon at the time of issue of such debt instruments.

In view of the volatile inflation rates, a few corporate bonds as well as government bonds have come up with “index-linked bonds”. This means that their par value is updated periodically in line with a price index and the coupon payment is also increased or decreased by the amount of change in the index. When such instruments are issued by public sector enterprises, they are called “variable rate bonds”. When they are floated by private sector enterprises, they are called “floating rate bonds”.

“Eurobonds” are another type of debt instruments. They are bonds issued in a currency that is not the currency of the country of issue. The eurobond market has been the most innovative of all debt instruments. Eurobonds are different from “foreign bonds” which are denominated in domestic currency and issued in that country by foreign corporate enterprises or non-residents. In London, foreign enterprises may issue “bulldog bonds”, while in the US, foreign enterprises may issue “yankee bonds”.

Usually, the ownership of the bonds is “registered”. This implies that the issuers of the debt instruments maintain a register of current owners and pay these registered bondholders the interest “coupon rate” either monthly, half-yearly, or annually. However, some of these bonds are also issued as “bearer bonds”. For instance, most eurobonds are bearer bonds. The term “bearer bonds” imply that whoever surrenders the coupons will be able to claim the interest amount. In this case, possession of the bond coupon establishes the proof of ownership. There are debt instruments that do not pay any interest. They are termed as “zero coupon bonds”. These bonds do not pay any interest, but are issued at a discount to their redemption value (at the time of maturity) to provide a return to the holders of these instruments. They are also called “deep discount bonds”. The deep discount bonds have the characteristic of longer maturity, no interest, low initial payment, and high face value.

A new bond instrument is the “strips”. Each strip is the single component cashflow of a bond traded separately. Strips are similar to “zero coupon bonds” in the sense that no interest payments are attached to them. For example, a bond maturing in four years for which half-yearly coupon interest are due, carries eight interest payments and one principal repayment. This instrument could be split into nine strips (eight coupon payments with different maturities plus one principle payment after four years). Each of these strips is traded in the market as separate zero coupon bonds maturing on their appropriate dates.

The traditional bonds that are issued with an interest rate are often termed as “straight bonds”, “plain vanilla bonds”, or “bullet bonds” to distinguish them from these improvised debt instruments. These traditional bonds are issued with a par value. This is the price at which they will be redeemed at maturity. Bonds may be issued at a discount or a premium to their par value. These bonds pay a fixed amount as interest to their holders at set intervals throughout the life of the bond. When this amount is set in absolute terms, it is known as a “coupon”. The coupon rate can be obtained by dividing the coupon amount by the par value of the bond. This coupon is paid till the maturity of the bond. The maturity of the bond is the time duration for which the bond document is valid. Residual maturity is the time till the repayment of the par value from the current date. Thus, a five-year bond, issued in January 2000, is due for redemption in January 2005. If the present time is worked out as January 2004, the residual maturity of this bond will be one year and not five-years.

Bonds are classified according to these residual maturities. The bonds may thus be classified as “short”, “medium”, or “long”. Though long-term debt instruments are those that have a maturity of at least one year, the sub-classification is usually as follows:

< 5 years’ residual maturity are called short-term debt instruments;

Debt instruments with a residual maturity of 5 to 15 years are called medium-term instruments; and Debt instruments with more than 15 years’ residual maturity are called long-term debt instruments.

PLAYERS IN THE DEBT MARKET

Both government and private enterprises that need long-term financial assistance issue fixed interest securities in all countries. Government participants could be the Central Government, state governments, or other regional administrations. When public sector enterprises issue bonds, they are also usually categorised as government bonds since they have characteristics similar to government bonds. Table 6.1 shows the value of bonds traded in the Wholesale Debt Market (WDM) segment of the National Stock Exchange in India.

 

 

New issues of corporate bonds are similar to the issue of equity instruments. Debt instruments are issued in the primary market of the stock exchange, just like equity instruments. It is usually a public offer for sale at a fixed price. They may also be issued by auction in some markets. In the auction method, no price is fixed. Debt instruments are sold to the highest bidder at the price they submitted at the auction.

The issue of government bonds, on the other hand, is often most crucial since it reflects the monetary policy of the country. The traditional method of issue of government bonds is a sale by tender in which the government offers a specified quantity of debt instruments for sale on a specific day at a minimum price and invites bids. If the offer is undersubscribed, all offers are accepted. In this instance, the government retains the unsold stock and releases it to the market subsequently when conditions are favourable. Such subsequent issues are known as “tap stock”. On the other hand, if it is oversubscribed, the highest bids are accepted, but at a common price. The government may also choose the auction method to issue the instruments in the market. In this method no minimum price is set. The bonds are sold to the highest bidders at the price they bid. From the point of money supply, sale by auction has the advantage that the issuer can set the volume, knowing that it will be fully subscribed, since the price will adjust to ensure this. The third method is to “buy” the bond itself and to release it to the market as and when market conditions permit. This is similar to the “tap stock” issue by the government.

After debt instruments are issued in the stock exchanges, they subsequently come into the secondary market for trading. The secondary market trading systems are either “quote-driven” or “order driven”, or have a mixed nature. Dealer markets are called quote-driven markets. In dealer markets, the dealer makes a market by holding an inventory of his stocks and announcing (continuously) a price at which he is prepared to buy and sell. Since the quotes are given first, the available price is known beforehand; this is known as a quote-driven market.

