, I i hand, if the receipts are at the beginning of the period, the value requirod i~ 1. If no value is given, the function takes the defau~ value to be 0, i.e. the ca~h flows are at the end of the period. In the present instance, the payments are at the beginning of the year. Accordingly. the parameter Type has to be given an input of 1, In addition, we have to give Rate = 8 %, Nper = 8, Pmt = 1,000. The parameter, FV need not be given any input. We derive the present value of the annuity as As. 6,206.37 with the help of these values. When an annuity is received forever, it is called as perpetuity. Interestingly, it is easy to calculate the present value of perpetuity. The present value of an annuity of As. a, discounted at r per cent, tends to Bfr as n tends to infinity PV 01 an end-period perpetuity = air (1 .16) This formula is valid when payments are received at the end of the period. In case payments are received at the beginning of the period, then Ihe present value of the perpetUity is expressed by the formula PV of a beginning-period perpetuity = 8(1 + r )Ir ( 1.17) Illustration 1.14 shows the calculation of present value of perpetuity. Illustration 1.14 Find the present value of perpetuity of Rs. 5,000, discounted at 5 per cent Present value of a perpetuity = air = 5,00010.05 = As. 1,00,000 APPUCATIONS 1.7 PROJECT EVALUATION An Important application of discounting is in the area of projacl evalualion. A project is evaluated on the basis of projected future cash flows. Sinoa all etlsh flows are not at the same point of time, ~ becomes necessary to find the present value of all such cash flows. In projacl evaluation, we have to consider both cash inflows and outflows, Normally, there is a cash outflow at the beginning of the project although there can be subsequent cash outflows. Inflows can be generally expected .. fter a gestation period. They need not be equal and may not be at regular intervals. NPV method is commonly used to evaluate individual projects. Net present value Is the sum of present values of all cash flows; with a negative sign to 13 cash outflows. In effsct. it becomes Ihe difference between the present values of cash inflows and cash outflows (Illustration 1.15). lIIu~tration 1.15 If an investment of As. 10,000 yields, As. 3,000, As. 4,000 and Rs.5,OOO at the end of first, second, and third years, find the net present value of the project. assuming a discounting rate of 7 per cent. Excel provides a function NPV to find net present value. The function reculres values lor two input parameters -Rate and Values. The cash flows include ~oth inflows and outflows. We must be carefUl to give opposite signs to them. By convention, we give positive sign (+) to Inflows and negative sign H to outflows. Thus, the cash flows are -10,000, 3,000, 4,000, 5,000 By inputting the values to parameters as Rate = 7% and values as {-10.OIJO, 3,000, 4,000, 5,OOO} the Excel function NPV gives the net present value as Rs. 354. Do not forget to give negative sign to outflows. Also remember that to use the NPV function, cash flows should be at equal inteIVals. If that were not the case, one has to find individual present values and sum them with appropriate signs. The utility of NPV to compare two projects is limned. NPV is an absolute value and depends on the size of the project. Larger projects are likely to have larger NPV whereas smaller projects may have smaller NPV and therefore, it may not be appropriate to use NPV to compare projects of different sizes (Illustration 1.16). '"ustrstlon 1.16 Evaluate projects A and 8 based on the cash flow projections presented as under, assuming a discounting rate of 9 per cent Today After 1 Year After 2 Years After 3 Year.s: ---------------After4 Years Affer5 Years Project A -60,000 25,000 30,000 35,000 40,000 45,000 Project B -2,00,000 1,20,000 1,20,000 -80,000 1,20,000 1,20,000 Using the NPV function, we can obtain the net present value for project A as RB. 49,437 and for project Bas 1,03,047. Prima facie, it looks Ihal project 8 is betler than project A as the net present value is higher. 14 In view the limitation of NPV, another method called the internal rate of return (IRR), is used in project evaluation. In this method, instead of finding the net present value of all cash flows using an assumed discounting rate. WG try to find the discounting rate where the net present value is zero. That is the discounting rate at which the present value of all cash inflows is equal to the present value of all cash outflows. That discounting rate is called the IRR and the project that has a higher IRR can be expected to yield better return. Let us try to understand the meaning of IRR with the help of an example. Assume that an investment of Rs. 1,00.000 today yields Rs. 1,50,000 after 3 years. An investment of Rs. 1,00,000 is expected to give a rate of return f every year. So the amount at the end of first year is 1,00,000 (1 + r) . Thus, the investment at the beginning of second year is not 1,00,000 but 1,00,000 (1 + f). Similarly, ij