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Bond Valuation A bond is a security that obligates the issuer to make interest and principal payments to the holder on the specified dates. Bond’s price is equal to the present value of future cash flows received in the form of coupon and principal payment. Important points to note in Bonds Valuation: Bond price moves up with the decrease in the interest rate and vice versa. Longer-term bonds are more sensitive to interest rate changes than short term ones. Valuation of Coupon Bond n CM V tn T1 (1 r) (1 r) Where V = Price of the bond n = number of years C = annual coupon payments M= maturity value T = time period when the payment is received r = yield to maturity Bond Yields The general definition of yield is the return an investor will receive by holding a bond to maturity. So if you want to know what your bond investment will earn, you should know how to calculate yield. Required yield, on the other hand, is the yield or return a bond must offer in order for it to be worthwhile for the investor. The required yield of a bond is usually the yield offered by other plain vanilla bonds that are currently offered in the market and have similar credit quality and maturity. The returns to an investor in bond are made up of three components: coupon, interest from re- investment of coupons and capital gains/loss from selling or redeeming the bond. When we are able to compare the cash inflows from these sources with the investment (cash our flows) of the investor, we can compute yield to the investor. Depending on the manner in which we treat the time value of cash flows and re-investment of coupons, we are able to get various interpretations of the yield on an investment in bonds. The return of a bond is largely determined by its interest rate. The interest that a bond pays depends on a number of factors, including the prevailing interest rate and the creditworthiness of the issuer, which, of course, is what is assessed by the credit rating companies. The higher the credit rating of the issuer, the less interest the issuer has to offer to sell its bonds. The prevailing interest rate—the cost of money—is determined by the supply and demand of money. Like virtually anything else, the greater the supply and the lower the demand, the lesser the interest rate, and vice versa. An often used measure of the prevailing interest rate is the prime rate charged by banks to their best customers. Most bonds pay interest semi-annually until maturity, when the bondholder receives the par value of the bond back. Zero coupon bonds pay no interest, but are sold at a discount to par value, which is paid when the bond matures.
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Once an investor has decided on the required yield, he or she must calculate the yield of a bond he
CHAPTER or she wants to buy. Let’s proceed and examine these calculations. Nominal Yield, Coupon Rate Nominal yield, or the coupon rate, is the stated rate of interest of the bond. This yield percentage is the percentage of par value. Thus, a bond with a Rs. 1,000 par value that pays 5% coupon pays Rs.50/- per year in 2 semi-annual payments of Rs. 25/-. The return of a bond is the return/investment, or in the example just, Rs.50/Rs.1,000 = 5%. Current Yield Because bonds trade in the secondary market, they may sell for less or more than par value, which will yield an interest rate that is different from the nominal yield, called the current yield, or current return. The current yield calculates the percentage return that the annual coupon payment provides the investor. In other words this yield calculates what percentage the actual coupon payment is of the price the investor pays for the bond. Current yield relates the annual coupon interest to the market price. It is the ratio of the annual interest payment to the bond’s current price. The current yield therefore refers to the yield of the bond at the current moment. The price of bonds moves in the opposite direction of interest rates. If rates go up, the price of bonds decrease; if the rates go down, then the bonds increase in value. To see why, consider this simple example. You buy a bond when it is issued for Rs. 1000 that pays 8% interest. Suppose you want to sell the bond, but since you bought it, the interest rate has risen to 10%. You will have to sell your bond for less than what you paid, because why is somebody going to pay you Rs. 1000 for a bond that pays 8% when they can buy a similar bond of equal credit rating and get 10%. So to sell your bond, you would have to sell it so that the Rs. 80 that is received per year in interest will be 10% of the selling price—in this case, Rs. 800, Rs. 200 less than what you paid for it. (Actually, the price probably wouldn’t go this low, because the yield-to-maturity is greater in such a case, since if the bondholder keeps the bond until maturity, he will receive a price appreciation which is the difference between Rs. 1000, the bond’s par value and what he paid for it.) Bonds selling for less than par value are said to be selling at a discount. If the market interest rate of a new bond issue is lower than what you are getting, then you will be able to sell your bond for more than par value—you will be selling your bond at a premium. Current Yield = (Annual coupon payment / Market price of the Bond) 100 Example: You purchased a bond with a par value of Rs. 100/- for Rs. 95/- and it paid a coupon rate of 5%, then calculate the current yield. ((0.05 100)/95) 100 = 5.26% Note that if the market price for the bond is equal to its par value then: Current yield = Nominal Yield. Example: A 10% coupon bond is trading at the price of Rs.1050. Here the coupon amount is Rs. 100(1000 0.10) Current yield= 100/1050 = 0.0952 or 9.52%. Notice how this calculation does not include any capital gains or losses the investor would make if the bond were bought at a discount or premium. Because the comparison of the bond price to its par value is a factor that affects the actual current yield, the above formula would give a slightly inaccurate answer - unless of course the investor pays par value for the bond. To correct this, investors can modify the current yield formula by adding the result of the current yield to the gain or loss the price gives the investor: [(Par Value – Bond Price)/Years to Maturity].