In an order-driven market, trade takes place when dealers can match orders to buy and sell. In this case they are responding to orders and trying to find a price that will match the maximum number of buy and sell orders. The matching process may be continuous or it may take place at specified times. Sometimes, these markets may also work on a call basis rather than on a continuous basis. In case of a call basis, there is a call auction at the beginning of the day where all orders accumulated since the last close of trading are executed, followed by continuous trading throughout the day. Buyers and sellers may specify with their order whether they wish to exercise their order immediately or at the next call. In such cases, auctioneers do not hold their own inventory level since they buy from a seller and simultaneously sell to a buyer.

The cost of buying and selling securities are twofold. The cost besides the price of the security includes the charge by the dealer as commission. This is calculated as a certain percentage of the value of the transaction of the deal, subject to a minimum/maximum amount. The price of the security will usually have a bid price and an offer price. The bid price is the price at which the dealer will buy. This is always below the price at which the dealer will sell or offer the security. The difference between these two prices is called the spread. The lower the spread, the better is the market mechanism and the equilibrium price is within this narrow range.

VALUATION OF DEBT INSTRUMENTS

Since debt instruments mature at a definite time duration in the future, the valuation of debt instruments is the present value of all future cash flows discounted at an expected rate of return. The general formula used for the valuation of debt instruments is:

where

V_D = value of debt instrument

I_j = Interest due at time j

P_n = Par value at redemption time

k = Discount rate per annum,

n = Maturity time of the debt instrument

 

Most debt instruments are priced using this formula since it accommodates for change in interest rates, change in par value, and also change in discount rates over a period of time. However, debt instruments such as the zero coupon bonds do not have the characteristic of interest payment. Hence, their valuation will be based on the discounted par value at the redemption time alone. The formula for the valuation of zero coupon bonds is

In the above methods, the value of ‘k’ is presumed. That is, the investors’ expectation of the return from the debt instrument is assumed considering the time value of future returns. When the return expectation varies from investor to investor, the price/value of the instrument also tends to be different across investors. To overcome this, the bond yield is computed considering the traded price of the debt instrument as the current value of the debt instrument. The yield to maturity (YTM) is the rate that equates the current price to the future cash flows from the debt instrument. In other words, the formula for computing the YTM is

YTM is the annual yield to maturity. When the debt instrument pays interest at monthly, quarterly or half-yearly time durations, then YTM in the equation is replaced by (YTM/m) where m is the number of times interest is paid in a year and n in the equation is replaced by (n × m).

The measure of yield to maturity itself is not consistent due to several factors. Each of the factors in the equation leads us to define the term “yield” differently. Besides these, the current market price can also take two forms. These are the “ex-coupon price” and “cum-coupon price”. These arise due to the regular time intervals in which interest or coupon is paid on the bond instrument. For instance, a half-yearly coupon payment bond will pay interest every six months while a monthly coupon bond will pay interest every month. The debt instrument could be traded either on the day the interest has been paid or on any day in between two interest payment dates. This is also referred as clean price and dirty price of the bond.

Assume, for example, interest payments are made every six months, on a 5 per cent Rs 100 bond, maturing in two years. If the first issue is in January 2003, interest payments will be respectively in the end of June and December 2003 as well as June and December 2004. An investor might buy the bond in June just after the coupon payment and wait for the next six months to receive the interest from the investment. Another investor might buy the bond from the market in the beginning of November and might receive the payment of interest for six months by holding it only for one month. Here, if there is no price adjustment, the investor who buys the instrument pays less for the interest while the investor who sells the bond loses the interest for a five-month holding period. In practice, hence, long-wait bonds are priced below short-wait bonds to compensate for the waiting time to receive the interest from the bond issuer.

Figure 6.1 shows the price movement between two interest payment dates. The cum-interest price increases as the accrued interest increases. Accrued interest is that part of interest which is yet to be paid to bondholders, but has been earned since the last payment date. For the example illustrated earlier, the accrued interest in November will be

Out of the Rs 2.50 interest that will be received every six months, since the first investor held the security for four months till October, the number of days has been computed as 123 out of the 184 days in that period. The amount of accrued interest, hence, is that proportion of Rs 2.50, that is, Rs 1.67. When this accrued interest is added to the clean price or ex-coupon price, the dirty price/cum-coupon price is arrived at. The general formula for the computation of accrued interest is

where

C = Coupon payment

m = No of coupon payments in a year

d_j = No of days since the last coupon payment

Td = Total number of days between two coupon payments

 

In the debt markets, the price quoted in the exchange is the ex-coupon price or the clean price, where the effect of interest is not taken into account. However, when the actual transaction takes place, it should be presumed that the price exchanged between the buyer and seller will not be the clean price but the dirty price. The buyer will pay the accrued interest on the bond to the seller of the debt instrument.