becomes 1,00,000(1 ... f)2 at the end of second year and 1,00,000(1 + tl at the end of third year. We know that the amount available at the end of 3 years is 1,50,000. So we can equate 1,00,000(1 + rj3 to 1,50,000 and find the value of r. Instead of doing the calculations, we can use the Rate function of Excel. We have to give values to the parameters -Nper = 3; PV = 1.00,000; FV = 1,50,000. There is no value for the parameter Pm!. Since payments are at end we need not give value to the parameter Type. There is a parameter Guess, which we can ignore. Rate is equal to 14.47 per cent. This is the internal rate of return (IRR). It is easy to understand the concept of IRR in single cash outflow and single cash inflow. But consider a case where there are multiple cash inflows spread over a few years -like an initial investment of Rs. 10,000 giving cash flows of Rs.3,000, 4,000 and 5,000 at the end of first, second, and third years. What is the IRR In such a project? Let it be f . Then Rs. 10,000 becomes Rs. 10,000 (1 + t) at the end of first year. Out of this amount Rs. 3,000 is withdrawn leaving an investment of [10,000 (1 + r) -3,000] at the beginning of second year. This investment becomes [10,000(1 + f) -3,000] (1 ... f) at the end of second year. An amount of Rs. 4,000 is withdrawn from this amount, leaving an investment of ([I 0,000(1 + r) -3,000)(1 + f) -4,OOO} at the beginning of third year. This investment becomes Ifl 0,000 (1 + f) -3,000J(1 + r) -4,000} (1 ... r) at the end of third year. We have to equate this to Rs. 5,000, the final cash flow. We have {[I 0,000 (1 + f) -3,000] (1 + r) -4,000} (1 .. f) " 5,000 i.e. {[10,OOO (1 + r) -3,000] (1 ... r) -4,000} = 5,000/(1 + r) i.e. [10,000 (1 ... f ) -3,000] (1 + r) = 4,000 + 5,000!( 1 ... f), i.e. [10.000 (1 .. r) -3,000] = 4,0001(1 ... r) ... 5,000!(1 ... r)2. i.e. 10,000 (1 + f) = 3,000 + 4,000/(1 + r) + 5,000/(1 + rl', i.e. 10,000 = 3,000/(1 ... r) + 4,000/(t + r}" + 5,0001(1 ... f)" We can easily see that r is nothing but the discounting rate at which the 15 present value of cash outflow, i.e. Rs. 10,000 equals the presenl value of all cash inflows, i.e. Rs. 3,000 at the end of first year, Rs.4,000 at the end of second year and Rs. 5,000 at the end of third year. This is nothing but IRR. There are no direct methods to find r from this equation. We have to use trial and error method following the steps given as under: 1. Start with a guess rate, r. 2. Find NPV and compare it with zero. It can be (a) = 0, (b) > 0, and (c) < 0. 3. If NPV is zero, then r is IRR. 4. If NPV > 0, then decrease r and go to Step 2 5. If NPV < 0, then increase r and go to Step 2. In any case, these steps are nat required as Excel provides a function called IRR. It requires two parameters -Values and Guess. Cash outflows and inflows are given as Values. Note to give negative sign to outflows. The cash flows should be at periodic intervals. The parameter Guess can be ignored. For the example under conSideration, IRR is 8.9 per cent. Now let us see how IRR can be used to compare two projects with the help of Illustration 1.17. Illustration 1.17 Compare projects A and B of Illustration 1.16 using I RR method. Today After 1 After 2 After 3 After 4 After 5 Year Years Years Years Years Project A -80.000 25.000 30,000 35.000 40,000 45,000 Project B -2,00,000 1.20,000-1,20,000 -80,000 1,20,000 1,20,000 We can use the IRR function of Excel to get the internal rates of return of projects A and B as both are more or less equal to 30 per cent -implying that both the projects are expected to give similar returns. One of the serious limitations of IRR is that it assumes same return throughout the life of the project. 1.8 BOND VALUATION Bonds are instruments that promise the holder certain cash flows; government securities, and corporate bonds are some examples of bonds. A bond promises a fixed lump sum amount on a predetermined day in future. The fixed lump sum amount is called the face value of the bond. The predetermined date on which the face value is paid is termed as the maturity date. The time period up to the maturity date Is term to maturity. Besides the face value, the bond also promises periodic payments unhl the 16 maturity date. These periodic payments are called coupon paymgnto. ThggQ coupon payments are usually calculated as some fixed percentagQ. called the coupon rate, of face value. As an illustration, consider a 5·year bond with the facg value of Rs. 1.000 at 8 per cent, with half-yearly coupon payments. Here the coupon ratg Is 8 pgr cent and the coupon payments are every half-year. Each coupon paymgnt is, therefore, half of 8 per cent of 1,000, i.e. Rs. 40. Accor.dingly. this bond promises Rs. 40 every half-year for 5 years -resulting in 10 coupon payments. Along with thg last coupon payment at Ihe end of 5 years, there is also a payment of Rs. 1,000, the face value on the maturity date. Therefore, the cash flows are 40. 40, 40, 40, 40, 40. 