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The modified current yield formula then takes into account the discount or premium at which the -
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investor bought the bond. This is the full calculation:
AnnualCoupon (100 MarketPrice) AdjustedCurrent Yield 100 MarketPrice Years toMaturity Let’s re-calculate the yield of the bond in our first example, which matures in 30 months and has a coupon payment of Rs.5/-: = ((5/95) 100 ) + ((100-95)/2.5) = 5.26 + 2 = 7.26% The adjusted current yield of 7.26% is higher than the current yield of 5.26% because the bond’s discounted price (Rs.95/- instead of Rs.100/-) gives the investor more of a gain on the investment. Now we must also account for other factors such as the coupon payment for a zero-coupon bond, which has only one coupon payment. For such a bond, the yield calculation would be as follows: 1 Future Value n 1 PurchasePrice Yield = n = Years left until maturity If we were considering a zero-coupon bond that has a future value of Rs.1,000/- that matures in two years and can be currently purchased for Rs.925/-, we would calculate its current yield with the following formula: 1 Rs.1000 2 1 Rs.925 Yield = Relationships in Bond Pricing Theory Bond prices and yields move in opposite directions. Bond prices are more sensitive to yield changes the longer their maturities. The price sensitivity of bonds to yield changes increases at a decreasing rate with maturity. High coupon bond prices are less sensitive to yield changes than low coupon bond prices. With a change in yield of a given number of basis points, the associated percent gain is larger than the percent loss. Yield to Maturity (YTM) If an investor buys a bond in the secondary market and pays a price different from par value, then not only will the current yield differ from the nominal yield, but there will be a gain or loss when the bond matures and the bondholder receives the par value of the bond. If the investor holds the bond until maturity, he will lose money if he paid a premium for the bond, or he will earn money if he paid for it at a discount. The yield-to-maturity (YTM), or true yield, of a bond that is held to maturity will have to account for the gain or loss that occurs when the par value is repaid. When a bond is bought at a discount, yield to maturity will always be greater than the current yield because there will be a gain when the bond matures, and the bondholder receives par value back, thus raising the true yield; when a bond is bought at a premium, the yield to maturity will always be less than the current yield because there will be a loss when par value is received, and this lowers the true yield. Because some bonds are callable, these bonds will also have a yield to call, which is calculated exactly the same as yield to maturity, but the call date is substituted for the maturity date and the call price or call premium is substituted for par value. When a bond is bought at a premium, the yield to call is always the lowest yield of the bond. Some bonds also have a put option, which allows the bondholder to receive the principal of the bond from the issuer when the bondholder exercises the put. This yield to put would be calculated like the yield to maturity, except that the date that the put is exercised is substituted for the
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maturity date, because the bondholder receives the par value on the exercise date just as if the bond
CHAPTER matured. Finally, there is the yield to worst, which simply calculates the bond’s yield if the bond is retired at the earliest possible date allowed by the bond’s indenture. The yield to maturity is the return an investor earns on a bond if he holds the bond till its maturity. It is the interest rate that equates the present value of all future payments to the current market price of a bond. The yield to maturity (YTM) is the discount rate which returns the market price of the bond. It is thus the internal rate of return of an investment in the bond made at the observed price. The YTM is always given in terms of annual effective rate. To achieve a return equal to YTM, the bond owner must: Reinvest each coupon received at YTM. Hold the bond until maturity. Calculating Yield to Maturity The current yield calculation we learned above shows us the return the annual coupon payment gives the investor, but this percentage does not take into account the time value of money, or, more specifically, the present value of the coupon payments the investor will receive in the future. For this reason, when investors and analysts refer to yield, they are most often referring to the Yield to maturity (YTM), which is the interest rate by which the