Bonds and debentures are terms used interchangeably; both represent long term fixed income securities. The cash flow stream (in form of interest and principal) as well as the time horizon (i.e. the date of maturity) are well specified and fixed. Bond returns can be calculated in various ways; Coupon Rate, Current yield, Spot interest rate, Yield to maturity (YTM), Yield to call (YTC), Realized YTM.
1. Coupon rate: It is the nominal rate of interest that is fixed and is printed on the bond certificate. It is calculated on the face value of the bond. It is the rate at which interest is paid by the company to the bondholder. It is payable by the company at periodical intervals of time till maturity. Example: A bond has a face value of Rs 1000 with an interest rate of 12% p.a. payable annually. It means that Rs 120 will be paid by the company on an annual basis to the bond holder till maturity.
2. Current yield or Market Yield: The current market price of the bond in the secondary market may differ from its face value (i.e. it may be currently selling at a discount or at a premium). Current yield relates the annual interest receivable on a bond to its current market price. Current yield = (Annual interest / Current market price)*100. It thus measures the annual return accruing to a bondholder who purchases the bond from the secondary market and sells it before maturity presumably at a price at which he bought the bond. Example: A bond has a face value of Rs 1000 and a coupon rate of 12%. It is currently selling for Rs 800. The current yield = (120/800)*100 = 15%. If Current yield > coupon rate: when bond is selling at a discount. If Current yield < coupon rate: when bond is selling at a premium. If Current yield = Coupon rate: when bond is selling at par.
3. Spot interest rate: Zero coupon bond or Deep discount bond is a special type of bond which does not pay annual interest. Rather such bonds are issued at a discount to be redeemed at par. The return comes in form of the difference between the issue price and the maturity value .Spot interest rate is the return on deep discount bonds when expressed in % terms on an annual basis. Mathematically, it is that rate of discount which makes the present value of the single cash inflow to the investor (on redemption of bond, no interest being payable annually) equal to the cost of the bond. A zero coupon bond has a face value of Rs 1000 and maturity period of five years. If the issue price of the bond is Rs 519.37, what is the spot interest rate? It is that rate of interest (discount) which makes the PV of 1000 = 519.37= 1000 = 0.14 or 14%.
4. Yield to maturity (YTM) or Yield: It is the rate of return that an investor is expected to earn on an annualized basis expressed in % terms from a bond purchased at the current market price and held till maturity. It is the internal rate of return earned on a bond if held till maturity. YTM is that rate of discount (r) which makes the present value of cash inflows from the bond (in form of interest and redemption value) equal to the cash outflow on purchase of the bond i.e. MP. Approximate YTM may be calculated as follows: YTM = (I + (RV – MP) / N) / ((RV + MP) / 2), where, I: Annual interest, RV: Redemption value, MP: Market price, N: Number of years remaining to maturity. Example: A bond of face value Rs 1000 and a coupon rate of 15% is currently available at Rs 900. Five years remain to maturity and bond is redeemable at par. Calculate YTM. MP = 900 RV = 1000, I = 15% of Rs 1000 = 150, YTM = (I + (RV – MP) / N) / ((RV+MP)/2) = (150 + (1000 – 900) / 5) / ((1000+900)/2) = 0.1789 or 17.89%. (Exact by excel= 18.21%)
5. Yield to call (YTC): Some bonds may be redeemable before their full maturity at the option of the issuer or the investor. In such cases, two yields are calculated: YTM (assuming that the bond will be redeemed only at the end of full maturity period). YTC (assuming that the bond will be redeemed at a call date before maturity). YTC is computed on the assumption that the bond’s cash inflows are terminated at the call date with redemption of the bond at the specific call price. And specific call price is treated as RP. Thus, YTC is that rate of discount which makes the present value of cash inflows till call equal to the current market price of the bond. Same method as YTM would be used except for ‘N’ now being years remaining to call. If YTC > YTM, it would be advantageous to the investor to exercise the redemption option at the call date. If YTM > YTC, it would be better to hold the bond till final maturity.