40, 40, 40, 1,040 You can notice that there are three important characteristics to a bond -face value, fixed coupon rate, and maturity date. There can be deviations to these characteristics. There can be bonds without any coupon payments but only a lump sum payment on the maturity date. They are called zero coupon bonds. There can be coupon payments. But they need not be necessarily fixed. The coupon rate can be detennined at the lime of coupon payment, by linking the rate to some eldemal indicator like inflation rate or a reference interest rate. Such bonds are called floating rate bonds. Some bonds may not have any maturity date. There may be promises to keep paying some fixed amount forever. Such bonds are perpetuities. There can be a combination of these deviations in diHerent ways, leading to plethora of hybrid instruments. But normally a bond has fhree intrinsic characteristics -face value, coupon payment, and maturity date. These characteristics are attached to the bond till the end. They do not change. However, there are two more characteristics of a bond that keep changing -one is its price and the other is its yield. The price of the bond is the price at which it is traded. It is not necessary that a bond is traded at Hs face value. A bond with a face value of As. 100 need not be traded at Rs. 100. It can be traded for an amount less than Rs. 1 ~O, say Rs. 98.50 or an amount greater than Rs. 1 ~O, say As. 102. If it Is traded below the face value, it is said to be traded al discount. If it is traded above, it is said to be traded at premium. The value at which a bond is traded is the mutually agreed value of the bond: agreed by both the buyer and the seller. How or why do they agree for such a vatue is the question. To understand this Question and Its answer, we have to understand the concept of yield of a bond. Yield, in its simplest terms, is the expected rate of return. An 8 per cent bond with the lace value of Rs. 100 gives the bondholder As,8 every year. If the bond is purchased at Rs. 100 and if the holder geG As. 8 every year, then the rate of return on the bond is B per cent. However, If the bond is purchased at As. 99 and not at Rs. 100, then Ihe return has to be 17 calculated on Rs. 99. Since Rs. 8 is received on an investment of Rs. 99, the rate of retum is (8199)'100 = 8.08 per cent But in this case, we are not taking into account the difference in the values received at different point of time. While Rs. 99 is invested today. Rs. 8 is received after 1 year, 2 years, 3 years and so on. To consider such differences, we need to U5e the present values of cash flows. The rate of retum on a bond is thus that rate of return, which equals the present value of all cash inflows on the bond to its price. This rate of return is called yield-to-maturity (YfM) of the bond. When we refer to yield of a bond, we usually refer to its YfM. Illustration 1. 18 Find the yield of a 5-year 8 per cent bond with the face value of As. 100 purchased at Rs. 99. The coupon payment is As. 8. The term to maturity is 5 years. The face value and the price are Rs. 100 and Rs. 99 respectively. By inputting the values to the four parameters -Pmt = 8, Nper = 5, FV = 100, and PV = -99 10 the RATE function of Excel, the rate can as be obtained 8.25 per cent. (See the negative sign to PV vis-it-vis the positive sign to Pmt and FV.) Yield or YfM, as it is called, is nothing but the IAR referred to in project evaluation. Like IAR, here also it is assumed that the rate does not change till the maturity of the bond, which may not be realistic. Another assumptiOn made is that the bond is held till maturity and is not sold in between. This is also not realistic. However, this assumption does not aner the yield-to-maturity concept. (Why?) Yield, as we have seen, is derived as that rate that equals present values of all cash inflows and outflows. What does it signify? It is the rate of return expected by the buyer of a bond. The buyer Is willing to pay a price (cash outflow) for a series of coupon payments plus face value (cash inflows). The buyer expects to get a rate of return on investment, which is nothing but the price paid for the bond. Yield can be understood to be the expec1ed rate of return. Since all characteristics of the bond -term to maturity, coupon payments, and face value -are fixed and predetermined, it depends mainly on the price. How is the price determined? From the concept of yield, it is easy to infer that pries is nothing but the present value of all future cash inflows discounted at the expected rata of return, which Is nothing but the yield-to-maturity or yield. The actual method of calculating price of a bond (also called as value of a bond) is shown in Illustration 1.19. 