CFA Level1 Jun2007 Fixed Income Analysis of Fixed

CFA Level-1 [Jun-2007] : Fixed Income Analysis of Fixed Income Investments CFA Level – 1 [June 2007] Study Sessions 15 -16 Lecture 3 of 3 Instructor: Asif Ali Qureshi, CFA

CFA, Level-1 [Jun-2007] : Fixed Income ANALYSIS & VALUATION Study Session No. 16 2

CFA, Level-1 [Jun-2007] : Fixed Income 2. Yield Measures, Spot Rates & Forward Rates 3

CFA, Level-1 [Jun-2007] : Fixed Income Sources of Return on Fixed Income Securities q Coupon – Generally in shape of periodic payments. q Capital Gain/Loss – Cash received at maturity/at sale > Purchase Price Capital Gain. q Reinvestment Income – Return from re-investment of intermediate cash flows from security prior to its final maturity. 4

CFA, Level-1 [Jun-2007] : Fixed Income Traditional Yield Measures 1. Current Yield 2. Yield to Maturity (YTM) 3. Yield to Call 4. Yield to Put 5. Yield to Worst 6. Cash Flow Yield 5

CFA, Level-1 [Jun-2007] : Fixed Income Current Yield q Definition: – Current Yield = Annual Dollar Coupon Price q Example: – 7% coupon, 8 - year bond, priced at $94. 17 – Current Yield = (0. 07 x 100)_ = 7. 43% 94. 17 6

CFA, Level-1 [Jun-2007] : Fixed Income Current Yield q Current Yield and Coupon Rate – Current Yield > Coupon Rate, if bond selling at discount – Current Yield < Coupon Rate , if bond selling at premium – Current Yield = Coupon Rate, if bond selling at par q Drawbacks – Only coupon taken into consideration. Other sources of return (capital gain/loss, reinvestment income) not taken into account. 7

CFA, Level-1 [Jun-2007] : Fixed Income Yield to Maturity (YTM) q Definition: – The interest rate that will make PV of a bond’s cash flows equal to its market price. – Concept similar to IRR. n Price = ∑ Coupon (1 + Y)i + Par Value (1 + Y)n i =1 8

CFA, Level-1 [Jun-2007] : Fixed Income Yield to Maturity (YTM) q YTM reported in Bond Equivalent Yield (BEY) Terms – BEY = semi-annual yield x 2 – Effective Annual Yield > BEY q Example: – 7. 0%, 8–year bond selling for $94. 172 16 94. 172 = ∑ 3. 50 (1 + Y)i 100 + (1 + Y)16 i =1 Y = 4. 0% YTM = Y x 2 = 8. 0% 9

CFA, Level-1 [Jun-2007] : Fixed Income Yield to Maturity (YTM) q Limitations – YTM inherently assumes that coupons shall be reinvested at rate to YTM. – Bond is held till maturity. q Important Differentiation: – Total Future Dollars: Coupons + Principal – Total Dollar Return: Coupons + Capital Gain/Loss + Reinvestment Income 10

CFA, Level-1 [Jun-2007] : Fixed Income Yield to Maturity (YTM) q Option – 1: Invest $94. 17 in bank’s CD maturing in 8 years offering semi-annual interest rate of 4. 0% (BEY of 8. 0%). No interim payments of interest and entire amount paid at maturity. q Total amount at maturity $94. 17 x (1. 04)16 Total Future Dollars = 176. 38 – Return of Principal = = Total Interest on CD 94. 17 = 82. 21 11

CFA, Level-1 [Jun-2007] : Fixed Income Yield to Maturity (YTM) q Option – 2: Invest $94. 17 in 8 year bond, carrying 7% coupon (BEY) paid semi-annually yielding 8. 0% YTM. Bond price is $94. 17. q Cash Flows from the Bond – 16 semi-annual coupon (3. 5 each) = 56. 00 – Principal (incl. $5. 83 capital gain) = 100. 00 – TOTAL = 156. 00 q Difference between cash flows from CD and Bond – 176. 38 – 156. 00 = 20. 38 Reinvestment return 12