There is an additional consideration that is to be evaluated by an investor besides the changes discussed above. This happens when there is a difference in the dates of coupon payments and book closure date for closure of the register books of the issuer. This implies that since bonds are traded, the issuer holds the register of bondholders to pay interest on the due dates. The buyer of the bond intimates the change to the issuer and the issuer duly incorporates the name of the new buyer into the register books and deletes the name of the old bondholder. This mechanism may require a few days; hence, the issuer might identify a date as a book-closure date. Beyond this date, the issuer will not recognise any trade. The trade will be recognised only after coupons are issued to registered bondholders. After the book closure date, the interest will go to the previous owner of the debt instrument and not the new owner. Hence, this trade is called ex-coupon, meaning without the benefit of the coupon payment. The new owner has to hold the bond for more than six months to receive interest from the issuer. During this period, instead of the dirty price, being made up of the clean price plus accrued interest, the dirty price will be now below the clean price. Figure 6.2 shows this situation. The dirty price is below the clean price on all book closure dates marked as Bd in the figure. The dirty price is equal to the clean price as on the date of coupon payment at the points C1, C2, C3, and C4.

The formula for accrued interest when there is a book-closure date in interest/coupon payments will be

where

d_xi = No. of days since the last book-closure date

d_xc = No. of days between the book-closure date and coupon-payment date

 

For example, if we assume that the 5 per cent interest is paid half-yearly on June 30 and December 31, and June 15 and December 16 are the respective book-closure dates, the accrued interest on the Rs 100 debt instrument as on June 16 will be as follows:

 

            d_xi = June 15 to June 16 = 1 day

            d_xc = June 15 to June 16 = 15 days

            T_d = 184 days

            C =5

            m = 2

 

Substituting these values in the formula gives us a value of -0.19 accrued interest. This negative value reduces the market value of the debt instrument to below the clean value. This is because the coupon payment of Rs 0.19 will go to the seller and the buyer needs compensation for holding the bond but not receiving the next coupon. This is illustrated in Figure 6.3.

On the other hand, assume that the debt instrument is to be bought on December 1. Here, the accrued interest will be computed as follows:

 

            d_xi = June 15 to December 1 = 169 days

            d_xc = June 15 to June 16 = 15 days

            T_d = 184 days

            C =5

            m = 2

 

The calculated accrued interest is Rs 2.09, a positive value that increases the dirty price of the debt instrument. This ensures that the seller is compensated for not receiving the coupon even though he had held the debt instrument for a total of 169 days. Figure 6.4 shows the point that results in increased market price of the debt instrument.

Examples

1. Essar Oil pays 12 per cent per annum quarterly interest rates due every March, June, September, and December end. The quoted price in the market is Rs 86 on January 24,2003. Determine the accrued interest as on this date.

   The accrued interest is (12/4) * (24/97) = 3 * 0.2474 = Rs 0.74

2. Hotel Lee’s 10 per cent per annum half-yearly interest rates are due every March and September end and has a current quoted price of Rs 44 on February 3. What should be the price at which the debentures will be exchanged in the market on this date?

   The accrued interest amount is (10/2) * (126/182) = 5 * 0.6923 = Rs 3.46

           Exchange price (Dirty price) = 44 + 3.46 = Rs 47.46.

   Figure 6.4 The effect of book-closure dates

   
3. The Tata Investment Bond 2002 was issued in January 2003 with a maturity of two years. The coupon payment is 6 per cent per annum made every six months (face value Rs 100). If the current market price on January 15,2003, is Rs 96, what is the yield to maturity?

           96 = (3/(1+x)^1) + (3/(1+x)^2) + (3/(1+x)^3) + (3/(1+x)^4 + (100/(1 + x)^4); x = 0.04

   (r/2) = x Hence, the annual return from the bond (r ) = 0.08(8%).

4. Jindal debt instruments (face value Rs 10) are zero coupon bonds issued at Rs 6.30, encashable at the end of two years. Compute the return for the debenture instrument.

   6.3 = (10/(1+r)^2); (1+r)^2 = 1.5873; 1+r = 1.2599; r = 0.2599−25.99%

5. Vidhi Dyeslu has its bonds traded in the Bombay Stock Exchange. The clean price of the instrument as on March 30 is Rs 188.50. An interest of 6 per cent is paid half yearly at the end of March and September. The book closure dates are March 28 and September 27. If the instruments are exchanged as on March 30, what will be the accrued interest? What is the dirty price of the instrument?

   Accrued interest = (6/2) * [(2−3)/184]) = (3 * −0.0054) = −0.0163

       Dirty price = clean price + accrued interest = 188.50 − 0.0163 = Rs 188.4837

Besides the concept of YTM, other yield measures are also useful to investors in identifying a possible investment opportunity. They are the running yield, simple yield to maturity, and holding period yield.

RUNNING YIELD

The simple rate of return relating the periodic coupon payments to the clean price of the debt instrument is called the running yield. It is also known as interest yield or current yield. The formula for the computation of running yield is

 

 

The assumptions behind the formula is that the debt instrument is bought, held for a period, and then sold for the same price. In this instance, the return on the bond consists of the periodic payments (coupon payments); hence, the rate of return relates only this income stream to the price that was paid for it. The formula uses the clean price rather than the dirty price in calculating the running yield. The explanation for this is that the dirty price of a debt instrument varies as a linear function of the number of days to the next coupon payment date. Since the coupon payment itself is a constant, and assuming the coupon payments could be paid on all days irrespective of the payment from the issuer through market mechanisms, the running yield replaces/uses the clean price to compute the return from the debt instrument.

This method is often used to estimate profits of debt instruments held for a short term. For instance, compared to other short term interest rates, if the running yield is higher, then those instruments could be used to fund the debt instrument that has a higher running yield. The borrowing rate hence is lower and the investor is able to make a short-term profit by trading in the debt instrument. However, the assumption is that the time to the maturity of the debt instrument is also short.

