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Session 8Convexity & Factors affecting Interest Rate1Convexity Analysis of Bond Price VolatilityBond Price Volatility2Price Yield Relationship for Six Hypothetical Bonds9% coupon bond with 5 years to maturity9% coupon bond with 25 years to maturity6% coupon bond with 5 years to maturity6% coupon bond with 25 years to maturity0% coupon bond with 5 years to maturity0% coupon bond with 25 years to maturity

3Bond Price Volatility3Req yield9%/5 yr9%/ 25 yr6%/5 yr6%/ 25 yr0%/5 yr0%/ 25 yr6%112.79138.5910010074.422.87%108.3123.4595.8488.2770.8917.908%104.05110.791.8878.5167.5514.079%10010088.1370.3564.3911.079.01%99.9699.9088.0970.2864.3611.04410%96.13990.8784.5563.4861.398.711%92.4683.0681.1557.658.546.812%88.9576.3577.9152.7155.835.524Bond Price Volatility4Percentage Price Change Req yieldChange in Basis Point9%/5 yr9%/ 25 yr6%/5 yr6%/ 25 yr0%/5 yr0%/ 25 yr6%-30012.8038.5913.4742.1315.56106.047%-2008.3223.468.7525.4610.0961.738%-1004.0610.744.2611.604.9127.109.01%1-0.04-0.10-0.04-0.11-0.05-0.2410%100-3.86-9.13-4.06-9.76-4.66-21.2311%200-7.54-16.93-7.91-18.03-9.08-37.8912%300-11.04-23.64-11.59-25.08-13.28-50.965Bond Price Volatility5Price Volatility Characteristics of Option- Free Bonds (Properties)Although the prices of all the option free bonds move in the opposite direction from the change in yield required, the percentage price change is not the same for all bonds.For very small changes in the yield required, the percentage price change for a given bonf is roughly the same, whether the yield required increases or decreases 6Bond Price VolatilityFor large changes in the required yield, the percentage price change is not the same for an increase in the required yield as it is for a decrease in the required yieldFor a given change in basis point, the percentage price increase is greater than the percentage price decrease. (Long position: Potential capital gain is more than the capital loss, Short position: ) 7Bond Price Volatility7Characteristics of a Bond that Affect its Price VolatilityThere are two characteristics of an option free bonds that determine its price volatility: Coupon and Term to MaturityFor a given term to maturity and initial yield, the price volatility of a bond is greater, than the lower the coupon rateFor a given coupon rate and initial yield, the longer the term to maturity, the greater the price volatility

8Bond Price Volatility8Long maturities in portfolio:Investors who wants to increase a portfolios price volatility because they expect interest rate to fall, all other factors being constant, should hold for long maturityShort maturities in portfolio:

9Bond Price Volatility9Duration Question?Calculate the Bond duration and Modified bond for the bond with the face value of Rs 1000 for 5 years coupon rate is 10% semi annually. YTM is 9%10Convexity & Factors affecting Interest RatePeriodCash FlowPV Cash FlowDuration Calc0($1,039.56)1 50.00 47.85 47.85 2 50.00 45.79 91.57 3 50.00 43.81 131.44 4 50.00 41.93 167.71 5 50.00 40.12 200.61 6 50.00 38.39 230.37 7 50.00 36.74 257.19 8 50.00 35.16 281.27 9 50.00 33.65 302.81 10 1,050.00 676.12 6,761.24 Total 8,472.07 11Convexity & Factors affecting Interest RateBond price = 1039.56Duration = 8472.07 / 1039.56 / 2 = 4.07Modified Duration = 4.07 / 1.045= 3.9012Convexity & Factors affecting Interest RateDuration AdjustmentModified duration x Yield change x Bond price

13Convexity & Factors affecting Interest RateConvexity & Factors affecting Interest Rate14Convexity The Importance of ConvexityConvexity is the difference between the actual price change in a bond and that predicted by the duration statistic

Convexity helps to approximate the change in price that is not explained by duration because of the convex nature of the yield curve.