18 //Iustration 1.19 Find the value of a 3-year 8 per cent bond with the face value of Rs. 100, ~ coupon payments are made every half-year. Assume lhat the prevailing interast rate on similar bonds is 9 per cent. We should always keep In mind the difference between the coupon rate and the expected rate of return. Coupon rate is tagge,e}to the bond. It is generally fixed and does not change over the life of 1M bond. On the other hand, expected rate of return depends on market realit~s and perceptions. It is market determined and keeps changing. Prevailing interest rate in the market on similar bonds becomes the expected return for any particular bond. For the bond under consideration, the coupon payments ara every halfyeea at the rate of 8 per cent per annum, which means Rs. 4 every half year. Thus, the cash flows are 4,4, 4,4, 4, 104 The appropriate discounting rate is the prevailing interest rate on a similar bond, which is 9 per cent per annum. As the coupon payments are made every half-year, the interest rate is half of 9 per cent. which is 4.5 per cent. There are in all 6 coupon payments. We can use the PVlunction as we have the values for all the parametersRaat ~ 4.5 %, Nper ~ 6, Pmt ~ 4; and FV ~ 100. The present value turns out to be Rs. 97.42. . We can see that the bond in the illustration 1 .19 is traded at discount. The reason is obvious. A bond that has a coupon rate of 8 per cent has an expected rate 01 return of 9 per cent. Since the coupon rate is less than the expected rate of return, price has to be lower to compensate the investor for the difference in expected rate of return and the coupon payments given by the bond. The bond has to be traded at discount. As a natural corollary, we can say that a bond gets traded at a premium if the prevailing interest rate is less than the coupon rate. Thus, • A bond is traded at discount if the prevailing interest rale is orOO16r than the coupon rate . • A bond is traded at premium if the prevailing interest rate is less than the coupon rate, From the illustrations 1.18 and 1.19, we can see that price and yield ar" tw~ sides 01 the same coin. Yield can be determined if price is given and if yield ig given price can be calculated. Will there be only one yield in the market? The answer is no. Yield lor a 5-year bond may not be equal to the yield for an B-year bond. TypicallY. yields 19 are different for bonds of different tenns to maturity. Based on the market forces that drive them, yields change. Prices change along with yields. Change in the price of a bond has a great significance to its holder. The holder gains If the price rises and loses if It falls. Prices are dependant on yields. So changes in yields (which are nothing but the prevailing interest rates) affect bondholder. The holder has to keep reassessing the value of the bondS held. Thus, the value of a bond depends on its yield. Let us see the relationship price has with yield. • Price goes up when yield goes down. • Price goes down when yield goes up. It is expected. When yield, which is nothing but the rate that can be expected in the market, goes uP. the buyer will not be interested in paying the same price. The buyer will be willing to pay a lesser price, bringing down the value of the bond. In a similar way, when Interesl rates go down, the seller will not be interested in receiving the same price. The seller will be willing to receive a higher price, pushing up the value of the bond. This inverse relationship between yield and price is brought out by Illustration 1.20. Illustration 1.20 Whal is the price of a 6-year 8 per cent bond with the face value of Rs. 100 ~ yield on a similar bond in the market is 8.5 per cent? What will happen to the price if yield goes up by 0.5 per cent to 9 per cent? What will happen if yield goes down by 0.5 per cent to 8 per cent? We have Pmt ~ Rs. 8, Nper ~ 6, FV = Rs.l00, and Aate = B.5%. Using PV function, we can get the price equal to Rs. 97.72. Now we can change Rate to 9 per cent and obtain price to be As. 95.51. We can see that as the Interest rate increases from 8.5 per cent to 9 per cent, price decreases from Rs. 97.72 to Rs. 95.51, i.e. a fall of Rs. 2.21. Now suppose instead of riSing by 0.5 per cent the interest rate decreases by 0.5 per cent whiCh is equal to 8 per cent. What Is the price when yield is 8 per cent? Without any calculation, we can tell that the price is Rs. 100 (why 7) As interest rate decreases from 8.5 per cent to 8 per cent, the price increases from Rs. 97.72 to Rs. 100, i.e. an increase of Rs. 2.28. From Illustration 1.20, we can clearly see that the decrease and increase in price are not the same for an equal rise and fall in yield. It leads to another interesting observation on the relationship between price and yield. • The fall in the price of a bond for a given rise in the yield is less than the rise in the price of a bond for exactly same fall in the yield. 