CFA, Level-1 [Jun-2007] : Fixed Income Yield to Maturity (YTM) OPTION -1 Value at Maturity OPTION -2 176. 38 Coupons 56. 00 Return of Principal 94. 17 Capital Gain Interest on CD 82. 21 Reinvestment Return 20. 38 Total Return 82. 21 5. 83 • Both options yield same returns but option-2 has higher reinvestment risk. • YTM limitation highlighted. Actual realized yield (on coupon bond) would equal YTM only if coupons reinvested at YTM 13

CFA, Level-1 [Jun-2007] : Fixed Income Yield to Maturity q Factors affecting Reinvestment Risk – For given YTM and a given non-zero coupon rate: Longer the maturity Higher the reinvestment risk Higher the coupon Higher the reinvestment risk 14

CFA, Level-1 [Jun-2007] : Fixed Income Semiannual vs. Annual Pay Bonds q Bond Equivalent Yield (BEY) of Annual Pay Bond = 2 [ (1 + Annual Yield)0. 5 – 1 ] q Example: – Yield on Annual Pay Bond = 6. 0% – BEY = 2 [ (1 + 0. 06)0. 5 – 1] = 5. 913% 15

CFA, Level-1 [Jun-2007] : Fixed Income Yield to Call q Yield to Call assumes that the issuer will call a bond on some assumed call date at the call price specified in the call schedule. q Types of Yields to Call – Yield to First Par Call – Yield to Refunding 16

CFA, Level-1 [Jun-2007] : Fixed Income Yield to Call q Yield to First Call – Example: A 7%, 8 -year bond with maturity value of $100 is current trading at $106. 36. First call is 3 years from now at $103. What is the Yield to First Call. n Price = ∑ Coupon + Call Price (1 + Y)i (1 + Y)n 3. 50 103. 00 i =1 6 106. 36 = ∑ (1 + Y)i + (1 + Y)6 i =1 Y = 2. 8% Yield to First Call = 5. 6% 17

CFA, Level-1 [Jun-2007] : Fixed Income Yield to Call q Yield to Refunding – Used when bond is currently callable but contains some restrictions on the source of funds used to buy back the debt when call is exercised. – Cash flows for yield calculation: coupon payments to first refunding date & call price at first refunding date q Yield to First Par Call – Cash flows for yield calculation: coupon payments to first par call date & par value at the first par call date. 18

CFA, Level-1 [Jun-2007] : Fixed Income Yield to Put q Example: – 6. 2%, 8 year bond is putable at par in 3 years. Current bond price is $102. 19. Calculate the Yield to Put n Price = ∑ Coupon + Put Price (1 + Y)i (1 + Y)n 3. 10 100. 00 i =1 6 102. 19 = ∑ + (1 + Y)i (1 + Y)6 i =1 Y = 2. 7% Yield to Put = 5. 4% 19

CFA, Level-1 [Jun-2007] : Fixed Income Yield to Worst q Lowest of all possible yields, i. e. , YTM, Yield to Call and Yield to Put. q Example: – Yield to Call for 4 call dates: 6%, 6. 2%, 5. 8% & 5. 7% – YTM is 7. 5% – Yield to Worst is 5. 7% i. e. , yield to 4 th call date 20

CFA, Level-1 [Jun-2007] : Fixed Income Cash Flow Yield q Applies in case of Mortgage Backed Securities q Cash flows for MBS are projected using certain assumption about prepayment rate/speed. q Cash flow yield is the interest rate that makes PV of the projected cash flows equal to MBS price. q MBS cash flows are typically on monthly basis 21

CFA, Level-1 [Jun-2007] : Fixed Income Cash Flow Yield q Calculation – MBS cash flows are typically on monthly basis – Effective semi-annual yield = (1 + monthly yield)6 -1 – Cash Flow Yield = 2 x effective semi-annual yield q Limitations – Assumes reinvestment of cash flows at a rate equal to the cash flow yield – Also assumes that security held till final pay-off. 22

CFA, Level-1 [Jun-2007] : Fixed Income Theoretical Spot Rates q Spot rates are yields on zero coupon securities. q Treasury yield curve is constructed using treasury spot rates. q Limitations: – >12 months treasury securities carry coupons – On-the-run securities of limited maturities available. q Theoretical treasury spot rate curve constructed from observed treasury yields Bootstrapping 23

