Tarheel Consultancy Services

1. Tarheel ConsultancyServices Manipal, Karnataka

2. Corporate Training and Consulting

3. Course on Fixed Income Securities For XIM -Bhubaneshwar

4. For PGP-II 2003-2005 Batch Term-V: September-December 2004

5. Module-I Part-IV Prices & Yields: Advanced Concepts

6. Valuation in between Coupon Dates • While valuing a bond we assumed that we were standing on a coupon payment date. • This is a significant assumption because it implies that the next coupon is exactly one period away. • What should be the procedure if the valuation date is in between two coupon payment dates?

7. The Procedure for Treasury Bonds • Calculate the actual number of days between the date of valuation and the next coupon date. • Include the next coupon date. • But do not include the starting date. • Let us call this interval N1.

8. Treasury Bonds (Cont…) • Calculate the actual number of days between the coupon date preceding the valuation date and the following coupon date. • Once again include the ending date but exclude the starting date. • Let us call this time interval as N2.

9. Treasury Bonds (Cont…) • The next coupon is then k periods away where

10. Illustration • There is a Treasury bond with a face value of $1,000. • The coupon rate is 8% per annum, paid on a semi-annual basis. • The coupon dates are 15 July and 15 January. • The maturity date is 15 January 2022. • Today is 15 September 2002.

11. No. of Days Till the Next Coupon Date

12. No. of Days between Coupon Dates

13. Treasury Bonds (Cont…) • K = 122/184 = .6630 • This method is called the Actual/Actual method and is often pronounced as the Ack/Ack method. • It is the method used for Treasury bonds in the U.S.

14. The Valuation Equation • Wall Street professionals will then price the bond using the following equation.

15. Valuation • In our example

16. The 30/360 Approach • The Actual/Actual method is applicable for Treasury bonds in the U.S. • For corporate bonds in the U.S. we use what is called the 30/360 method. • In this method the number of days between successive coupon dates is always taken to be 180. • That is each month is considered to be of 30 days.

17. The 30/360 Approach (Cont…) • The number of days from the date of valuation till the next coupon date is calculated as follows. • The start date is defined as • D1 = (month1, day1,year1) • The ending date is defined as • D2 = (month2,day2,year2)

18. The 30/360 Approach (Cont…) • The number of days is then calculated as • 360(year2 – year1) + 30(month2 – month1) + (day2 – day1)

19. Additional Rules • If day1 = 31 then set day1 = 30 • If day1 = 30 or has been set equal to 30, then if day2 = 31, set day2 = 30 • If day1 is the last day of February, then set day1 = 30

20. Examples of Calculations

21. Pricing of A Corporate Bond • Let us assume that the bond considered earlier was a corporate bond rather than a Treasury bond.

22. Pricing (Cont…)

23. 30/360 European Convention • In this convention, if day2 = 31, then it is always set equal to 30. • So the additional rules are: • If day1 = 31 then set day1 = 30 • If day2 = 31 then set day2 = 30 • If day1 is the last day of February, then set day1 = 30

24. Examples of Calculations

25. Other Conventions • Actual/365 Convention • In this case the year is considered to have 365 days, while calculating the denominator, even in leap years. • Actual/365 Japanese • This is used for Japanese Government Bonds (JGBs) • It is similar to the Actual/365 method. • The only difference is that in this case, the extra day in February is ignored in leap years, while calculating both the numerator and the denominator.

26. Accrued Interest • The price of a bond is the present value of all the cash flows that the buyer will receive when he buys the bond. • Thus the seller is compensated for all the cash flows that he is parting with. • This compensation includes the amount due for the fact that the seller is parting with the entire next coupon, although he has held it for a part of the current coupon period.

27. Accrued Interest (Cont…) • This compensation is called Accrued Interest. • Let us denote the sale date by t; the previous coupon date by t1; and the following coupon date by t2 • The accrued interest is given by

28. Accrued Interest (Cont…) • Both the numerator and the denominator are calculated according to the conventions discussed above. • That is for U.S. Treasury bonds the Actual/Actual method is used, whereas for U.S. corporate bonds the 30/360 method is used.

29. Why Accrued Interest? • Why should we calculate the accrued interest if it is already included in the price calculation? • The answer is that the quoted bond price does not include accrued interest. • That is, quoted prices are net of accrued interest.