6. Realized YTM: The calculation of YTM assumes that cash flows received through the life of the bond are being reinvested at a rate equal to YTM. However, the reinvestment rate may differ over time. In such cases, Realized YTM is a more appropriate measure. Example: A Rs 1000 par value bond carrying a coupon rate of 15% maturing after 5 years is being considered. The present market price of this bond is Rs 850. The reinvestment rate applicable to future cash flows is 16%. Calculate realised YTM. Total FVs = (271.5 + 234 + 202.5 + 174 + 150 + 1000) = 2032. Reinvestment period @ 16%, 4, 3, 2, 1, 0. FV of this CFs 271.5, 234, 202.5, 174, 150, Maturity value 1000. Separately FVs of reinvestment of Rs. 150 are calculated. For calculating Realized YTM, Calculate Rate using excel with PV=850, FV=2032, Period 5 years. (Exercise-answer 19.04%). Example ; A bond has a face value of Rs 1000 and was issued five years ago at a coupon rate of 10%. The bond had a maturity period of 10 years. If the current market interest rate is 14%, what should be the PV of the bond? (Rs 862.71). Example If in the above question, interest is payable semi-annually, what would be the intrinsic value of the bond? (Rs 859.48.) (exercise)
7. Bond pricing: Bonds are issued at a fixed rate of interest payable on the face value which is referred to as COUPON RATE. At the time of issue, coupon rate is representative of the then prevailing market interest rate. However, subsequent changes in the market interest rates may have its effects on the bond prices. If market interest rate rises above the coupon rate. The existing bonds would start providing lower return. Thus becoming unattractive. Hence the price of the bond would fall below its face value i.e. the bond would start selling at a discount. If market interest rate falls below the coupon rate. The existing bonds would start providing relatively higher return. Thus becoming very attractive. Hence the price of the bond would rise above its face value i.e. the bond would start selling at a premium. A change in interest rate structure would result in a relatively large price changes in a long maturity bond.
8. Bond risks: Risk is the possibility of variation in returns. The actual returns realized from investing in bond may vary from what was expected on account of: Default or delay on part of the issuer to pay interest and/or principal, Change in market interest rates. Thus there are two broad sources of risk associated with bonds: Default risk and Interest rate risk.
9. Default risk: It refers to the possibility that a company may fail to pay the interest or principal on the stipulated dates. Poor financial performance of the company may lead to such default. Credit rating of debt securities is a mechanism adopted for assessing the credit risk involved. Credit rating process involves: Qualitative assessment of company’s business and management. Quantitative assessment of company’s financial performance. Specific features of the bond being issued. Credit rating is an opinion of the credit rating agency regarding the relative ability of issuer of debt to fulfill the debt obligations in respect of interest and repayment.
10. Interest rate risk: It refers to variation in returns of bond because of a change in market interest rates. Interest rate risk is composed of two risks: Reinvestment risk and Price risk. Reinvestment risk. An investor in bonds receives interest annually or semi-annually. He reinvests it each year at the then prevailing interest rate. Thus interest is earned on the interest received from the bonds each year. If the market interest rate moves up, the investor would be able to reinvest the annual interest received from the bond at a higher rate than expected. Thus he would gain from the reinvestment activity. When the market interest rates move down, the investor would be able to reinvest the interest only at a lower rate than expected. Thus he would lose on reinvestment activity. Price risk: The price of the bond is inversely related to changes in market interest rate: If the market interest rate moves up, bond price may decline below its face value. Thus the investor would suffer a loss while selling the bond. If the market interest rate goes down, the existing bonds may start selling at a premium. Thus the investor would gain from sale of such bond.
11. Interest rate risk: When the market interest rate rises, an investor can reinvest the interest at a higher rate, thus gaining from reinvestment. However the future bond price would decline, thus losing on sale of the bond. If the gain on reinvestment > loss on sale of bond: Net Gain. If the gain on reinvestment < loss on sale of bond: Net loss. When the market interest rate falls, Investor can reinvest the interest at a lower rate, thus losing from reinvestment. However, the future bond price would rise, thus gaining on sale of the bond. If the loss on reinvestment > gain on sale of bond: Net loss. If the loss on reinvestment < gain on sale of bond: Net gain. Thus, reinvestment risk and price risk are inversely related. Together they constitute interest rate risk.