In practice, the effects of convexity are minor15Convexity & Factors affecting Interest RateConvexityMeasures how much a bonds price-yield curve deviates from a straight line

Convexity refers to the shape of the price/yield function, and is a measure used in conjunction with duration to estimate bond price sensitivity to changes in interest rates

16Convexity & Factors affecting Interest Ratemeasure of the sensitivity of the price of a bond to changes in interest rates Positive convexity corresponds to curvature that opens upward. Negative convexity corresponds to curvature that opens downward.

ConvexityConvexity is useful for comparing bonds with same duration and yield.Convexity is a measure of the curvature of the price-yield relationship.

17Convexity & Factors affecting Interest RateThe relationship between the convexity, coupon, maturity and yield of bond:The coupon of a bond and convexity (when coupon and maturity are constant) are inversely related ie lower coupon, higher convexity.The maturity of a bond and convexity (when coupon and yields are constant) are directly related ie Longer maturity, higher convexity.The yield of a bond and convexity (when coupon and maturity are constant) are inversely related. The price yield curve is more convex at its lower yield

18Convexity & Factors affecting Interest Rate18Convexity Calculation Convexity & Factors affecting Interest Rate19Question Calculate convexity measure for 5 year semi annually bond face value Rs 100 coupon rate is 9% semi annually, YTM 9%.

Cash flow = 4.5 for the period of 1020Convexity & Factors affecting Interest RatePeriodCash flow (Rs) 1 (1.045)t+2t(t+1)CFt(t+1)CF * 1/ (1.045)t+2 14.50.87629697.88624.50.8385612722.64134.50.8024515443.33244.50.7678959069.11054.50.73482813599.20164.50.703185189132.90174.50.672904252169.57184.50.643927324208.63294.50.616198405249.56010104.500.589663114956778.186Total7781.02021Convexity & Factors affecting Interest RateConvexity measures = conv cal / bond price / (2)^Frequency of payment in a year

= 7781.020 / 100 = 77.8102 / 4=19.4522Convexity & Factors affecting Interest RateConvexity Adjustment=0.5*Convexity*Yield_Change^2*Bond_Price

0.5* 19.45* (0.02^2)*1000.5* 19.45* 0.0004*100

Yield change is the difference between the initial yield and current market yield.0.38923Convexity & Factors affecting Interest RateQuestion 1 Calculate:DurationModified DurationConvexityAdjustment of Bond price if yield will change by 3% (+ & -)For 5 year bond face value Rs 1000 coupon rate is 8% compounding semi annually, YTM 10%.

24Convexity & Factors affecting Interest RateDuration calculation PeriodCash FlowPV Cash FlowDuration Calc0($922.78)1 40.00 38.10 38.10 2 40.00 36.28 72.56 3 40.00 34.55 103.66 4 40.00 32.91 131.63 5 40.00 31.34 156.71 6 40.00 29.85 179.09 7 40.00 28.43 198.99 8 40.00 27.07 216.59 9 40.00 25.78 232.06 10 1,040.00 638.47 6,384.70 7714.08 25Convexity & Factors affecting Interest RateDuration = 7714.08 / 922.78 / 2 = 4.18Modified Duration = 4.18 / 1.05= 3.98If 3% change in yield means bond price will change:Modified duration x Yield change x Bond price3.98* 0.03* 922.78= 110.20

26Convexity & Factors affecting Interest RateConvexity calculation PeriodCFPV on YTMt(t+1)CF3x41 40.00 0.863837599 80.00 69.11 2 40.00 0.822702475 240.00 197.45 3 40.00 0.783526166 480.00 376.09 4 40.00 0.746215397 800.00 596.97 5 40.00 0.71068133 1,200.00 852.82 6 40.00 0.676839362 1,680.00 1,137.09 7 40.00 0.644608916 2,240.00 1,443.92 8 40.00 0.613913254 2,880.00 1,768.07 9 40.00 0.584679289 3,600.00 2,104.85 10 1,040.00 0.556837418 114,400.00 63,702.20 Total 72,248.57 27Convexity & Factors affecting Interest RateConvexity measures = conv cal / bond price /4= 72248.57 / 922.78 = 78.295 / 4

19.5728Convexity & Factors affecting Interest RateConvexity Adjustment=0.5*Convexity*Yield_Change^2*Bond_Price