CFA, Level-1 [Jun-2007] : Fixed Income Theoretical Spot Rates q Interpolation: – Yields of missing data points on yield curve estimated thorough linear interpolation. – Yield at higher maturity – Yield at lower maturity No. of Years between two observed maturities – Example: 5 year yield is 3. 25%, 10 -year yield is 4. 35% (4. 35% - 3. 25%) / 5 = 0. 22% would be sequentially added to 5 year security for estimating yields for 6, 7, 8 and 9 year bonds 24

CFA, Level-1 [Jun-2007] : Fixed Income Theoretical Spot Rates: Calculation q Basic Principle: – Value of treasury coupon security SHOULD BE EQUAL to the value of the package of zero-coupon treasury securities that duplicates the bond’s cash flows. q Spot rate (zn) for period ‘n’ Pricen = Cn (1 + z 1)1 + Cn (1 + z 2)2 + + Cn+ Par (1 + zn)n 25

CFA, Level-1 [Jun-2007] : Fixed Income Theoretical Spot Rates: Calculation 6 -month period Years Yields Price 1 0. 50 3. 00% - Z 1 3. 00% 2 1. 00 3. 30% - Z 2 3. 30% 3 1. 50 3. 50% 100. 00 Z 3 4 2. 00 3. 90% 100. 00 Z 4 Price 3 = C 3 + (1 + z 2)2 (1 + z 1)1 100. 00 = 1. 75 (1. 015)1 z 3 = 1. 7527% + 1. 75 + (1. 0165)2 Spot Rates C 3+ Par (1 + z 3)3 + 1. 75+100 (1 + z 3)3 Z 3 = 3. 5053% 26

CFA, Level-1 [Jun-2007] : Fixed Income Theoretical Spot Rates: Calculation 6 -month period Years Yields Price 1 0. 50 3. 00% - Z 1 3. 00% 2 1. 00 3. 30% - Z 2 3. 30% 3 1. 50 3. 50% 100. 00 Z 3 3. 50% 4 2. 00 3. 90% 100. 00 Z 4 Price 4 = C 4 + (1 + z 2)2 (1 + z 1)1 100. 00 = 1. 95 (1. 015)1 z 4 = 1. 9582% C 4 + 1. 95 (1. 0165)2 + C 4 Spot Rates + (1 + z 3)3 + 1. 95 C 4+ Par (1 + z 4)4 + (1. 017527)3 1. 95+100 (1 + z 4)4 Z 4 = 3. 9164% 27

CFA, Level-1 [Jun-2007] : Fixed Income Treasury Yield Curve 28

CFA, Level-1 [Jun-2007] : Fixed Income Nominal Spread q The spread between YTMs of a non-treasury bond a treasury bond of same maturity. q Measures the compensation for additional risks i. e. , credit risk, option risk, and liquidity risk. q Drawbacks – YTM does not take into account term structure of interest rates, and – In case of bond carrying an option, interest rate volatility may change cash flows of non-treasury bond. 29

CFA, Level-1 [Jun-2007] : Fixed Income Zero Volatility Spread (Z-Spread or Static Spread) q A measure of spread that the investor would realize over the entire treasury spot rate curve if the bond is held to maturity. q Calculated as the spread that will make PV of cash flows of non-treasury bond equal to its price when discounted at the treasury spot rate plus the spread. 30

CFA, Level-1 [Jun-2007] : Fixed Income Nominal Spread vs. Z-Spread q Nominal spread is a spread off one point on the treasury yield curve. q While Z-Spread is a spread over the entire treasury spot rate curve. q Divergence between Nominal Spread & Z-Spread – Typically low for bonds with bullet maturity – Higher for amortizing securities – Function of: 1) term structure and 2) security characteristics 31

CFA, Level-1 [Jun-2007] : Fixed Income Option Adjusted Spread (OAS) q Z-Spread does not take into account interest rate volatility which could change the cash flows with embedded options. q OAS values a security using “valuation models” which take into account interest rate volatility. q OAS is the spread (over treasury spot rate curve) that would equate the ‘fair price’ of a security to its market price. OAS is model dependent. q Z-Spread = OAS + Option Cost 32

CFA, Level-1 [Jun-2007] : Fixed Income Spread Measures - Summary Spread Benchmark Risks Accounted For: Nominal Treasury Yield Curve Credit, Option, Liquidity Zero Volatility Treasury Spot Rate Curve Credit, Option, Liquidity OAS Treasury Spot Rate Curve Credit, Liquidity 33