present values of all the future cash flows are equal to the bond’s price. An easy way to think of YTM is to consider it the resulting interest rate the investor receives if he or she invests all of his or her cash flows (coupons payments) at a constant interest rate until the bond matures. YTM is the return the investor will receive from his or her entire investment. It is the return that an investor gains by receiving the present values of the coupon payments, the par value and capital gains in relation to the price that is paid. The yield to maturity, however, is an interest rate that must be calculated through trial and error. Such a method of valuation is complicated and can be time consuming, so investors (whether professional or private) will typically use a financial calculator or program that is quickly able to run through the process of trial and error. If you don’t have such a program, you can use an approximation method that does not require any serious mathematics. To demonstrate this method, we first need to review the relationship between a bond’s price and its yield. In general, as a bond’s price increases, yield decreases. This relationship is measured using the price value of a basis point (PVBP). By taking into account factors such as the bond’s coupon rate and credit rating, the PVBP measures the degree to which a bond’s price will change when there is a 0.01% change in interest rates. The charted relationship between bond price and required yield appears as a negative curve:
Exhibit 1: Relationship between Bond Price and Yield This is due to the fact that bond’s price will be higher when it pays a coupon that is higher than prevailing interest rates. As market interest rates increase, bond prices decrease.
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The second concept we need to review is the basic price-yield properties of bonds: -
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Premium bond: Coupon rate is greater than market interest rates. Discount bond: Coupon rate is less than market interest rates. Thirdly, remember to think of YTM as the yield a bondholder receives if he or she reinvested all coupons received at a constant interest rate, which is the interest rate that we are solving for. If we were to add the present values of all future cash flows, we would end up with the market value or purchase price of the bond. The calculation can be presented as: Cashflow1 Cashflow2 Last Cashflow BondPrice 1 2 n (1 yield) (1 yield) (1 yield) OR 1 1n (1 interestrate) 1 BondPrice Cashflow Maturity Value interestrate (1 interestrate)n This equation shows that the bond price is equal to the present value of all bond payments with the interest rate equal to the yield to maturity. YTM can also be calculated by following approximation formula: (M P) C YTM n 0.4M 0.6P Where, C = coupon payment M = maturity price (price to be received on the maturity of the bond) P = current market price n = no. of years to maturity Note: YTM computed by approximation formula will only be an approximate value of actual YTM. Relationship between Bond prices and yield to maturity: Bond prices are inversely related to interest rates (or yield to maturity). A bond sells at par only if its yield to maturity equals the coupon rate. A bond sells at a premium if its coupon rate is above the yield to maturity. A bond sells at a discount if its coupon rate is below the yield to maturity. Variants of Yield to Maturity Bonds with callable or put table redemption features have additional yield calculations. A callable bond’s valuations must account for the issuer’s ability to call the bond on the call date and the put table bond’s valuation must include the buyer’s ability to sell the bond at the pre-specified put date. The yield for callable bonds is referred to as yield-to-call, and the yield for put table bonds is referred to as yield-to-put. Yield to call (YTC) is the interest rate that investors would receive if they held the bond until the call date. The period until the first call is referred to as the call protection period. Yield to call is the rate that would make the bond’s present value equal to the full price of the bond. Essentially, its calculation requires two simple modifications to the yield-to-maturity formula:
or, expressed in summation, or sigma, notation:
