Bond Theory

Bonds_Pricing PRICE GIVEN YTM FACE VALUE REDEMTION VALUE YEARS TO MATURITY SETTLEMENT MATURITY COUPON RATE MARKET YIELD (YTM) FREQUENCY CURRENT PRICE

COUPON BOND ZERO COUPON BOND 100 100 100 100 7 7 1-Jan-96 1-Jan-96 31-Dec-02 31-Dec-02 8.00% 0 12.00% 12.00% 2 2 81.41

44.24

YTM GIVEN PRICE FACE VALUE REDEMTION VALUE YEARS TO MATURITY SETTLEMENT MATURITY COUPON RATE CURRENT PRICE FREQUENCY YTM CURRENT YIELD

COUPON BOND ZERO COUPON BOND 100 100 100 100 7 7 1-Jan-96 1-Jan-96 31-Dec-02 31-Dec-02 8.00% 0 81.41 44.24 2 2 12.00% 9.83%

12.00% 0

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Bond_Principles 1.THE LEVEL OF YTM LOWER HIGHER FACE VALUE 100 100 100 100 REDEMTION VALUE 100 100 100 100 YEARS TO MATURITY 7 7 7 7 SETTLEMENT 1-Jan-96 1-Jan-96 1-Jan-96 1-Jan-96 MATURITY 31-Dec-02 31-Dec-02 31-Dec-02 31-Dec-02 COUPON RATE 8.00% 8.00% 8.00% 8.00% MARKET YIELD (YTM) 8.25% 8.00% 11.00% 10.75% FREQUENCY 2 2 2 2 CURRENT PRICE % CHANGE IN PRICE

98.69

100.00 1.33%

85.62 1.28%

Market Yield (YTM) cell value has been changed

86.71

LOWER THE INITIAL YTM, HIGHER THE PRICE CHANGE FOR A CHANGE IN YTM 2. THE DIRECTION OF CHANGE IN YTM INCREASE IN YTM DECREASE IN YTM FACE VALUE 100 100 100 100 REDEMTION VALUE 100 100 100 100 YEARS TO MATURITY 7 7 7 7 SETTLEMENT 1-Jan-96 1-Jan-96 1-Jan-96 1-Jan-96 MATURITY 31-Dec-02 31-Dec-02 31-Dec-02 31-Dec-02 COUPON RATE 8.00% 8.00% 8.00% 8.00% MARKET YIELD (YTM) 8.25% 8.50% 8.25% 8.00% FREQUENCY 2 2 2 2 CURRENT PRICE % CHANGE IN PRICE

98.69

97.40 -1.30%

98.69

Market Yield (YTM) cell value has been changed

100.00 1.33%

PRICE MOVEMENTS RESULTING FROM EQUAL ABSOLUTE INCREASES AND DECREASES IN YIELDS ARE ASYMMETRIC. A DECREASE IN YIELD RAISES BOND PRICES MORE THAN INCREASES IN BOND YIELD DECREASES PRICES 3. THE COUPON RATE SMALLER LARGER FACE VALUE 100 100 100 100 REDEMTION VALUE 100 100 100 100 YEARS TO MATURITY 7 7 7 7 SETTLEMENT 1-Jan-96 1-Jan-96 1-Jan-96 1-Jan-96 MATURITY 31-Dec-02 31-Dec-02 31-Dec-02 31-Dec-02 COUPON RATE 8.00% 8.00% 9.00% 9.00% MARKET YIELD (YTM) 8.25% 8.00% 8.25% 8.00% FREQUENCY 2 2 2 2 CURRENT PRICE % CHANGE IN PRICE