CFA, Level-1 [Jun-2007] : Fixed Income Implied Forward Rates q Implied forward rates are extrapolated from the default free theoretical spot rate curve. 3 fn 3 f 4 z 1 z 2 z 3 z 4 z 5 zn 34

CFA, Level-1 [Jun-2007] : Fixed Income Implied Forward Rates (1 + z 4)4 z 3 z 2 z 1 z 4 (1 + 3 f 4) (1 + z 3)3 (1 + z 4)4 3 f 4 = (1 + z 3)3 = (1 + z 4)4 x (1 + 3 f 4) – 1 (1 + z 3)3 35

CFA, Level-1 [Jun-2007] : Fixed Income Implied Forward Rates Semiann ual Annual Z 1 3. 0000% z 1 1. 5000% Z 2 3. 3000% z 2 1. 6500% Z 3 3. 5053% z 3 1. 7527% Z 4 3. 9164% z 4 1. 9582% 3 f 4 = (1 + z 4)4 – 1 (1 + z 3)3 3 f 4 = (1. 019582)4 – 1 = 2. 577% 5. 15469% (BEY basis) (1. 017527)3 36

CFA, Level-1 [Jun-2007] : Fixed Income Implied Forward Rate q Formula: 1/t tfm = (1 + zm+t)m+t – 1 (1 + zm)m 37

CFA, Level-1 [Jun-2007] : Fixed Income 3. Measurement of Interest Rate Risk 38

CFA, Level-1 [Jun-2007] : Fixed Income Full Valuation Approach q Valuation of bond based on different interest rates scenario is called full valuation approach. q Also called scenario analysis. q Measures the interest rate risk exposures of a bond or portfolio of bonds. 39

CFA, Level-1 [Jun-2007] : Fixed Income Price Volatility: Option Free Bond q Price and Yields move in opposition direction but %age price changes not same for all bonds q For small changes in yield, %age price change for a given bond is roughly the same for increase and decrease (Duration) q For large changes in yield, %age price change is not symmetrical. q For a given large change in yield, the %age price increase is greater than %age price decrease (convexity) 40

CFA, Level-1 [Jun-2007] : Fixed Income Convexity Illustrated Price Y – Y 1 = Y 2 - Y P 1 – P > P – P 2 (P 1– P) P P 2 (P – P 2) Y 1 Y Y 2 Yield 41

CFA, Level-1 [Jun-2007] : Fixed Income Price Volatility: Callable Bond Price a Option Free Bond a – a’ b b’ For a given change in yield: Price Appreciation < Price Decline Callable Bond a’ – b a’ Negative Convexity y* Yield Difference between the price of option free bond and the price of callable bond = value of call option 42

CFA, Level-1 [Jun-2007] : Fixed Income Price Volatility: Bond with Put Option Price a Option Free Bond a – a’ For a given change in yield: Price Decline < Price Appreciation c’ Putable Bond a–c c a’ y* Yield 43

CFA, Level-1 [Jun-2007] : Fixed Income Duration q Duration – Approx %age price change for 100 bp change in yield Duration = V– – V + 2 x V 0 x ∆y ∆y: change in yield in decimal V 0 : initial price V– : price if yields decline by ∆y V+: price if yields increase by ∆y 44

CFA, Level-1 [Jun-2007] : Fixed Income Duration q Example: 9%, 20 -year bond, YTM: 6. 0%. Estimate duration using 20 bp change in yield Duration = V– – V + 2 x V 0 x ∆y ∆y: 0. 002 V 0: 134. 6722 V –: 137. 5888 V+: 131. 8439 Duration = 137. 5888 – 131. 8439 = 10. 66 2 x 134. 6722 x 0. 002 45

CFA, Level-1 [Jun-2007] : Fixed Income Duration: Graphical Illustration Price Duration underestimates new bond price for large change in yield P 1’ P P 1 P 2’ P 2 200 bp Y Yield 46