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n C P k B kn CHAPTER k1 (1 Y) (1 Y) B = current bond price C = coupon payment per period P = par value of bond or call premium; n = number of years until maturity or until call or until put is exercised; Y = yield to maturity, yield to call, or yield to put per pay period depending on which values of n and P are chosen. Yield to Put It is same as Yield to Call with the only difference that here the bond holder has the option to sell the bond back to the issuer at a fixed price on a specified date. Yield to put (YTP) is the interest rate that investors would receive if they held the bond until its put date. To calculate yield to put, the same modified equation for yield to call is used except the bond put price replaces the bond call value and the time until put date replaces the time until call date. For both callable and puttable bonds, sharp & shrewd investors will compute both yield and all yield- to-call/yield-to-put figures for a particular bond, and then use these figures to estimate the expected yield. The lowest yield calculated is known as yield to worst, which is commonly used by conservative investors when calculating their expected yield. Unfortunately, these yield figures do not account for bonds that are not redeemed or are sold prior to the call or put date. Now you know that the yield you receive from holding a bond will differ from its coupon rate because of fluctuations in bond price and from the reinvestment of coupon payments. In addition, you are now able to differentiate between current yield and yield to maturity. Valuation of Zero Coupon Bonds FV v n (1 i) Where, V = Value of the bond FV = Value of the bond at time t i = Coupon rate n = Term period Example: For instance, if a one year zero-coupon bond is trading at Rs.950 and has a par value of Rs.1, 000 (paid at maturity in one year), the bond’s rate of return at the present time is approximately 5.26% ((1000-950) / 950 = 5.26%). Although it is difficult to solve for the yield using the above equation, it can be readily solved by using Microsoft Excel, as shown below. Yield-to-Maturity (YTM) Formula for Bonds using Microsoft Excel YTM = Yield(settlement, maturity, rate, price, redemption, frequency, basis) Settlement = Date in quotes of settlement. Maturity = Date in quotes when bond matures. Rate = Nominal coupon interest rate. Price = Redemption value as a percent of par value. Redemption = Price as a percent of par value. Frequency = Number of coupon payments per year. 1 = Annual 2 = Semiannual 4 = Quarterly
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Basis = Day count basis. -
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0 = 30/360 (U.S. NASD basis). This is the default if the basis is omitted. 1 = actual/actual (actual number of days in month/year). 2 = actual/360 3 = actual/365 4 = European 30/360 Note that yield to call (YTC) and yield to put (YTP) can also be calculated using this formula. To calculate the yield to call: Maturity = Date of earliest possible call. Redemption = Call price. To calculate yield to put: Redemption = Date that put can be exercised. Yield to Worst, Yield to Sinker, and Yield to Average Life can be calculated by substituting the appropriate date for the maturity date. Yield to Maturity (YTM) Example If Settlement date = 3/31/2008 Maturity = 3/31/2018 (10 year bond) Nominal coupon rate = 5% Price = 92.56 (as a percent of par value which equals 92.56% x Rs.1,000 = Rs. 925.60 Bond Price) Redemption = Value received at maturity as a percentage of par value = 100 (100% x par value = Rs.1,000) Frequency = 2 semi-annual coupon payments Then YTM = Yield (“3/31/2008”,”3/31/2018”, 0.05,92.56,100,2) = 6.00% Realized Yield to Maturity The realized yield to maturity is the actual yield obtained by investor holding a debt instrument. The realized yield depends upon the rate at which all coupon receipts are reinvested for additional interest income and it also depends on the bond price as and when it is sold. The realized compound yield is the yield obtained by reinvesting all coupon payments for additional interest income. It will also depend on the bond price if it is sold before maturity. What this yield ultimately is depends on how interest rates change over the holding period of the bond. Although future interest rates and bond prices cannot be predicted with certainty, horizon analysis is the forecasting of interest rates and bond prices over a specific time period to yield an expectation of the realized compound yield. Example: Ram purchased a Rs. 10000/- par value bond, carrying a coupon rate of 15% (payable annually) and maturing after 5 years. The present market price of this bond is Rs. 8500/-. The reinvestment rate applicable to the future cash flows of this bond is 16%. Ram receives Rs. 20320/- after maturity then calculate the realized yield to maturity. Future value= Present value(1 + r )n Hence, 8500 (1 + r)5 = 20320 (1 + r)5 = 20320/8500 = 2.39 (1 + r) = (2.39)(1/5) (1 + r) = 1.19, r = 19% It can be calculated by using Casio BOND Mode (Fc-200V)
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Press BOND to enter BOND Mode.