20 Now we will look at the relationship between the price of a bond and ils term to maturity. Let us first state the relationship and then di~cuss its rationale . • As term to maturity decreases, the premium or discount on a bond decreases at an increasing rate. We can make one obvious observation about the price of a bond on the maturity date. On that day, price is equal to the face value; there is no premium or discount. On any day prior to the maturity date, the bond is sold at a premium ~ coupon rate is higher than YTM and is sold at discount if coupon rate is lower than YTM. The quantum of premium or discount depends on YTM to a large extent. It also depends on term to maturing, I.e. the time between the present day and date of maturity. Farther the maturity date, greater the quantum of premium or discount. It decreases as it approaches the maturity date. The rate of decrease increases as the maturity date approaches. Illustrations 1.21 and 1.22 serve as examples to undersland the relationship between price and term to maturity of a bond. Table 1.,1 summarises the observation clearly. The quantum of premium or discount decreases. The decreases are higher as term to maturity approaches. Illustration 1.21 Consider a bond maturing in 3 years with face value of Rs. 100 and coupon rate 6 per cent. What is Its price today if prevailing interest rate is 8 per cent. Assuming that the interest rate remains same, what is the price ot the bond after 1 year, after 2 years, and after 3 years? Let us first find the price at the end of 3 years. At the end of 3 years, term to maturity is zero. The present value of cash flows is the value of cash flows themselves. Therefore, the bond is sold at par -no discount and no premium. Now let us find the price of the bond at the end of 2 years. This means that term to maturity is 1 year. We can use the function PV. The values to be provided are Pmt = 6, FV = 100, Nper = 1, and Rate = 8%. We get the price as Rs. 98.15. The bond is sold at a discount of Rs. 1.85. Similarly, after 1 year, when the term to maturity is 2 years, WIl gilt thll price of the bond as Rs: 96.43. The bond is sold at a discount of Rs. 3.57. As on today, when the term to maturity is 3 years, the price is Rs. 94.85. The disoount is Rs. 5.15. A summary of the resuns is provided in Table 1.1 . Illustration 1.22 Solve the problem relating to the bond in illustration 1.21 with tne assumpTion that the interest rate is 4 per cent. 21 The price of the bond on the maturity date (when term to maturity is zero) is its face value, i.e. Rs. 100. When the term to maturity is 1 year, the price can be obtained as Rs. 101.92. It is sold at a premium of Rs. 1.92. When the term to maturity is 2 years, Ihe price becomes Rs. 103.77. The premium increases to Rs. 3.77. The term to maturity today is 3 years. The price today would be Rs. 105.55, quoted at a premium of Rs. 5.55. A summary of Ihese results is presented in Table 1.1 . Table 1.1 Term to Maturity and Premium/Discount (Amount in rupees) .-Term to Yi61d == 4 Per Cent Yield: 8 Per Cent -. --Maturity Premium Decrease in DIscount Decrease in Pr6mium Discount 3 5.15 5.55 2 3.57 1.58 3.77 1.78 1 1.85 1.72 1.92 1.85 0 0.00 1.85 0.00 1.92 From Table 1.1 we can observe that when yield is 4 per cent, the premium on bond decreases from Rs. 5.15 to Rs. 3.57, to Rs. 1.85, to Re. a as the term to maturity decreases from 3 years to 2 years, to 1 year to a year. However, the difference in decreases in premium from 3 years to 2 years is Rs. 1.58. This difference increases to Rs. 1.72 between 2 years and 1 year. It further increases to Rs. 1.85 between 1 year and 0 year. Similarly, when yield is 8 per cent, the discount on bond decreases from Rs. 5.55 to Rs. 3.77, to Rs, 1.92 and a as the time to maturity decreases from 3 to 2 to 1 to a years. However, the amount of decrease in dlscounl increases from Rs. 1.78 to Rs. 1.8510 Rs. 1.92. Before we go further on interesl rale changes and risks associated with such changes, let us consider one specific type of bond called zero coupon bond (ZeB). Normally, bonds have two types of cash flows -coupon payments, and face value. In the case of zess, there are no coupon payments. There is only one lump sum payment, the face value, towards the end ofterm to maturity. illustrations 1.23 and 1.24 relate to valuation of ZeBs and its comparison with valuation of coupon bonds. //Justration 1.23 FincJ the price of 7 -year zero coupon bond with the face value of Rs. 100 it the IntereSI rate is 8%. Compare the price of this bond with a similar bond paying coupons at the rate of 7 per cent. 22 The price of a zero coupon bond is the present value of Rs. 100 received after 7 years, discounted at 8 per cent. By supplying values to parameters Rate; 8 %, Nper ~ 7, and FV ; 100 to PV function, we get the price oHhe 2CB as Rs. 58.35. It the bond has coupons paid at the rate of 7 per cent, there are yearly payments of Rs. 7 for 7 years. In addition to the values supplied to Rate, Nper, and FV, supply the value of 7 to the parameter Pmt. We get the price of the coupon paying bond as Rs. 94.79. You can see that the price of the zero coupon bond is less -in fact far less -than a similar coupon bond. It is understandable as there is only one cash flow in the case of a 2GB, which is its face value and that too at the end of the term to maturity. IIlus/ration 1.24 For the bonds in Illustration 1.23 calculate prices if interest rate goes up to 8.5 per cent. Find out the per cent change in the price of zero coupon and coupon bond for the