A debt instrument with a clean price of Rs 100 and coupon payment of Rs 6 will have a running yield of .06 or 6 per cent (coupon/clean price = 6/100), irrespective of the date of computation.

Running yield has the limitation that it does not take into account any capital gains that could be realised by an investor from investing in the debt instrument. For instance, an investor might profit from buying the Rs 100 bond stated earlier at a market price of Rs 90 and sell it after the receipt of coupon at par value. Here, the investor would realise not only the coupon rate but also the capital gain due to the difference in market price and par value plus interest accrual.

Examples Anil buys a Rs 100 bond with a coupon interest of 5 per cent. The ex-coupon (clean) price is Rs 97.45. The running yield on the bond will be

 

 

The running yield of a 7 per cent coupon bond (face value Rs 100), clean price Rs 104.58, and dirty price Rs 104.76 will be

 

 

Arvind buys a 10 per cent bond at a price of Rs 109.43 with borrowed funds at 9 per cent interest and holds the bond for three months. What is Arvind’s profit from the transaction, ignoring transaction costs and capital gains/loss?

The running yield on the bond is (10/109.43)*100 = 9.14%. The cost of borrowing funds is 9%. Hence, the profit from the transaction would be 0.14%. This, however, is not the exact benefit for Arvind since he would also have incurred a capital gain or loss for the three months from the sale of the bond.

SIMPLE YIELD TO MATURITY

Simple yield to maturity takes into account not only the coupon payment but also the capital gains that are realised from holding the debt instrument over a period of time. The formula for computation of simple yield to maturity is

where

C = coupon amount

F = face value of the bond

P = clean price

n = number of years to maturity.

 

Simple yield to maturity thus considers the running yield plus the capital gain (the difference between face value and clean price). The assumption is that capital gains accrue evenly throughout the years of the investment.

Simple yield to maturity is useful since investors can compute the return from their bond investments till the date of redemption of these instruments. This method gives only an approximate return from the bond instrument and not the exact return from holding the instrument. This yield measure does not consider the fact that the coupon payments received can be reinvested. Hence the measure understates the yield that actually accrues to the investor.

Examples Kanika buys a five-year 7 per cent bond at Rs 99.48. The simple yield to maturity of this bond will be

 

 

Note that if the running yield had been computed, the return would have been merely 7.04%.

A 10-year 9 per cent bond bought at Rs 108.32 can be said to have a simple yield to maturity of 7.54%. The respective calculation is

 

 

A 15-year 12 per cent bond is bought in the market at Rs 112.45 with five years to maturity. The simple yield to maturity of this instrument is computed as follows:

 

 

Deepa buys a 10-year 8 per cent bond with six months to maturity at Rs 99.89. The simple yield to maturity for Deepa will be

 

 

REDEMPTION YIELD

Redemption yield considers the fact that coupon payment received over the lifetime of the bond can be reinvested, as well as any capital gain or loss that might occur between purchase and redemption. The redemption yield is the return that equates the discounted values of the bond’s cash flows back to its dirty price. This can also be called the internal rate of return for the debt instrument. This can be stated mathematically as

where M-market exchange price of the debt instrument.

This equation gives an approximate redemption yield and is a good approximation for long-term debt instruments having a maturity of more than 15 years.

Assume a Rs 100 20-year 5 per cent debt instrument has a market exchange price of Rs 114.67, while the clean price of the instrument is Rs 112.59. The redemption yield can be determined approximately as follows:

 

 

For short-or medium-term debt instruments, a trial and error method could be used to determine the yield to maturity. This method uses the following equation to determine the

Here,

M_d = market exchange price (dirty price) of the debt instrument

n = number of days between the current date and the next coupon payment

Y_r = redemption yield

q = number of coupon payments before redemption

m = number of times coupon payments are made

Td = total number of days between two coupon payments

RP = redemption price.

 

The first part of the equation uses the redemption yield to discount the future cash flows back to the next coupon payment date but, the next coupon payment date could be any fraction of a coupon payment duration. Hence, it is discounted back to the present.

Examples Anup buys a five-year 6 per cent bond with a maturity period of three years on October 13 for Rs 98.56. The coupon payments are made semi-annually at the end of April and October. By trial and error it can be seen that the redemption yield will be between 0.0754 and 0.0756. The redemption yield can be identified as 7.8%.

 

 

Rama buys a 10-year 4 per cent bond with a maturity period of seven years on September 15, for Rs 103.42. The coupon payments are made semi-annually, at the end of June and December. The redemption yield for the bond will be:

 

 

A 14-year 8 per cent bond is bought at Rs 97.34 on May 7. The bond can be redeemed after 12 years. The interest is paid at the end of every March, June, September, and December. The redemption yield for the bond is

 

 

The redemption yield makes two assumptions that may not hold well in all situations. One, it assumes that the investors will hold the debt instrument till maturity which, may not be true. It takes into account the effect of compound interest on the coupons when they are reinvested and in this it assumes that the coupons can be reinvested at the constant redemption yield. This assumption might not hold when the interest rates are fluctuating in the market.

HOLDING PERIOD YIELD

The holding period yield further refines the return computations by considering different reinvestment rates and by recognising a holding time period which can be different from the maturity date. The equation for computing the holding period yield is as follows:

 

where,

Y_h = holding period yield

i₁ and i₂ = rates of interest at which the first coupon, second coupon, and so on can be reinvested

P₁ = the price at which the debt instrument will be sold by the investor.