CHAPTER Press Set: Periods/Y: Annual n = 5 (Number of Coupon payments until Maturity) RDV = 20320 (Redemption or maturity price) CPN = 0(Coupon amount) PRC = –8500 (Price of the BOND) YLD = Annual Yield Solve = 19.04 Theoretically, Realized yield to maturity looks superior to the conventional YTM. But conventional yields to maturity on coupon bonds are difficult to interpret, realized yields to maturity are difficult to implement. Holding-Period Return YTM is the single discount rate at which the present value of payments received from the bond equals to its price. It is the average yield over the term of the bond. If a bond is sold before maturity, then its actual yield will probably be different from the yield to maturity. If interest rates rise during the holding period, then the bond’s sale price will be less than the purchase price, decreasing the yield, and if interest rates decrease, then the bond’s sale price will be greater. The holding-period return is the actual yield earned during the holding period. Holding period of return is the income earned over a given holding period as percentage of its price at the beginning of the period. It can be calculated using the same formula for yield to maturity, but the sale price would be substituted for the par value, and the term would equal the actual holding period. Note that, unlike yield to maturity, the holding-period return cannot be known ahead of time because the sale price of the bond cannot be known before the sale, although it could be estimated. Example: A 10 year Rs. 1,000/- par bond paying an annual coupon of Rs. 80 is bought for 1,000/-, its YTM is 9%. If the bond price increases to Rs. 1080 by year end, its YTM will fall below 9% because it is selling at a premium, but its holding-period return for the year exceeds 9% Holding period return = (80 + (1080–1000))/1000 = 16% We can also prepare summary of Bond Yield Relationships as per below: Summary of Bond Yield Relationships When the bondholder pays... Bond Yield Relationships less than par value (discount). Yield to Maturity > Current Yield > Nominal Yield par value. Nominal Yield = Current Yield = Yield to Maturity more than par value (premium). Nominal Yield > Current Yield > Yield to Maturity
Yield Curve The graphical depiction of the relationship between the interest rates (or yields) on bonds of the same credit quality and different maturities is known as the yield curve. In other words, the yield curve is a depiction of debt market interest rates across maturities, with time being plotted on the X- axis and yields on the Y-axis. The curve graphically demonstrates the rate at which market participants are willing to transact debt capital for short-term, medium-term and long-term periods. Yield curves are used by fixed income analysts, who analyze bonds and related securities, to understand conditions in financial markets and to seek trading opportunities. Normal Curve: This is the curve that is observed most commonly. The yield curve slopes gently upward - reflecting higher future rates (see figure). In the absence of economic disruptions, investors who risk their money for longer periods expect higher yields than those who risk their money for
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shorter-time periods. The underlying reasoning is that investors have a greater liquidity preference -
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and, therefore, they attach lesser risk to shorter-term securities as they are closer to cash. Steep Curve: This curve is normally observed at the beginning of an economic expansion or just at the end of a recession. The slope of the yield curve increases as the difference between long-term yields and short-term yields become wider (see figure). The inherent assumption behind such a curve could be that while short-term economic conditions warrant lower rates, factors like inflation, etc. could rise in the medium / long-term justifying much higher long-term rates. Inverted Curve: This curve is downward sloping. In other words, interest rates are higher for shorter periods than those for longer periods (see figure). Typically, they are caused by short-term monetary imbalances in the economy whereas the long-term conditions are expected to normalize. In the past, such curves have been experienced during times when Central Banks have raised short-term rates to ward off speculative pressures on currency, etc. Flat Curve: For a yield curve to change from normal to inverted, it may pass through a phase where long-term rates are more or less equal to short-term rates (see figure). However, not all flat curves become inverted. In other words, flat curves do not necessarily guarantee an economic slowdown, but the odds can still be pretty good. Clean/Dirty Price of Bonds This concept applies mainly to the coupon paying bonds but even zero coupon bonds are prone to such pricing anomalies. A bond’s dirty price is its clean price plus accrued interest. If you were to graph the bond’s dirty price minus clean price each day, you’d see it rise during the accrual period from zero to the period’s coupon amount. For example, a Rs. 1000 bond might have an annual coupon rate of 6 per cent, for a coupon amount of Rs. 60. If it pays interest semiannually, the bond’s dirty price reaches a maximum value every six months equal to the clean price plus Rs. 30 and then falls back to the clean price on payment date. Price-Yield Relationship A basic property of a bond is that its price varies inversely with yield. The reason is simple. As the required yield increases the present value of the cash flow decreases, hence the price decreases. Conversely, when the