12. Bond duration (D): When considering the reinvestment risk and price risk, loss in one may be exactly compensated for by the gain in the other, thus completely eliminating the interest rate risk. This particular holding period at which the interest rate risk disappears is referred to as Bond Duration. Bond duration is calculated as the weighted average measure of the bond’s life. Duration is a measure of interest rate risk of a bond. It shows the sensitivity of a bond’s price to interest rate changes and also takes into account the timing of the bond’s cash flows. In other words, duration is a measure of the length of time at the end of which the investor would get his investment returned. In finance, duration has a specific connotation. It measures (in number of years) the time taken by all expected future cash flows of the bond to repay the time adjusted true value of the bond. Mathematically, duration is the 1st derivative of the price-yield curve, which is a line tangent to the curve at the current price-yield point. Duration has several simple properties: duration is proportional to the maturity of the bond, since the principal repayment is the largest cash flow of the bond and it is received at maturity; duration is inversely related to the coupon rate, since there will be a larger difference between the present values for the earlier payments over the lesser value for the principal repayment; duration decreases with increasing payment frequency, since half of the present value of the cash flows is received earlier than with less frequent payments, which is why coupon bonds always have a shorter duration than zeros with the same maturity.
Durations are of two types: Macaulay Duration and Modified Duration. While former is explained with the help of an example as below, the later is arrived at by applying the Formula: Modified duration can be used to find out the percentage change in bond price due to change in yield.
Modified Duration= D*= D/ (1+YTM), where D is Macaulay’s Duration and YTM is Yield to Maturity
Example: A bond with face value of Rs 1000 with 8% coupon rate is due for redemption at par in 3 years. The YTM is 10%. Calculate bond duration.
Time (Years) |
Cash Flows |
PVF(10,n) |
PV |
PV*Time |
1 |
80 |
.909 |
72.72 (80*0.909) |
72.72*1=72.72 |
2 |
80 |
.826 |
66.08 (80*0.826) |
66.08*2=132.16 |
3 |
1080 |
.751 |
811.08(1080*0.751) |
811.08*3=2433.24 |
Sum: 949.88 |
Sum: 2638.12 |
Duration: D = Sum ((PV*Time)/Market rate= 2638.12/950= 2.77 Years. (Macaulay Duration) Long term debt funds such as gilt or income are most vulnerable to the interest rate movement. A recent action by RBI on the interest rates followed by the sudden rise in the returns of debt funds has also seen long term gilt funds getting the highest rise among all categories. Even within this category there were some which were impacted most while some had a lesser impact. In such situations, for a layman investor, selecting a debt fund becomes a difficult task considering the lack of knowledge on the market itself. A measure like Modified Duration, when used with other parameters, helps in analyzing the impact of interest rates on debt mutual funds schemes. This tool is mentioned in the fact sheet across debt schemes and by understanding it one can simplify the task of selecting debt funds for their requirement.
For investors, by comparing the different debt mutual funds scheme within a category, they can analyze which has more interest rate risk. There are also categories like dynamic bond funds where there are frequent changes in the portfolio. These funds rely more on fund manager skills. The modified duration analysis can also indicate how the fund manager is taking a view on the movement of interest rates. But no investment decision should rest on this analysis alone. Thus, by using this measure investors can analyze and compare different debt mutual funds scheme to see which matches their objective of the investment. If one is looking to invest for a shorter period, taking a higher risk will not be in the list of parameters and so low duration funds will find a place in the portfolio. On other hand if one has a higher risk appetite and a longer investment horizon, the preference for taking an extra risk for higher returns will be there which can be met by including higher duration fund in the portfolio. But do remember that this is not the only factor which can impact a portfolio of a debt scheme. Credit quality of the securities and other factors are there which are equally important and so should be given equal weightage when you are comparing schemes to create a debt portfolio.
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