same change in interest rate from 8 per cent to 8.5%. Using PV function, we can calculate the prices when yield is 8.5 per cenl. The price of the 2CB is Rs. 56.49. The price of 7 per cent coupon bond is Rs. 92.32. Table 1.2 presents prices, decrease in prices and per cent decreases for both zero coupon and coupon bonds for the two interest rates -8 per cent and 8.5 per cent. Tabla 1.2 Impact of Interest Rate Change on lCe and Coupon Bond Interest Rate Price Decrease in Per Cent Price Decrease Zero coupon B% 58.35 8.5% 56.49 1.66 3.18 Coupon 8% 94.79 8.5% 92.32 2.47 2.61 It can be seen from Illustration 1.24, and more specifically from Table 1.2, that per cent decrease in the price of a zero coupon bond for a given per cant increase in the interest rate is higher than the corresponding per cent decrease in the price of a coupon bond for the same per cent increase in the intaregt rate. This observation leads to two questions: Firstly, is holding a zero coupon bond more risky than holding a coupon bond? Secondly. if so, how much risk is involved? We will try to answer these questions in the next section on interest rate risk. 23 1.9 INTEREST RATE RISK We have seen that the value of a bond decreases as Interest rate increases. But an increase in interest rate has an advantage too. The coupon payments the investor receives can be reinvested at a higher rate. Similarly, when the interest rate decreases, the value of the bond increases. But the coupons can be reinvested at the lower interest rate. ThUS, there are two risks associated with changes in interest rate. The value of the bond decreases when interest rate increases. But the decrease is off set to some extent because there Is a possibility of earning higher retums by reinvesting the coupon receipts at a higher rate of Interest. When interest rate decreases, returns on reinvestment ot coupon receipts also fall. But they are off set to a great extent by the inc reese in the value of the bond due to decrease in interest rates. This helps us in understanding answers to the two questions raised in the previous section. In the case of leB, there are no coupon payments to off set the loss due to reduction in the value of a bond when the prices increase. So they are more sensitive to interest rate changes compared to coupon bonds. Thus, holding a leB is more risky than holding a coupon bond. The statement is qualitative and answers the first question -compared to a coupon bond, whether leB is riSky or not. To answer the second question -how much riskier it is, we require a quantitative measure of risk; to be specific interest rate risk. For deriving such a measure, let us first look at two extreme situations -a cash flow of Rs. 100 today and a cash flow of Rs. 100 after 3 years. A cash flow of Rs. 100 today carries no interest rate risk al all. While cash flow of Rs. 100 after 3 years carries the risk of changes in the Interest rate for 3 years. Now compare this with cash flow of Rs. 100 after 5 years, which carries the risk for another 2 years. It is natural to conclude that a bond that has cash flows farther away carry greater risk. Not all real life cases are extreme cases like the one mentioned above. It is not easy to assess how tarther cash flows are in the case of bonds that have different cash flows al different points of bme. In such cases, it is useful to find some sort of average time poinl at which all cash flows are received. The average should be able to indicate how far the cash flows are from today on an average. For example, consider the cash flows -8, 8, 8, 8, and 108. There are 5 cash flows, at the end of years I, 2, 3, 4, and 5. The average of these 5 years is 3. But we cannot say that all cash flows, on an average, are received at the end of third year, In fact by that time only Rs. 24 is received and Rs. 116 is still due. So calculating simple average does not work. One needs to look at weighted average with suitable weighls (see Box 1.4). 24 1 • , I , I Box 1.4 Weighted Average Simple average is the sum of values divided by number of values. Suppo,", 6 percent and 8 per cent are rates of retums from two inve:Jtment~, then the average retum can be obtained as (6 + 8)12. which equals 7 per cent. Such an average is meaningful only if the sizes of the two investments are equal. ~ they are unequal, say 2 lakhs and 5 lakhs. then simple average does not represent true average retum. We have to give appropriate weights to the investments -21akhs to the first investment. and 5 lakhs to the second investment. To get the weighted average, 6 per cent is muniplied by 2 lakhs and 8 per cent by 5lakhs. The two products are added and the sum is divided by sum of weights -2 lakhs and 5 lakhs. Thus, the weighted average IS: (6' 2,00,000 ... 