98.69

100.00 1.33%

103.93 1.30%

Market Yield (YTM) cell value has been changed

105.28

PERCENTAGE CHANGE IN PRICE IS LARGER FOR BONDS WITH SMALLER COUPON RATE FOR ANY GIVEN CHANGE IN YTM 4. THE DISCOUNT OR PREMIUM FROM FACE VALUE PREMIUM DISCOUNT FACE VALUE 100 100 100 100 REDEMTION VALUE 100 100 100 100 YEARS TO MATURITY 7 7 7 7 SETTLEMENT 1-Jan-96 1-Jan-96 1-Jan-96 1-Jan-96 MATURITY 31-Dec-02 31-Dec-02 31-Dec-02 31-Dec-02 COUPON RATE 8.00% 8.00% 7.00% 7.00% MARKET YIELD (YTM) 7.75% 8.00% 7.75% 8.00% FREQUENCY 2 2 2 2 CURRENT PRICE % CHANGE IN PRICE

101.33

100.00 -1.31%

96.01 -1.34%

Market Yield (YTM) cell value has been changed

94.72

BONDS SELLING AT A DISCOUNT ARE MORE SENSITIVE TO CHANGES IN MARKET YIELDS, ALL OTHER FACTORS REMAINING EQUAL, COMPARED TO BONDS SELLING AT OR ABOVE PAR. DEEPER THE DISCOUNT, GREATER THE SENSITIVITY OF A BOND'S PRICE TO A CHANGE IN YIELDS 5. THE MATURITY OF THE BOND LONGER FACE VALUE 100 100 REDEMTION VALUE 100 100 YEARS TO MATURITY 7 7 SETTLEMENT 1-Jan-96 1-Jan-96

SHORTER 100 100 100 100 5 5 1-Jan-96 1-Jan-96

Market Yield (YTM) cell value has been changed

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Bond_Principles Market Yield (YTM) cell value has been changed MATURITY 31-Dec-02 31-Dec-02 31-Dec-00 31-Dec-00 COUPON RATE 8.00% 8.00% 8.00% 8.00% MARKET YIELD (YTM) 7.75% 8.00% 7.75% 8.00% FREQUENCY 2 2 2 2 CURRENT PRICE % CHANGE IN PRICE

101.33

100.00 -1.31%

101.02 -1.01%

100.00

LONGER THE REMAINING PERIOD TO MATURITY, HIGHER WILL BE THE VOLATILITY IN PRICE FOR A GIVEN CHANGE IN YIELDS

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Duration Duration measure of the waiting period (average) for the holder to receive cash payment The duration of a zero maturing in n years = n The duration of a couppn bond maturing in n years < n Bond price (P) calculation P = Sum (C*E -yt) - where cash flows C are discounted at their resp. interest (y) and year fractions (t) - and E = euler constant The Duration (D) is defined D = [Sum (T*C*E -yt)]/P - where t = year fractions, P = bond price - the present values of cash flows divided by bond price gives the prorportional flows - duration is therefore the weighted time average of all cash flows (proportional) - the weighting of timing of flows is done by the cash flows - the sum of wieghts is 1.0 Relationship between bond price and duration dP/dy = -PD for small shifts in interest rates delta P = -P*D* deltay or in % terms deltaP/P =D* deltay Duration (D) assumes that y is expressed with continous compounding if y is expressed in a frequency of m times a year Modified Duration = D/(1+y/m) Convexity For small changes in interest rates change in value est by duration For larger changes the linear relation between % change and yield changes breaks down The risk is substantial over or underestimates of value changes The convexity is the 2nd difference which changes specific to a bond/portfolio The convexity is the non-linear move in change in value More than one estimation is available for convexity

(delta P)/P

porfolio

1

deltay

portfolio 2

One of them based on Taylor series expansion of portfolio value changes is C = [Sum (T2*C*E -yt)]/P - where t = year fractions, P = bond price - the present values of cash flows divided by bond price gives the prorportional flows - duration is therefore the weighted time average of all cash flows (proportional) - the weighting of timing of flows is done by the cash flows - the sum of wieghts is 1.0 - the convexity C has to be adjusted for the frequency of cash flows - deltaP/P =D* deltay + 0.5 C* deltay^2 - the second part of the equation is the convexity adjustment factor Non parallel shifts / Curve rotation - calculate the bond duration - based on bond duration map into different duration buckets - sum all bonds in each duration bucket - use above formula to estimate the change in the value of the bucket - each bucket can use a different shift hence curve rotation possible Problems with Duration and Convexity the ytm problem - two bonds with different cash flow pattern have the same ytm - reinvestment risk not factored in - cash flow mapping as a way out of ytm problem and curve rotation