 

The above equation can be restated as follows:

Comparison of Yield Computations

The graph in Figure 6.5 shows the movement of the different yield curves for the same debt instrument that are traded at different prices. The assumption of 6 per cent Rs 100 face value bond maturing four years, interest payments being made half-yearly in the end of June and December, is being made for the debt instrument. The computations are based on the assumption that the instrument has been bought on October 15 at varying prices. The X-axis shows the Market Price and Y-axis yield. The running yield is within the range of 6.5% to 5% while the simple and redemption yield have higher variation. When the market price is low at Rs 95 these yields are 7.5% while they are as low as 4.5% when the market price is Rs 104.6. At Rs 100, all the yields are 6%.

The simple and redemption yields are very close to each other at different price levels.

YIELDS TO CALL/PUT

A yield to call is a yield on a security calculated by assuming that interest payments will be made until the “call” date, at which point the security will be redeemed at its “call” price. This is the percentage rate of a bond if the investor buys and holds the security until the call date. This yield is valid only if the security is called prior to maturity. Generally, bonds are “callable” over several years and normally are called at a slight premium. The calculation of yield to call is based on the coupon rate, length of time to call, and market price. Yield to call is the yield that would be realised on a callable bond in the event that the issuer redeemed the bond on the next available call date.

where

Yc = yield to call

C = coupon amount

Pc = Call price/put price

M = Market exchange price (dirty price) of the debt

n = the number of years to call/put date.

 

Similar to call bonds, a company may also issue put bonds. The valuation of put bonds is similar to call bonds. The price at call will be replaced by the price at put. The put bond gives an investor the right to redeem the bond prior to maturity.

Examples A 15-year 8 per cent (quarterly interest) bond, issued on call after five years, has the following call prices for the Rs 100 face value instrument. Year 5, call price Rs 103. Year 6, call price Rs 102. Year 7, call price Rs 101. Year 8 onwards call price is Rs 100. The current cum coupon price is Rs 171. What is the yield on call if the company exercises its call option in the sixth year?

 

        171 = (∑(2/(1+{Yc/4})^i) + (102/(1+Yc)^6)

              i = 1 to 24

Solving for the above equation, Yc =.09 or 9%

A 10-year 6 per cent (half-yearly interest) bond issued on call after three years has the following put prices for the Rs 100 face value instrument. Year 3, put price Rs 101. Year 4, put price Rs 100.50. Year 5 onwards, put price Rs 100. The current market price is Rs 124. What is the yield on put if the investor exercises its put option in the third year?

 

        124 = (∑(3/(1+{Yp/2})^i) + (101/(1+Yp)^3)

              i = 1 to 6

Solving for the above equation, Yp =.07 or 7%

YIELD ON INDEX LINKED BONDS

In certain bond markets, the government and/or corporate enterprises issue bonds that have either or both of their coupon and principal linked to a price index, such as the retail price index, or a consumer index, or a commodity index, or a stock market index. To compute the yield on such index-linked bonds, it is necessary to make forecasts of the relevant index first, which are then used in the yield computation.

The linkage of the bond index could vary, that is, in some instances, only coupon rates will be linked to the index while in others only the principal payment will be linked to the index. There are also instances where both principal and coupon payments are linked to the index.

When the principal and coupon payments are linked to an index, these initial values will be scaled up or down in accordance with the movement of the index. For instance, if a bond linked to a Retail Price Index, (RPI) is issued the coupon payment will be

where,

C = coupon payment

m = times the coupon payment is made

RPI_base = base of the index

RPI_C-d = index value before the coupon payment time

 

In case of half-yearly payments, ‘d’ will be six months plus the delay in publication of index and computation of coupon payment.

The computation of principal amount (assuming a principal amount of Rs 100) is

Here, the ratio is the proportion of retail price index value ‘d’ duration earlier to maturity time ‘M’ to base index value.

Two types of yield measures are computed for index-linked bonds—nominal yield and real yield. The nominal yield forecasts all future cash flows from the bond. Since future cash flows for the bond are linked to the index, the yield requires a forecast of future index values. In case of price indexes, the forecast is made on the basis of the expected inflation rate. For example, the forecast of retail price index will be

where,

RPI_t = the forecasted “retail price index”

RPI _base = the current “retail price index”

i = the assumed future annual inflation rate

m = the number of months between base index and forecasted index.

 

The money/nominal yield is computed using the following equation.

where,

M _d = market exchange price of the bond

m = number of times coupon payments are made

Y_ri = nominal yield

M = maturity payment

n = total number of coupon payments till maturity.

 

Real yield is related to nominal yield. The computation of real yield is as follows:

Here Y_ri is the nominal yield and ‘i’ is the forecasted inflation rate and Y_ry is the real yield.

The equation used for computing the real yield can be restated as follows:

where,

RPI_a = RPI₁(1+i)^(1/m)

where,

RPI₁ = index in the first year

i = the inflation rate

m = the number of times interest is compounded

Examples An index-linked bond (eight years) with a coupon rate of 3 per cent per annum (half-yearly) was issued in April 1994. The base RPI for 1994 was 146.3, while the RPI for 2002 was 183.9. Calculate the actual coupon payment and principal repayment.