CFA, Level-1 [Jun-2007] : Fixed Income Modified Duration vs. Effective Duration q Modified Duration – Approx %age change in a bond’s price to 100 bp change in yield, assuming that the bond’s expected cash flows do not change when the yield changes. – Applicable to option free bonds q Effective Duration – Approx %age change in a bond’s price to 100 bp change in yield, with due consideration that bond’s cash flows could change due to change in yield. – Valuation based on models. Applied to bonds with embedded options. 47

CFA, Level-1 [Jun-2007] : Fixed Income Modified Duration vs. Effective Duration DURATION Generic description of the sensitivity of a bond’s price (as %age of initial price) to a change in yield Modified Duration Effective Duration measure in which it is assumed that yield changes do not change the expected cash flows. Duration measure in which recognition is given to the fact that yield changes may change the expected cash flows. 48

CFA, Level-1 [Jun-2007] : Fixed Income Macaulay Duration and Modified Duration q Modified Duration: 1 1 x PVCF 1 + (1 + yield/k) 2 x PVCF 2 + …. + n x PVCFn k x Price k: no. of period/year yield: YTM n: no. of periods until maturity PVCF: PV of cash flow q Modified Duration = Macaulay Duration/(1+yield/k) 49

CFA, Level-1 [Jun-2007] : Fixed Income Portfolio Duration q Weighted avg. duration of all bonds in the portfolio w 1 D 1 + w 2 D 2 + … + w k Dk wi: weight of bond ‘i’ in the portfolio Di: duration of bond ‘i’ K: no. of bonds in the portfolio q Limitation – Assumes parallel shift in yield curve i. e. , bonds of all maturities change by same amount. 50

CFA, Level-1 [Jun-2007] : Fixed Income Portfolio Duration: Example Bond Price Yield Par Amount Mkt. Value Duration 10% 5 -yr 100. 0000 10. 00% 4, 000, 000 3. 861 8% 15 -yr 84. 6275 10. 00% 5, 000 4, 231, 375 8. 047 14% 30 -yr 137. 8586 10. 00% 1, 000 1, 378, 586 9. 168 10, 000 9, 609, 961 TOTAL w 1: 4, 000 / 9, 609, 961 = 0. 416 D 1 : 3. 861 w 2: 4, 231, 375 / 9, 609, 961 = 0. 440 D 2 : 8. 047 w 3: 1, 378, 586 / 9, 609, 961 = 0. 144 D 3 : 9. 168 Portfolio Duration = 0. 416 x 3. 861 + 0. 440 x 8. 047 + 0. 144 x 9. 168 = 6. 47 51

CFA, Level-1 [Jun-2007] : Fixed Income Convexity Adjustment q Convexity adjustment to %age price change = C x 100 (∆y*)2 x V+ + V- – 2 V 0 C= 2 V 0 (∆y)2 ∆y: change in yield in decimal V 0: initial price V–: price if yields decline by ∆y V+ : price if yields increase by ∆y ∆y*: change in yield for which price change is sought 52

CFA, Level-1 [Jun-2007] : Fixed Income Convexity Adjustment q Example: 9%, 20 year bond, YTM: 6%, ∆y: 20 bp ∆y: V 0: 0. 002 V –: 137. 5888 V+: 131. 8439 C= C= 134. 6722 V+ + V- – 2 V 0 (∆y)2 131. 8439 + 137. 5888 – 2 (134. 6722) = 81. 95 2 (134. 6722) (0. 002)2 Duration = V– – V + 2 x V 0 x ∆y = 137. 5888 – 131. 8439 = 10. 66 2 x 134. 6722 x 0. 002 53

CFA, Level-1 [Jun-2007] : Fixed Income Convexity Adjustment q Estimate the %age price change for 200 bp change in yield: – Total Estimated Price Change = Estimated Change Using Duration – Convexity Adjustment Price Change using Duration = Convexity Adjustment = = Total Price Change = 0. 02 x 10. 66 = 21. 32% C x (∆y*)2 x 100 81. 95 x (0. 02)2 x 100 21. 32% – 3. 28% = 18. 04% 54

CFA, Level-1 [Jun-2007] : Fixed Income Price Value of a Basis Point (PVBP) q Dollar value of 1 bp change in yield q PVBP = Bond Price x Duration x 0. 0001 q Example: 9%, 20 year bond, 6% YTM – Bond Price = $134. 6722 – PVBP = 134. 6722 x 10. 66 x 0. 0001 = $0. 1435 55