VALUATION OF FIXED INCOME SECURITIES. Presented By Sade Odunaiya Partner, Risk Management Alliance Consulting

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1 VALUATION OF FIXED INCOME SECURITIES Presented By Sade Odunaiya Partner, Risk Management Alliance Consulting

2 OUTLINE Introduction Valuation Principles Day Count Conventions Duration Covexity Exercises

3 INTRODUCTION Valuation determination of fair value i.e. Price Yield and Yield spread is relative measure of value

4 TERMINOLOGIES Fixed Income Security A financial obligation that promises to pay specified sums at specified future dates Debt obligation borrower/lender with a promise to pay interest and principal Preferred stock ownership interest with fixed dividend payment from after tax profit

5 Terminologies (Contd) Indenture & Covenants Rights & obligations managed by a trustee Affirmative & Negative covenants Term to Maturity No of years issuer promises to pay Same as tenor only on issue date Par Value Amount payable at maturity Principal, face value, redemption value and maturity value Assumed 100

6 Terminologies (Contd) Trading Value Below Par Value Discount Above Par Value Premium Coupon Rate Interest rate issuer pays each year Called nominal rate Coupon is the actual amount paid/payable annually No coupon instruments are Zero Coupon instruments

7 BASIC PRINCIPLES OF VALUATION Value or Price Present Value (PV) of expected cash flows 3 basic steps Step 1: Estimate expected cash flows (coupon & principal) Step 2: Estimate appropriate discount rate using comparable bonds Step 3: Calculate the PV of cash flows in Step 1 using discount rate in Step 2

8 STEP 1: ESTIMATE EXP CASH FLOW Cash expected to be received in the future Principal & interest Difficult to estimate for some instruments where- Option to change contractual date e.g. callable/putable bonds, mortg backed sec. Coupon reset based on reference rates, prices / exch rate Investor has choice to convert or exchange security

9 STEP 2: DETERMINING APPR RATE Minimum rate should be credit-risk free rate with same maturity i.e. Treasury rate Premium should reflect additional risk embedded in the instrument & issuer Comparative risk or alternative investment return Differing rate for each of the expected cash flows

10 STEP 3: DISCOUNTING THE CASH FLOWS In general, the price of a bond is given by P = C + C + C + + C + M 1+ r (1+ r) 2 (1+ r) 3 (1+ r) n (1+ r) n or: P = C + M (1+ r) t (1+ r) n where: C = coupon payments usually semiannual P = price of the investment n = number of periods r = periodic interest rate M = maturity value t = time period when the payment is received

11 STEP 3: DISCOUNTING THE CASH FLOWS Annuity Concept - valuation with the same discount rate The present value of an annuity is equal to 1-1 Annuity payment X (1 + r) n r

12 STEP 3: DISCOUNTING THE CASH FLOWS Present Value Properties A bond that matures in 4 years time has a coupon rate of 10% with an annual interest payment frequency. a. Assuming applicable discount rate for similar security is 8%, what is the price of the bond today? b. If the discount rate is changed to 12%, what will the price be?

13 STEP 3: DISCOUNTING THE CASH FLOWS NOTE: TIMING: The longer the term to maturity, the lower the value of the cash flow today DISCOUNT RATE: The higher the discount rate, the lower the security value RELATIONSHIP OF COUPON RATE, DISCOUNT RATE & PRICE RELATIVE TO PAR: Coupon rate = yield required; price = par value Coupon rate < yield required; price < par value Coupon rate > yield required; price > par value CHANGE IN VALUE AS TIME MOVES CLOSE TO MATURITY: A bond s valueo Decreases over time if the bond is selling at a premium o Increases over time when the bond is selling at a discount o Is unchanged if the bond is selling at par value o At maturity is equal to its Par Value

14 VALUATION USING MULTIPLE RATES Proper way to value cash flows Illustration Suppose we have a 4-year 10% coupon bond with annual interest payment and the appropriate discount rates are as follows: Year 1 6.8% Year 2 7.2% Year 3 7.6% Year 4 8.0%

15 VALUATION USING MULTIPLE RATES Practice: What is the value of a 5-year 7% coupon bond assuming the payments are received annually and the discount rates for each year are as follow: Year 1 3.5% Year 2 3.9% Year 3 4.2% Year 4 4.5% Year 5 5.0%

16 VALUING SEMI-ANNUAL CASH FLOWS Same Computation Each period is now 6 months No of periods will double Exercise 5 assuming semi-annual cash flows

17 VALUING ZERO COUPON BONDS One cash flow at maturity Maturity value (1 + i) n x 2 Where i is the semi-annual discount rate n is the number of years to maturity Note: The number of periods is n X 2 for consistency with the pricing of coupon bearing bonds in the market.