8' 5,00,000)/(2,00,000 + 5.00,000) = 7.4 percent. Uke average, weighted average lies between the extreme values; it is nearer to the value whose weight is higher. In the present example, 7.4 per cent lies between 6 per cent and 8 per cent and is nearer to 6 per cent. Anhough 2 lakhs and 5 lakhs are used as weights in this example, normally weights are ratios that lie between 0 and 1, with their sum equalling 1. So the weights In the present example are 2,00,0001(2,00,000 + 5,00,000) and 5,00,000/(2,00,000 + 50,00,000). i.e. 2/7 and 5/7. The weights lie between 0 and 1 and add to 1. Actually. weights indicate the relative importance given to the values. They should, therefore, be similar and comparable in all respects. Since there is a larger cash flow towards 1he end and smaller cash flows at the beginning, probably cash flows themselves can serve as suitable weights. Thus, the years 1, 2, 3, 4, and 5 can be averaged with weights as 8, 8, 8, 6 and 108, the cash flows received in the corresponding years. Are cash flows appropriate weights? Weights have to be comparable in all respects. Cash flows are in the form of rupees and are comparable in units. But their values across years are not the same. To make values comparable across years, we have to use the present values of the cash flows as weights and not just the cash flows. So, the weights are present values of the cash flows 8, 8, 8, 8, and 108. The average time point at which the cash flows can be considered to have been received therefore, is: 1 • PV (8) + 2' PV (8) + 3 • PV (8) + 4' PV (8) + 5' PV (108) PV (8) + PV (8) + PV (8) + PV (8) + PV (108) 25 Assuming a discounting rate of 5 per cent, we can find that the average period is 4.35 years. This average point at which the cash flows are received is called the duration of the bond. Thus, we can formally define duration (D) as D= It; PV (G)lIpv (G;) (1.18) where I; are the time periods and G; are the corresponding cash flows. Incidentally, denominator is the price of Ihe bond. There are some quick observations about duration: 1. Duration is measured in terms of years and months. 2. Duration of any coupon bond is less than or equal to its term to maturity. 3. Duration of a zero coupon bond is its term to maturity. In addition to this, we can make some more observations about duration: 1. Higher the duration, more risky the bond; it is a measure of risk involved (Illustration 1.25). 2. Duration serves as a measure of sensitivity of bond's price to variations in yield. 3. Matching duralions of assets and liabilities helps in immunising a portfolio (Illustration 1.27). '"ustration 1.25 Which bond carries the lowest interest rate risk: 1. 3D, September 2011, 8 per cent (half-yearly) Rs. 100 bond. 2. 31, December 2013, 5 per cent Rs. 100 bond. 3. 30, June 2012, Rs. 100 bond. il the prevailing interest rates on similar bonds are 9 per cent, 10 per cent and 9.5 per cent as on 15, June 2006. Duration for the three bonds can be calculated using the duration formula. But calculating of duration with the help of this formula is little tedious. There is an EKcel Function DURATION. It gives duration of a bond with the face value of Rs. 100 after taking inputs to the following parameters: Settlement (day on which duration is calculated). Maturity (day of maturity of the bond). Coupon (coupon rate). Yld (yield). Frequency (no. at times coupon is paid). BasIS (type of day count). Selliemeni date is same for all three bonds and so the input to the parameter Settlement is same, which is t 5-Jun-2006. Values to other parameters are different for th" three differ"nl bonds: 26 1. Maturity = 3O-Sep-2011; Coupon = 8%; Yld = 9%, Frequency = :1. The duration 0/the bond is 4.32. 2. Maturity = 31-0ec-2013; Coupon = 5%; Yld = 10"10; Frequency:; 1. The duration of the bond is 6.08. 3. The duration of a 2CB is term to maturity "self, which is 6 years 15 days, which is approximately equal to 6.04. Since bond (f) has the least duration it carries the least interest rate risk. ----------------------_ .. Duration 0/a portfolio is a weighted average of durations of individual assets in the portfolio; with weights being proportional to investments in individual assets (Illustration 1.26). Illustration 1.26 Find the duration of a portfolio comprising bonds (1), (2), and (3) mentioned In Illustration 1.25 with 30 per cent investment in (1), 40 per cent investment in (2), and 30 per cent investment in (3). Durations of bonds (I), (2), and (3) are 4.32, 6.08, and 6.04 years. The weights are 0.3, 0.4, and 0.3, proportions of investments in (1), (2), and (3). The duration of the portfolio is the weighted average of durations, with weights as proportions of investments. Thus, duration of the portfolio is: 4.32 • 0.3 + 6.08 • 0.4 + 6.04 • 0.3 = 5.53 years Duration is very useful in immunising a portfolio. Immunising means neutralising the affects of interest rate changes on both assets and liabllnles. A bond portfolio can be said to be immunised if it is not very much affected by changes in interest rates. Illustration 1.27 helps in understanding the concept of immunisation as also the process of immunlsing a portfolio. with the help of duration. .