 

 

The computed nominal yield of a bond is 4.3 per cent per annum (half-yearly interest payment) and the expected inflation rate is 2 per cent. Compute the real yield.

 

        Y_ry/2 = {[1 + (.043/2)]/[(1 + .02)^^(1/2)]}− 1

                    = {1.0215/1.0099 } − 1 = 1.01149 − 1 = .01149

           Y_ry = 0.01149 * 2 = 0.02298 = 2.3%

YIELD CURVES

A yield curve is a graphic representation of the relationship between yield and maturity of securities. The curve captures the relationship between yield and maturity at a particular time. The shape of the curve changes over time. Yield curves are used to compare yields of different securities, benchmark rates, and discover yield curve divergence. To be representative, the instruments plotted on the yield curve must have common characteristics, such as same credit risk and same tax treatment.

Ascending Yield Curve

Higher returns as time progresses are shown by ascending yield curves. This could be the shape of bond yields in developed countries. Such a curve shows that yield rises for longer maturities. The risk related to such long duration bonds also are high. The ascending yield curve could show a steady, steep increase or a steep increase in the beginning followed by a slower rate in the latter years. An ascending yield curve could also start at a very slow pace in the beginning and increase in later years. These formations are shown in the Figure 6.6.

Descending Yield Curve

The descending yield curve depicts a reduction in return as time progresses. Short-term interest rates are high compared to long-term yield rates. Longer maturities, hence, have low yields. This shape is often seen when the market expects interest rates to fall, either due to control of interest rates or due to economic changes. These yield curves can also show different slopes as shown in Figure 6.7.

RISK MANAGEMENT IN BONDS

A bond ensures a fixed return over a period of time, defined by the bond agreement. In the market-place, the rates of interest are not constant and vary with time. Owing to the changes in interest rate from the date of entering into a debt contract to the date of maturity of the debt contract, the investor faces the risk. Since bonds are exchanged in the market, when interest rates in the market changes the price received/ paid for the bond in the market would be below/above the expectations. Besides this price risk, bond investors also face a reinvestment risk. This risk is inherent since there is an assumption that interest rates are reinvested at the same rate as defined in the bond contract. However, with changing interest rates, the reinvestment rates also differ and this would result in below/above expectations benefit at the time of maturity. Besides debt instruments might face purchase power risk and liquidity risk.

Default Risk

Default risk identifies the uncertainty prevalent in the repayment of interest and principal to the bondholders. There is always a possibility on the part of the issuer of the debt instrument that might make the repayment schedule uncertain. To offset this fear and to improve the trading volume in the debt market, several credit rating agencies have sprung up. They offer their services of measuring the repayment capacity of the issuer of debt instruments.

The safety of capital depends on the issuer’s credit quality and ability to meet its financial obligations. Issuers with lower credit ratings usually have to offer investors higher coupon rates to compensate for the additional credit risk. A change in either the issuer’s credit rating or the market’s perception of the issuer’s business prospects will affect the value of its outstanding securities. In most cases, these securities carry investment-grade ratings from two or more rating agencies such as CRISIL and ICRA.

Reinvestment Risk

The maturity period of bonds are spread over a fixed time duration. Since the par value or repayment value is also fixed, the investor purchasing the debt instrument for such a long duration is holding it for a known fixed rate of interest, irrespective of fluctuations in interest rates in the market.

Movements in interest rates will inversely affect the market value of the securities. When interest rates rise, market prices of debt securities fall, as demand increases for new-issue securities with the higher rates. As prices decline, yields are brought in line with the prevailing rates. When interest rates fall, market prices will rise, because higher rates on prevailing debt securities will make them more valuable. Downward trends in interest rates also create reinvestment risk—the risk that income or principal repayments will have to be invested at lower rates. Reinvestment risk is an important consideration for investors in “callable” securities.

Purchasing Power Risk

Debt instrument investors have to look at the real rate of return, or the actual return minus the rate of inflation. Rising inflation has a negative impact on the real rates of return, because inflation reduces the purchasing power of the investment income and principal.

Price Risk

Investors who need their principal prior to maturity have to rely on the available market for the securities. Although investors in debt securities may take advantage of the exchange listing to sell their instruments prior to maturity, the price received may be more or less than the purchase price as a result of market risk factor (demand and supply for funds).

Liquidity Risk

The exchange listing of debt securities does not guarantee liquidity. The liquidity in the market is mostly influenced by the demand and supply situation for that instrument by the market players.

The differential demand and supply might induce the price received in a sale prior to maturity to be more or less than the liquidation value or principal amount, and more or less than the amount an investor originally paid.

A bond investor can try to minimise these risks in a bond portfolio by a process called immunisation. In other words, if an investor gets back a yield from the bond which is at least the computed yield from holding the bond till its maturity period, then the bond investment is said to have been immunised. The measures of bond immunisation lie in duration analysis, convexity analysis, and dispersion analysis.

Maturity of bonds play a vital role in the measurement of bond yield. However, the inherent risk of the instruments makes coupon rates much more important than the maturity period. For instance, a 10-year bond with a 5 per cent coupon rate will be considered more sensitive to interest rate changes than a 10-year bond with an 8 per cent coupon rate. Similarly, a five-year zero coupon bond may be viewed as more sensitive than an eight-year 6 per cent bond.

Since, for a given maturity period, coupon rates are subject to different volatility, the yield to maturity becomes an easy way of comparing bond performance. Similarly, for similar yield instruments, the bond duration becomes an easy way of comparing bond performance.