0.5* 19.57* (0.03^2)*922.780.5* 19.57* 0.0009*922.788.13

29Convexity & Factors affecting Interest Rateif 3% change in yield means bond price will change by Duration adjustment is 110.20Convexity adjustment is 8.13Total adjustment on bond price if yield will increase by 3% is 110.20 8.13 = 102.07Total adjustment on bond price if yield will reduce by 3% is 110.20 + 8.13 = 118.33

30Convexity & Factors affecting Interest RateYieldDurationConvexityAdjustmentBond price13%110.208.13102.07820.717%110.208.13118.331041.11Question 2 Calculate convexity measure for 5 year bond face value Rs 100 coupon rate is 10% semi annually, YTM 12%.Calculate:DurationModified DurationConvexityAdjustment of Bond price if yield will change by 2% (+ & -)Cash flow = 5 for the period of 1031Convexity & Factors affecting Interest RatePeroidCFPV on YTMt(t+1)CF3x41 5.00 0.839619283 10.00 8.40 2 5.00 0.792093663 30.00 23.76 3 5.00 0.747258173 60.00 44.84 4 5.00 0.70496054 100.00 70.50 5 5.00 0.665057114 150.00 99.76 6 5.00 0.627412371 210.00 131.76 7 5.00 0.591898464 280.00 165.73 8 5.00 0.558394777 360.00 201.02 9 5.00 0.526787525 450.00 237.05 10 105.00 0.496969364 11,550.00 5,740.00 Total 6,722.81 32Convexity & Factors affecting Interest RateConvexity measures = conv cal / bond price /4= 6722.81 / 92.64 = 72.569 / 4=18.1433Convexity & Factors affecting Interest RateConvexity Adjustment=0.5*Convexity*Yield_Change^2*Bond_Price

0.5* 18.14* (0.02^2)*92.640.5* 18.14* 0.0004*92.64

It means if yield will change by 2% then the bond price will change by modified duration adjustment convexity adjustment

Yield change is the difference between the initial yield and current market yield.0.3434Convexity & Factors affecting Interest RateQuestion- Duration Calculate:Macaulay's durationModified durationConvexity Bond price with duration and convexity adjustment if yield will increase and decrease by 3%.

BondFace ValueCoupon rateYTMMaturity A100010%12%5 yrs B100010%9%7 yrs35Convexity & Factors affecting Interest RateBond AYearCash flowPV of cashflowDuration cal150 47.17 47.17 250 44.50 89.00 350 41.98 125.94 450 39.60 158.42 550 37.36 186.81 650 35.25 211.49 750 33.25 232.77 850 31.37 250.96 950 29.59 266.35 1050 586.31 5,863.15 TOTAL926.40 7,432.07 36Convexity & Factors affecting Interest RateMac duration = 4.01Modified duration = 3.78Duration adjustment @ 3% YTM changed= 3.78 x 0.03 x 926.40105.10

37Convexity & Factors affecting Interest RatePeriodCash flow (Rs) 1 (1.06)t+2t(t+1)CFt(t+1)CF * 1/ (1.045)t+2 1 50.00 0.83961928100 83.96 2 50.00 0.79209366300 237.63 3 50.00 0.74725817600 448.35 4 50.00 0.704960541000 704.96 5 50.00 0.665057111500 997.59 6 50.00 0.627412372100 1,317.57 7 50.00 0.591898462800 1,657.32 8 50.00 0.558394783600 2,010.22 9 50.00 0.526787534500 2,370.54 10 1,050.00 0.49696936115500 57,399.96 Total 67,228.10 38Convexity & Factors affecting Interest RateConvexity = (67228.10 / 926.40) /4=18.14Convexity adjustment= 0.5 x18.14 x (0.03)2 x 926.40=7.56

39Convexity & Factors affecting Interest RateyieldBond price @ 12%Duration adjustmentConvexity adjustmentTotal AdjustmentChanged BP9%926.40105.107.56112.661039.0615%926.40105.107.5697.54828.8640Convexity & Factors affecting Interest RateBond BMacaulay's Duration4.07Modified Duration3.90Duration Adjustment121.61Convexity19.06Convexity Adjustment8.92Bond price @ 12%1039.56 (121.61 -8.92) = 909.03Bond price @ 6%1039.56 + (121.61 +8.92) = 1170.09Rs 100010% SA9%7 yrs41Convexity & Factors affecting Interest Rate