required yield decreases, the present value of the cash flow increases, hence the price increases. The graph of the price-yield relationship for any callable bond has a convex shape as shown below: Why bond prices are inversely related to interest rates? Unlike stock market where an upward movement of market leads to upward movement in stock prices, it is a fall in the market yield that pushes up the prices of debt securities. This happens because there exists an inverse relationship between the yield and the price of a bond. So, if there is an upward movement of interest rates after one has invested in a bond fund, the prices of bonds will go down leading to a corresponding fall in the net asset value (NAV) of the bond funds. Let us take an Example: Suppose a person buys a bond for Rs. 100 having a coupon rate of 10 percent. In other terms, the person should get Rs. 110 at the end of the year. If the RBI announces a hike in the bank rate and the market yield for the duration of the bond increases, say to 11 percent, the prices of the bond will fall around to Rs. 90.91 in order to adjust to the market yield. This is termed as interest rate risk in financial jargon. An investor stands to benefit in the opposite scenario, when the interest rates are cut as then the prices go up leading to better returns from the fund. If the interest rate in the above example falls to 9 percent, a person still gets Rs. 10 in interest but in order to align the amount received to the prevailing market yield, the price of the bond adjusts to Rs. 111.11. In this case, the investor is better off by selling it at Rs. 111.11 than holding it to its maturity, as then he will only get Rs. 110. This risk is also dependent upon the maturity and duration of the bond and generally, the longer a fund’s duration or average maturity, the higher its interest-rate risk, or the more sensitive the NAV
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of the debt mutual fund will be to changes in interest rates. One can reduce the interest rate risk by
CHAPTER choosing a bond fund with a shorter duration or average maturity. Yield Spread A bond’s yield relative to the yield of its benchmark is called a spread. The spread is used both as a pricing mechanism and as a relative value comparison between bonds. The higher the yield spread, the greater the difference between the yields offered by each instrument. The spread can be measured between debt instruments of differing maturities, credit ratings and risk. For example, a trader might say that a certain corporate bond is trading at a spread of 75 basis points above the 10- year Treasury. This means that the yield to maturity of that bond is 0.75% greater than the yield to maturity of the on-the-run 10-year Treasury. If a different corporate bond with the same credit rating, outlook and duration was trading at a spread of 90 basis points on a relative value basis, the second bond would be a better buy. Yield spreads on lower quality issues are more volatile There are different types of spread calculations used for the different pricing benchmarks. The four primary yield spread calculations are: Nominal Yield Spread: The difference in the yield to maturity of a bond and the yield to maturity of its benchmark. Consider the following two 10-year bonds: o A Treasury bond having a YTM of 6.5% o A non-Treasury bond having a YTM of 8% Nominal Spread = Yield of non-Treasury Bond – Yield of Treasury Bond Nominal Spread = 8% – 6.5% = 1.5% The difference between the YTM for the two bonds is 1.5% (150 bps). This is the nominal spread. A non-Treasury bond usually provides a higher yield compared to a Treasury bond because of the additional risk involved, especially the credit risk and the liquidity risk. Zero-Volatility Spread (Z-spread): The constant spread that, when added to the yield at each point on the spot rate Treasury curve where a bond’s cash flow is received, will make the price of a security equal to the present value of its cash flows. The Z-spread is also known as a “static spread”. For example, take the spot curve and add 50 basis points to each rate on the curve. If the two year spot rate is 3%, the rate you would use to find the present value of that cash flow would be 3.50%. After you have calculated all of the present values for the cash flows, add them up and see whether they equal the bonds price. If they do, then you have found the Z-spread, if not, you have to go back to the drawing board and use a new spread until the present value of those cash flows equals the bonds price. Discount Margin (DM): Bonds with variable interest rates are usually priced close to their par value. This is because the interest rate (coupon) on a variable rate bond adjusts to current interest rates based on changes in the bond’s reference rate. The DM is the spread that, when added to the bond’s current reference rate, will equate the bond’s cash flows to its current price. Factors that Affect Yield Spreads 1. Credit cycle - At top of cycle, bonds have lower credit risk and are bullish so credit spread narrows as the cycle improves 2. Economic conditions - Credit spread narrows as economy strengthens 3. Financial market performance - Credit spread narrow in strong performing markets and widen in weak ones. Also narrow in steady markets with low volatility. 4. Broker-dealer capital - Yield spreads are narrower when broker dealers provide enough capital but can widen when market making capital becomes scarce 5. General market demand and supply - Credit spread narrows in time of high demand for bonds and excess supply leads to widening spreads.
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