BOND DURATION

A measure used to make a comparison across different coupon rates is the bond duration. Bond duration compares the sensitivity of the instruments to changes in interest rates. Bond duration is the average amount of time required by a security to receive the interest and the principal. Duration, hence, is a weighted average of the times that interest payments and the final return of principal are received. The weights are the present values of the payments, using the bond’s yield to maturity as the discount rate.

The duration thus calculates the weighted average of the cash flows (interest and principal payments) of the bond, discounted to the present time. Fund/portfolio managers typically use this calculation when they are planning for cash flows that will be required over time. The bond duration helps in determining the need for additional cash flows or surplus cash positions. As duration increases, the risk of recovering the full value of the bond also increases.

Duration is stated in terms of years. Meaningful interpretation of duration analysis is done when duration is multiplied by the percentage change in interest rates. The duration measure will predict how much a bond’s price should change given a 1 per cent change in interest rates. Thus, a bond with a duration of 5 years, will decrease 5 per cent in price if yields rise by 1 per cent. For example, if interest rates rise from 6 per cent to 7 per cent, an investor holding a 6 per cent bond priced at Rs 100 with a duration of 5 years, will see the price of that bond drop by 5 per cent to 95.

Duration thus helps an investor to identify the percentage change in the price of a bond. For example, if a bond has duration of 10 years and if interest rates fall from 8 per cent to 6 per cent (a drop of 2 percentage points), the bond’s price is expected to rise by 20 per cent (10 × 2).

The duration of the bond can be computed using the following formula:

 

Duration =

∑(Present value of cash flows * times to cash flows)/∑(Present value of cash flows)

 

There are two types of duration—Macaulay duration and modified duration. Macaulay duration (Figure 6.8) is useful in immunisation, where a portfolio of bonds is constructed to fund a known liability. Modified duration is an extension of the Macaulay duration and is a useful measure of the sensitivity of a bond’s price (the present value of its cash flows) to interest rate movements.

Example A bond has a face value of Rs 100 and a 7 per cent coupon. The yield to maturity is 5 per cent and it matures in five years. The bond thus pays Rs 7 a year for five years and the principal amount of Rs 100 in the fifth year. Compute the Macaulay duration.

Macaulay duration is computed as follows:

A 15-year Rs 100 zero coupon bond is sold at Rs 36.25. The yield to maturity is 7 per cent. Calculate the duration.

 

All zero coupon bonds will have time to maturity as duration since there is only one cash flow at the end of maturity. The duration of a perpetual bond is (1 + y)/y. Y is the yield to maturity.

The price yield curve explains clearly the utility of the duration measure. The following price-yield curve shows that as yields fall, bond prices rise, and vice versa. However, at low yields, prices rise at an increasing rate as yields fall; at higher yields, prices fall at a decreasing rate as yields rise. This price behaviour is shown in Figure 6.9.

The duration formula estimates a bond’s movement along this price-yield curve. However, the duration formula is only a linear approximation of movement along the curve. It follows that for large changes in yields (interest rates), the duration formula will consistently underestimate the amount of price movement. Duration overestimates the price decline associated with a large upward change in yield (interest rates) and underestimates the price rise associated with a large downward change in yield.

Graphically duration can be shown as the slope of the tangential line drawn at a price against the price-yield curve. Thus, the slope shows the percentage change in price for a given change in yield rates. This is shown in Figure 6.9.

Modified duration is calculated as shown below.

 

 

where,

y = yield to maturity

m = number of discounting periods in year.

 

Modified duration indicates the percentage change in the price of a bond for a given change in yield. The percentage change applies to the price of the bond including accrued interest. For the illustration given previously with a Macaulay duration of 4.42 years, the modified duration will be 4.42/1.05 or 4.21 years.

Therefore, a change in the yield of +/− 2.5 per cent should result in a percentage change in the price of the bond of −/+ (4.21 *.025) −/+ 0.10525 (−/+ 10.525 %). For the bond with the market priced at Rs 108.71; YTM −5%; duration −4.42 years, the estimated prices are Rs 120.152 at 4.875 per cent yield and Rs 97.268 at 5.125 per cent yield.

CONVEXITY

Though modified duration will not be able to predict change in prices for a large increase/decrease in yield rates, it is still a good indication of the potential price volatility of a bond. The discrepancy between the estimated change in the bond price and the actual change for a large change in yield is due to the convexity of the bond, which must be included in the price change calculation when the yield change is large. Convexity accounts for the additional price movements associated with changes in yield rates.

Convexity is the rate at which price variation changes for a change in yield. Owing to the shape of the price-yield curve, for a given rise or fall in yield, the gain in price for a drop in yield will be greater than the fall in price due to an equal rise in yield. This “upside capture, downside protection” is what convexity accounts for. Mathematically, modified duration is the first derivative of price with respect to yield, and convexity is the second derivative of price with respect to yield. Convexity can also be stated as the first derivative of modified duration.

Convexity accounts for the curvature of the line. The convexity formula measures the rate of change of modified duration as yield rates change, fully accounting for the dynamic relationship between prices and interest rates. By using convexity in the yield change calculation, a much closer approximation can be achieved.