30. Why Accrued Interest? (Cont…) • The rationale is as follows. • On July 15 the price of the Treasury bond using a YTM of 10% was $829.83. • On September 15 the price using a yield of 10% is $843.5906. • Since the required yield on both the days is the same, the increase in price is entirely due to the accrued interest.

31. Why Accrued Interest (Cont…) • On July 15 the accrued interest is zero. • This is true because on a coupon payment date, the accrued interest has to be zero. • On September 15 the accrued interest is

32. Why Accrued Interest? (Cont…) • The price net of accrued interest is • $843.5906 - $13.4783 = $830.1123$, which is very close to the price of $829.83 that was observed on July 15. • We know that as the required yield changes, so will the price. • If the accrued interest is not subtracted from the price before being quoted, then we would be unsure as to whether the observed price change is due to a change in the market yield or is entirely due to accrued interest.

33. Why Accrued Interest? (Cont…) • However if prices are reported net of accrued interest, then in the short run, observed price changes will be entirely due to changes in the market yield. • Consequently bond prices are always reported after subtracting the accrued interest.

34. Clean versus Dirty Prices • Quoted bond prices are called clean or add-interest prices. • When a bond is purchased in addition to the quoted price, the accrued interest has also to be paid. • The total price that is paid is called the dirty price or the flat price.

35. Yield Measures • The yield or the rate of return from a bond can and is computed in various ways. • We will discuss various yield measures and their relative merits and demerits.

36. The Current Yield • This is very commonly reported. • Although it is technically very unsatisfactory. • It relates the annual coupon payment to the current market price.

37. Example of the Current Yield • A 15 year 15% coupon bond is currently selling for $800. • The current yield is given by

38. Current Yield (Cont…) • If you buy this bond for $800 and hold it for one year you will earn an interest income of $150. • So your interest yield is 18.75% • However, if you sell it after one year you will either make a Capital Gain or a Capital Loss.

39. Current Yield (Cont…) • What is a Capital Gain? • If the price at the time of sale is higher than the price at which the bond was bought, the profit is termed as a Capital Gain. • Else if there is a loss, it is termed a Capital Loss. • The current yield does not take such gains and losses into account.

40. Current Yield (Cont…) • On the other hand if you choose to hold on to the bond at the end of one year, you will earn additional coupons. • This aspect too is ignored by the Current Yield computation.

41. Yield to Maturity (YTM) • The YTM is the interest rate that equates the present value of the cash flows from the bond (assuming that the bond is held to maturity), to the price of the bond. • It is exactly analogous to the concept of the Internal Rate of Return (IRR) used in project valuation.

42. YTM (Cont…) • Consider a bond that makes an annual coupon of C on a semi-annual basis. • The face value is M, the price is P, and the number of coupons remaining is N.

43. YTM (Cont…) • The YTM is the value of y that satisfies the following equation.

44. YTM (Cont…) • The YTM is a solution to a non-linear equation. • We generally require a financial calculator or a computer to calculate it. • However it is fairly simple to compute the YTM in the case of a coupon paying bond with exactly two periods to maturity. • In such a case it is simply a solution to a quadratic equation.

45. YTM for a Zero Coupon Bond • The YTM is easy to compute in the case of zero coupon bonds. • Consider a ZCB with a face value of $1,000, maturing after 5 years. • The current price is $500. • The YTM is the solution to

46. Features of YTM • The YTM calculation takes into account all the coupon payments, as well as any capital gains/losses that accrue to an investor who buys and holds a bond to maturity.

47. Sources of Returns From a Bond • A bondholder can expect to receive income from the following sources. • Firstly there are coupon payments which are typically paid every six months. • There will be a capital gain/loss when a bond matures or is called before maturity or is sold before maturity.

48. Returns From a Bond (Cont…) • The YTM calculation assumes that the bond is held to maturity. • Finally when a coupon is received it will have to be reinvested till the time the bond eventually matures or is sold or is called. • Once again the YTM calculation assumes that the bond is held till maturity. • The reinvestment income is nothing but interest on interest.

49. YTM • A satisfactory measure of the yield should take into account all the three sources of income. • The current yield measure considers only the coupon for the first year. • All the other factors are totally ignored.

50. YTM (Cont…) • The YTM calculation takes into account all the three sources of income. • However it makes two key assumptions. • Firstly it assumes that the bond is held till maturity. • Secondly it assumes that all intermediate coupons are reinvested at the YTM itself.