Dur_MDur DURATION AND MDURATION EXAMPLES 1. YIELD CHANGE BY BASIS POINTS 10 Duration calculation uses proportion I.e., 1% =0.01 COUPON ZERO COUPON ZERO FACE VALUE 100 100 100 100 COUPON 6% 0% 6% 0% TERM 5 5 5 5 SETTLMNT 1-Jan-90 1-Jan-90 1-Jan-90 1-Jan-90 MATURITY 31-Dec-94 31-Dec-94 31-Dec-94 31-Dec-94 YIELD 9% 9% 9.10% 9.10% FREQUENCY 2 2 2 2 BASIS 1 1 1 1 PRICE 88.14 64.41 87.77 64.10 DURATION 4.34 5.00 MDURATION 4.16 4.78 R_PRICE CH (10bp) 0.366 0.308 price*duration*1/1000….for 10 bp R_PRICE CH -0.366 -0.308 price*duration*bp/10000 SUMMARY & COMPARISON OF DURATION vs. ACTUAL % CHANGE IN PRICE ACTUAL PREDICTED BY MDURATION RUPEE PRICE CHANGE ACTUAL PREDICTED BY R_PRICE CH

-0.41% -0.42% -0.37 -0.37

-0.48% difference in prices (%) -0.48% -duration*(diff in yields) -0.31 difference in prices (Rs.) -0.31 price*duration*bp/10000

2. DURATION AS TIME ELAPSES FACE VALUE COUPON TERM SETTLMNT MATURITY YIELD FREQUENCY BASIS PRICE DURATION MDURATION

100 100 12% 12% 100 10 1/1/1950 1/1/2040 12/31/2049 12/31/2049 0.11 0.11 1 1 1 1 109.09 105.89 10.09 6.43 9.09 5.79

3. DURATION OF A PORTFOLIO OF BONDS BOND A BOND B BOND C FACE VALUE 100 100 100 COUPON 12% 10% 11% TERM 100 10 10 SETTLMNT 1-Jan-50 1-Jan-40 1-Jan-40 MATURITY 31-Dec-49 31-Dec-49 31-Dec-49 YIELD 11.00% 11.00% 11.00% FREQUENCY 1 1 1 BASIS 1 1 1 PRICE 109.09 94.11 100.00 DURATION 10.09 6.65 6.53 MDURATION 9.09 5.99 5.88 DUR_PORT 7.85 = (Price of bond1/sum of prices of all bonds)* Duration of bond 1 +… +… MDUR_PORT 7.07 = (Price of bond1/sum of prices of all bonds)* M_Duration of bond 1 +… +…

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Yield_Change BASE FACE VALUE COUPON SETTLMNT MATURITY REDEMPTION FREQUENCY BASIS CUR.YLD PRICE % CHANGE

1% INC 1% DEC 2% INC 2% DEC 1000 1000 1000 1000 1000 11% 11% 11% 11% 11% 1-Jan-90 1-Jan-90 1-Jan-90 1-Jan-90 1-Jan-90 31-Dec-02 31-Dec-02 31-Dec-02 31-Dec-02 31-Dec-02 1000 1000 1000 1000 1000 1 1 1 1 1 1 1 1 1 1 9% 10% 8% 11% 7% 1149.71 1071.01 1237.07 1000.00 1334.25 -6.84% 7.60% -13.02% 16.05%

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b1

0.07

Page 7

b2

0.08

Page 8

b3

0.09

Page 9

b4

0.1

Page 10

b5

0.11

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Scenario Summary Current Values:

b1

b2

Changing Cells: $B$9 9% 7% 8% Result Cells: $B$10 1149.71 1334.25 1237.07 Notes: Current Values column represents values of changing cells at time Scenario Summary Report was created. Changing cells for each scenario are highlighted in gray.