18 VALUATION COMPLICATIONS Next Coupon less than 6 months away Interest earned by seller Interest earned by buyer Last coupon Settlement Next coupon payment date date payment date Accrued Interest (AI) interest earned by seller Full or Dirty Price based on discounted value Clean Price = Full price Accrued Interest

19 VALUATION COMPLICATIONS Full or Dirty Price Present value t = expected cash flow (1 + I ) t -1+w Where: w period = days btw settlement and next coupon date no of days in coupon period

20 DAY COUNT CONVENTIONS Different day-year conventions in different markets Eurodollar (LIBOR) market Actual/360 Eurobond market (AIBD) 360/360 US Treasury/ Ghanaian Money Market Actual/365 Nigeria Money Market (NMM) Actual/Actual The numerator depicts the number of days interest is to be earned while the denominator depicts the number of days in a year in the relevant market.

21 DAY COUNT CONVENTIONS A coupon bearing US Treasury security whose previous coupon payment was March 1 and the next is September 1. Suppose this Treasury security is purchased with a settlement date of July 17 th.

22 CASH FLOWS ARE UNKNOWN Mainly in respect of options embedded securities Valuation models Binomial Monte Carlo Simulation

23 Fixed Instrument Price Volatility Measures Bond price moves in an inverse relationship to yield changes Price Yield

24 Bond price volatility Measured in terms of percentage price change in price Influenced by more than yield behavior alone Factors such as: Par value Coupon rate Term to maturity Prevailing interest rate

25 Relationship between Yield/Price Price moves inversely to yield Price volatility is directly related to term to maturity i.e. longer maturity bonds post larger price changes Price volatility increases at a diminishing rate as term to maturity increases Price movements from equal absolute increases or decreases in yield are not symmetrical Higher coupon issues show smaller percentage fluctuations for a given change

26 Illustration 1: Effect of Maturity on Bond Price Volatility Present Value of an 8% Bond ($1,000 Par Value) Year 10 Years 20 Years 30 Years Yield to Maturity 7% 10% 7% 10% 7% 10% 7% 10% Present value: - interest , principal Total value 1, , , , % change -2.9% % -28.7%

27 Illustration 2 Effect of Coupon on Bond Price Volatility Present Value of an 20 Year Bond ($1,000 Par Value) Coupon 3% Coupon 8% Coupon 12% Coupon Yield to Maturity 7% 10% 7% 10% 7% 10% 7% 10% Present value: - interest ,287 1,030 - principal Total value , ,544 1,172 % change -44.7% % -24.1%

28 Trading Strategies What type of portfolio of bonds will you seek to have if you expect a major decline in interest rate going forward? Zero coupon ( or very low coupon) with long term to maturity in order to post maximum gains

29 Measures of bond price volatility Measure of interest rate sensitivity is Duration Three key measures: Macaulay duration Modified duration Effective duration

30 Macaulay Duration Measure of time (xteristics) flow from a bond Weighted average number of years over which total cash flows occur Weightings used are the market value of cash flows Σ C t (t) D = (1 + i) t Σ C t Price of the bond (1 + i) t

31 Example: Consider two bonds with the following features: Face value $1,000 $1,000 Maturity 10 years 10 years Coupon 4% 8% Calculate the Macaulay duration for each of the bonds assuming an 8% market yield.

32 Example: Bond A Year Cash flow 8% PV of flow PV as % of Price PV as % of Price time weighted , Total Duration = 8.12 years

33 Example: Bond B Year Cash flow 8% PV of flow PV as % of Price PV as % of Price time weighted , Total 1, Duration = 7.25 years

34 Macaulay Duration Characteristics Less than term to maturity due to interim cash flows Inverse relationship between coupon and duration I.e. higher coupon lower duration Positive relationship between term to maturity & Macaulay duration but duration increases at a decreasing rate with maturity Sinking fund and call provisions have dramatic impact on bond s duration

35 Modified Duration Measures the price volatility of a non-callable bond Modified Duration = Macaulay duration 1 + YTM/n Greater modified duration, the greater the price volatility for small price changes Therefore dp 1 = dy P - modified duration

36 Modified Duration Example: Consider a bond with Macaulay Duration of 8 years, yield of 10%. If you expect YTM to decline by 75 basis point (from say 10% to 9.25%) What is the expected percentage price change?

37 Modified Duration If you expect a decline in interest rate, you should increase the average modified duration of your portfolio Note that duration changes in a nonlinear fashion with yield changes a concept called convexity Necessary to recalculate and rebalance the portfolio as rate changes

38 Effective Duration Direct measure of interest rate sensitivity of any asset (bond inclusive) Based on observed market prices surrounding interest rate changes %age change in Price = - Modified Duration X change in Yield - D* = %age change in Price Change in yield The D* obtained this way is the effective duration.

39 Effective Duration Note: Effective duration greater than maturity is possible with CMO Negative effective duration as in the case of bonds with embedded options

40 THANK YOU ANY QUESTIONS??

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