-.... --------Illustration 1.27 A portfolio manager has only one cash outflow to make from her portfolio -an amount equal to Rs. 10,00,000 to be paid after 2 years. The manager has two choices -a bond that matures after 1 year and the other that matures after 3 years. There is only one cash flow of Rs. 1,070 at the end of 1 year for the l-year bond. In the case of 3-year bond there are three coupon payments of Rs. 80 each and a payment of Rs. 1,000 at the end of 3 years. Assuming a yield of 10 per cent, construct an immunised portfolio that meets the Rs. 10,00,000 requirement at the end of 2 years. 27 The manager can invest all her funds in I-year bond. The proceeds of the bond received can be reinvested after t year in a bond that has I-year maturity. In doing so, the manager faces reinvestment risk. If interest rates were to decline after 1 year, the funds realised from the first bond have to be reinvested at lower rates of interest. Altemately, the funds manager can invest in 3-year bonds and sell them at market price after 2 years. The manager in such circumstances faces the price risk. If interest rates were to go up at the time of selling the bond, its price would decline. Both the strategies -investing in I-year bond and 3-year bond -entail risks, i.e. either reinvestment risk or price risk. Whereas reinvestment risk occurs when interest rates decrease, price risk occurs when interest rates increase. The two risks are there because of the changes In the interest rate which move in opposite directions. It would be possible to construct a portfolio of these two bonds in such a way that the affects are nullified. It is achieved by constructing the portfolio in such a way that its duration is equal to 2 years. There is a single cash inflow of Rs. 1,070 at the end of I year in case of I-year bond. Its duration, naturally, is 1 year. In the case of 3-year bond, there are annual payments of Rs. 80 for 3 years and a payment of Rs. 1,000 at the end of 3 years. Duration of the 3-year bond can be catculated to be 2.78 years. We have to find out the proportion of investment to be made in these two bonds. If w, and w2 are proportions invested in I-year and 3-year bonds, then w, + w2 = 1 w, • 1 + w2 • 2.78 = 2 years The first equation states that sum 01 weights is 1. The second equation stipulates that the duralion of the portfolio is 2 years. We can get the proportional investmenls in the two bonds by solving these two equations. From the firsl equation we can get w1 to be equal to 1 -w2. Substlluting this value of w, in the second equation, we get I -w2 + 2.78 • w2 = 2 1.78w2 =1 w2 = 1'1.78 = 0.5618 Substituting this value of w2 in firsl equation, we get w, -1 -0.5618 = 0.4382 For immunisation, the portfoliO manager has to invest 58.18 per cent in S-year bond and 43.82 per cent in I-year bond. As the yield is 10 per cent, the amount to be invested today to get a caSh flow of Rs. 10,00,000 after 2 years is obtainad as 10,00,000/1 .12 , which is equal to Rs. 8,28,446. Using these percentages one can calculate the investments made into l-yMr and s-year bonds to be Rs. 3,62,149(8,26,446·0.4382) and I'ls. 4,64 ,297(8,26,446 • 0.5618) respectively. 28 At 10 per cent yield, the prices of l-year and 3·year bonds oan be workM out to be Rs. 972.73 and As. 950.25. At these prices, the numb"r of 1-yaa, and 3-year bonds purchased would be 372(3,62 ,149/972.73) and 489(4,64 ,297/950 .25 I. All the three possible scenarios regarding yield -decreases, remains same, increases -are considered. Table 1.3 illustrates portfolio immunisation rt yield at the end of 1 year -decreases to 9 per cent, remains at 10 per cent and increases to 11 per cent. In all the three cases, you can see that the aggrogate portfolio is more or less the same as Rs. 10,00,000. The portfolio has been constructed with the two objectives -~ gives a cash flow of Rs. 10.00.000 and it is immunised. Table 1.3 Immunisation of a Portfolio: a Scenario AnQlyoio (Am<>unt in _I Yield at the End 011 Yo., (y) 9% 10% (1% -----------_ .. .... -Value of 372 I -year bonds at the end of 2 years. Value at the end of 2 years after reinvesting proceeds of 1·year bonds at the end of 1 yeet = [1 ,070 • 372 -(1 + J1J Value of 489 3-ysar bonds a/the end of 2 years. Value from reinvesting coupons received at 4 ,33,864 4,37.644 4.41 ,824 the end of firsl year = [80 -489 • (1 + J11 42.64 1 43,032 43,423 Value of coupons received at the end of 2 years . (SO , 489j 39,120 39,120 39,120 Sale proceeds al Ihe end of 2 years = [1,080 ' 489/(1 + J1J 4,84.51 4 4,80,109 4,75,784 Aggregale portfolio value 81 the end of 2 years 10,00,138 10.00,105 10,00,151 When yield increases, the portfolio's losses owing to the selling of the loyear bonds at a discount after 2 years is offset by the gains from reinvesting the maturing l -year bonds and first year coupons of the 3-year bonds at the higher rate. Similarly, when the yield falls, the loss from reinvesting the l-year bonds and first year coupons on the 3-year bonds at a lower rate is offset by selling the 3·year bonds after 2 years at a premium. Thus, the portfolio is immunised. from the effect of changes in interest rates in the future. 29