b3

b4

9%

10%

1149.71

1071.01

b5

11% 999.98

Dur_Convexity DURATION AND CONVEXITY OF BONDS - EXAMPLE YIELD CHANGE BY BASIS POINTS 100 COUPON ZERO COUPON ZERO FACE VALUE 100 100 100 100 COUPON 3.66% 0.00% 3.66% 0.00% TERM 10 8 10 8 SETTLMNT 1-Jan-90 1-Jan-90 1-Jan-90 1-Jan-90 MATURITY 31-Dec-99 31-Dec-97 31-Dec-99 31-Dec-97 YIELD 9.00% 9.00% 10.00% 10.00% FREQUENCY 2 2 2 2 BASIS 1 1 1 1 PRICE 65.30 49.46 60.52 45.82 DURATION 8.00 8.00 MDURATION 7.66 7.65 CONVEXITY 71.35 62.26 COUPON ZERO % CHANGE IN PRICE ACTUAL PREDICTED BY MDURATION DIFFERENCE

-7.31% -7.66% 0.34%

-7.35% -7.65% 0.30%

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Convexity_Calc COMPUTATION AND ADJUSTMENT FOR CONVEXITY YIELD CHANGE BY BASIS POINTS -300 COUPON 8% 8% FACE VALUE 100 100 REMEMPTION 100 100 YIELD 10% 7.00% SETTLMNT 1-Jan-90 1-Jan-90 MATURITY 31-Dec-94 31-Dec-94 FREQUENCY 2 2 BASIS 1 1 PRICE 92.28 104.16 DURATION 4.18 MDURATION #ADDIN? 1. ITERATIVE FORMULA period (t) cashflow pvcf 1 4 2 4 3 4 4 4 5 4 6 4 7 4 8 4 9 4 10 104

% CHANGE IN PRICE ACTUAL 12.87% PREDICTED BY DURATION #ADDIN? ADJUSTMENT FOR CONVEXITY 0.88% CONVEXITY ADJUSTED PRICE CHANGE #ADDIN? CONVEXITY ADJUSTED PRICE CHANGE IS CLOSER TO ACTUAL PRICE CHANGE

t*(t+1) 3.81 3.63 3.46 3.29 3.13 2.98 2.84 2.71 2.58 63.85 92.28

PREDICTED BY DURATION CONVEXITY IN HALF YRS CONVEXITY IN YEARS ADJUSTMENT FOR CONVEXITY CONVEXITY ADJUSTED PRICE CHANGE

2 6 12 20 30 42 56 72 90 110

pvcf*t(t+1) 7.62 21.77 41.46 65.82 94.02 125.36 159.19 194.93 232.06 7023.17 7965.4 #ADDIN? =M_Duration * (bp difference)=M_Duration% 78.29 =pvcf*t*(t+1)/{[(1*ytm/f)^f]*pvcf} 19.57 =convHY/(f*f) 0.88% =0.5*convY/((bp/10000)*(bp/10000)) #ADDIN? =M_Duration% + Convexity adj

2. TALYOR SERIES EXPANSION EXPANSION period (t) cashflow pvcf pvcf % t2 =t2*pvcf% 1 4 3.81 1 0.04 0.04 2 4 3.63 4 0.04 0.16 3 4 3.46 9 0.04 0.34 4 4 3.29 16 0.04 0.57 5 4 3.13 25 0.03 0.85 6 4 2.98 36 0.03 1.16 7 4 2.84 49 0.03 1.51 8 4 2.71 64 0.03 1.88 9 4 2.58 81 0.03 2.26 10 104 63.85 100 0.69 69.19 92.28 77.96