 

Given the same example used for explaining computation of modified duration, convexity can be calculated as follows:

 

The expected change in prices for a 2.5 per cent change in yield are (4.01 * +/−0.025 = −/+ 0.1002 = −/+ 10.02%. The expected change in prices will be Rs 119.60 at 4.875 per cent yield and Rs 97.81 at 5.125 per cent yield.

The concept of convexity is also important in hedging. In hedging a portfolio of bonds, investors prefer instruments with less convexity than the target portfolio. This is because the short-hedged instrument will experience less volatile pricing in terms of adverse interest rate movements.

Example Determine the convexity of an 8 per cent coupon bond A (Rs 100) sold at Rs 96.480 with two years to maturity and a zero coupon bond B (Rs 100) with 20 years to maturity sold at Rs 14.9. The yield to maturity on these bonds is 10 per cent per annum.

If the yield changes to 9 per cent, then the price of the bond computed using only duration is

 

 

The new price computed using modified duration will be

 

 

The new price computed using adjustment for convexity is

 

DISPERSION

Dispersion measures the variance in the timing of a bond’s cash flows around its duration date using the present value of cash flows as weights. When cash flows are volatile, dispersion will be very large. The computation of dispersion is as follows:

 

Example The convexity and duration of a bond at 10 per cent yield is 1.59 and 1.92 respectively. Compute the dispersion.

        Dispersion = 1.59 * (1.1)^2 − (1.92)^2 − 1.92 = 1.9239 − 3.6864 − 1.92

                            = 1.9239 − 5.6064 = − 3.6825.

SUMMARY

Debt instruments are varied in nature and characteristics. Some innovative instruments in the market are the convertible debt securities, index-linked bonds, eurbonds, and callable or puttable bonds.

Bond investments also carry risks such as the interest rate risk and reinvestment risk. These risks are managed by investors using the concepts of duration, convexity, and dispersion. The investor can hold a portfolio of bonds with diversified risk return maturities that would maximise the investment yields and minimise the risk composition of the portfolio.

CONCEPTS

• Floating rate bonds	• Index linked bonds
• Yield to maturity	• Running yield
• Holding period yield	• Yield to call
• Yield curve	• Bond duration
• Modified duration	• Convexity
• Dispersion	• Fixed rate capital securities

SHORT QUESTIONS

1. What are the innovations in debt instruments?
2. What is zero coupon bond?
3. What are variable rate bonds?
4. What is yield to maturity?
5. What is bond duration?
6. How are bond duration, convexity, and dispersion related to each other?

ESSAY QUESTIONS

1. Describe the characteristics of corporate bonds.
2. What are the bond management strategies?
3. Describe the features and risks of fixed capital securities.
4. Explain the significance of the yield curve.
5. Explain the importance of bond convexity.

PROBLEMS

1. Compute the yield to maturity of Essar Oil non-convertible debentures if the market is Rs 88.50. Assume that a debenture’s face value (Rs 100) carries an interest rate of 10 per cent per annum (interest due every quarter) and is redeemable in March 2007.
2. A Rs 25,000- face value bond with a four year maturity and a11 per cent coupon rate sells in the market for Rs 26,201. Calculate the yield to maturity.
3. A 6 per cent bond (face value Rs 100) matures next month. If the yield on the bond is 10 per cent, what is the price of the bond?
4. A bond pays an interest of 10 per cent on the face value of Rs 100 every quarter, beginning January. An investor buys the bond on February 25. What is the accrued interest?
5. A debt instrument is bought on March 31. Half yearly interest payments of Rs 100 are due on the instrument at the end of every December and June. Compute the accrued interest on the instrument. Suppose the clean price of the instrument is Rs 87.4, what is the dirty price of the instrument?
6. An 8 per cent debt instrument is traded in the market at Rs 123. What is the running yield of the instrument?
7. A 9 per cent bond (maturity three years, face value Rs 100) is bought at Rs 104.3. Compute the running yield and simple yield to maturity.
8. A 10-year 6 per cent debenture instrument has a market quote of Rs 106.34. The accrued interest is Rs 4.08. Compute the redemption yield on the bond.
9. A five-year 8 per cent callable bond (face value Rs 100) gives the investor the right to call the bond from the fourth year onwards at Rs 100. The current market price of the bond is Rs 98.40. Compute the yield to call.
10. The nominal yield on a bond (half-yearly payments) is 6 per cent. The inflation rate is 3 per cent per annum. Compute the real yield on the bond.
11. A bond has a face value of Rs 100 and carries a 6 per cent per annum coupon payment. The yield to maturity is 5 per cent and the maturity period is five years. Compute the duration of the bond.
12. Compute the duration and the modified duration of a half-yearly coupon bond (7 per cent per annum) with a face value of Rs 1000, maturing at the end of the sixth year. The yield to maturity of the bond is 5 per cent.
13. A 7.5 per cent bond matures in five years. The bond price is Rs 100. Coupon payments are made every six months. The bond repayment price at the end of the fifth year is Rs 100. Assume a yield to maturity of 7.5 per cent and compute Macaulay’s duration.
14. Determine the convexity of an 8 per cent coupon bond with three years to maturity, coupon payments per year. The yield on the bond is 10 per cent per annum. If the yield changes to 11 per cent, what will be the price of the bond (after adjusting for duration and convexity)?
15. A company has a perpetual working capital liability of Rs 10 crore every year. In what proportions should the company invest in zero coupon bonds of 1-, 15- and 20-year maturities so that the debt is immunised?