PREDICTED BY DURATION CONVEXITY IN HALF YRS CONVEXITY IN YEARS ADJUSTMENT FOR CONVEXITY CONVEXITY ADJUSTED PRICE CHANGE

#ADDIN? =M_Duration * (bp difference)=M_Duration% 77.96 =sum(pvcf%*t2) 19.49 =convHY/(f*f) 0.88% =0.5*convY*(delta_Y2) #ADDIN? =M_Duration% + Convexity adj

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Portfolio Analysis 1. Individual Bond Duration change in Price = Price of bond * Duration * Interest Rate change(Prop) change in Price% = Duration * Interest Rate change (Prop) summary and comparison of actual and predicted price changes -% change in price actual -0.41% -0.48% difference in prices (%) predicted by R_PRICE change -0.42% -0.48% -duration*(diff in yields) -rupee price change actual -0.37 -0.31 difference in prices (Rs.) predicted by R_PRICE change -0.37 -0.31 price*duration*bp/10000 2. Portfolio Sensitivity based on Duration DURATION TOTAL (I%) D*% BOND 1 4.23 10% BOND 2 7.11 13% BOND 3 5.67 12% BOND 4 8.22 10% BOND 5 7.97 10% BOND 6 2.11 8% BOND 7 6.47 7% BOND 8 8.42 2% BOND 9 3.34 12% BOND 10 6.37 16% Portfolio Duration

0.42 0.92 0.68 0.82 0.8 0.17 0.45 0.17 0.4 1.02 5.86

predicted by R_PRICE change 3. Porfolio Sensitivity considereing Convexity and Curve Rotation - use convexity adjustment for individual bonds - map the bond value to the duration bucket - calculate the bucket sensitivity

FlRate_Bonds BASIC DATA TYPE OF BOND FRB FV/RV 100 MATURITY (YRS) 6 COUPON REF+80BPS MARGIN 80 RESET EVERY 6 MONTHS CURRENT REF.RATE 10% COUPON FREQ. 2 CURRENT COUPON 10.80% MARKET PRICE 99.3098 PERIOD 1 2 3 4 5 6 7 8 9 10 11 12

REFERENCE RATE CFLOW 10% 5.4 10% 5.4 10% 5.4 10% 5.4 10% 5.4 10% 5.4 10% 5.4 10% 5.4 10% 5.4 10% 5.4 10% 5.4 10% 105.4 PRESENT VALUE

PRESENT VALUE OF CASH FLOW ASSUMED ANNUAL YIELD SPREAD 80 84 88 96 100 10.80% 10.84% 10.88% 10.96% 11.00% 5.12 5.12 5.12 5.12 5.12 4.86 4.86 4.86 4.85 4.85 4.61 4.61 4.61 4.6 4.6 4.38 4.37 4.37 4.36 4.36 4.15 4.15 4.14 4.14 4.13 3.94 3.93 3.93 3.92 3.92 3.74 3.73 3.73 3.72 3.71 3.55 3.54 3.53 3.52 3.52 3.36 3.36 3.35 3.34 3.34 3.19 3.19 3.18 3.17 3.16 3.03 3.02 3.02 3 3 56.07 55.95 55.82 55.56 55.44 100 99.83 99.65 99.31 99.14

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COUPON AT ISSUE MATURITY COUPON RESET & PAYMENT DATES RESET SPREAD BASE RATE PRICE TODAY'S BASE RATE RESET COUPON TIME REMAINING TO MATURITY (ASSUMING TODAY IS 1-SEP-95) SIMPLE CURRENT YIELD RESET CURRENT YIELD ADJUSTED SPREAD TO BASE ZERO COUPON BASIS SPREAD RESET OR ADJUSTED YTM YTM OVER BASE RATE SETTLMNT MATURITY COUPON PRICE RV FREQ YTM

5% 1-Sep-99 MAR1, SEP1 100 6 MONTH USTB 99 5.50% 6.50% 4 YEARS

5.05% 6.57% 1.07% 1.250% 6.79% 1.29% 1/1/1997 12/31/2000 6.50% 99 100 2 6.79%

-99 0 0 0 100 0.25%