BOND PRICE VOLATILITY

PRICE YIELD PRICE YIELD RELATIONSHIP CONVEX SHAPE

WHAT IS VOLATILITY ? Volatility, a statistic similar to standard deviation, measures the uncertainty of the annualised underlying asset return. More precisely, volatility is the annualized standard deviation of the natural logarithm of the underlying asset return.

PROPERTY 1 : THE PERCENTAGE CHANGE IN THE PRICE OF THE BOND IS NOT THE SAME FOR ALL BONDS (NOT LINEAR) PROPERTY 2 : FOR A VERY SMALL CHANGE IN THE YIELD, THE PERCENTAGE PRICE CHANGE OF THE BOND IS ROUGHLY THE SAME. PROPERTY 3: FOR A LARGE CHANGE IN THE YIELD, THE PERCENTAGE PRICE CHANGE IS NOT THE SAME FOR AN INCREASE AS IT IS FOR A DECREASE. (Handout) PROPERTY 4: FOR A GIVEN LARGE CHANGE IN BASIS POINTS, THE PERCENTAGE INCREASE IN PRICE IS GREATER THAN THE PERCENTAGE DECREASE IN PRICE.

COMPONENTS OF A BOND THAT AFFECTS ITS VOLATILITY COUPON RATE TERM TO MATURITY

MEASURES OF BOND PRICE VOLATILITY INTEREST RATE SENSITIVITY OF A BOND

MONEY MANAGERS, ARBITRAGEURS AND TRADERS NEED TO HAVE A WAY TO MEASURE A BOND’S PRICE VOLATILITY TO IMPLEMENT HEDGING AND TRADING STRATEGIES. PRICE VALUE OF A BASIS POINT YIELD VALUE OF A PRICE CHANGE DURATION 3 techniques

PRICE VALUE OF BASIS POINT CHANGE IN PRICE OF THE BOND IF YIELD  BY 1BP (DOLLAR PRICE CHANGE NOT %) FROM THE HANDOUT (#3), YOU CAN NOTICE THERE IS NO GREAT CHANGE FOR ANY BOND WITH SUCH AN INCREMENTAL MOVE IN RATES. (1BP = 0.01 %) P63 OF OBLI

YIELD VALUE OF A PRICE CHANGE CALCULATE THE YTM OF THE BOND IF THE BOND DECREASES BY X DOLLARS. YIELD VALUE = NEW YIELD - THE OLD YIELD YIELD VALUE OF THE PRICE CHANGE

STOCKS BETA BONDSDURATION OPTIONS DELTA Sensitivity analysis

DURATION DURATION IS A MEASURE OF SENSITIVITY OF A BOND’S MARKET PRICE TAKING INTO CONSIDERATION ITS COUPON AND TERM TO MATURITY. (A ZERO-COUPON BOND THAT MATURES IN n YEARS HAS A DURATION OF n YEARS ) MACAULEY DURATION MODIFIED DURATION

Σ WEIGHTED PRESENT VALUE OF CASH FLOWS DURATION mac = Σ PRESENT VALUE OF CASH FLOWS MACAULEY DURATION Bond Price

IIIII 10 coupon + principal Consider this 7-year bond 10% coupon priced at 95 with a YTM of 11.06% I 10 Approx. (10+(5/7)/ year 110 Σ WEIGHTED PRESENT VALUE OF CASH FLOWS DURATION = Σ PRESENT VALUE OF CASH FLOWS (9x1) + (8.11x2) + (7.30x3)…………….(52.77x7) DURATION = Macc. Duration = 5.31

What is the Macauley duration of a 20 year zero coupon bond ? 20 years !!

MODIFIED DURATION = Sensitivity MACAULEY DURATION 1 + y Y = required yield APPROXIMATE PERCENTAGE CHANGE IN PRICE FOR A GIVEN CHANGE IN YIELD

SENSITIVITY Mc CauleyDURATION SENSITIVITY = y See page 78 for an approximate calculation of duration

Duration S = ( 1 + y ) Measure of Sensitivity (modified duration) For every « i » increase in rate, the sensitivity of the bond will decrease by S

Consider our 7-year bond 10% coupon priced at 95 with a YTM of Macc. Duration = % Modified duration or Duration = 5.31 / ( ) = 4.78 For each 100BP change in rates, the bond will vary by 4.78%

WHAT IS THE Modified DURATION OF A ZERO COUPON BOND ? ITS MATURITY y

Duration of a Bond Portfolio BONDMKT VALUEWEIGHTMODIFIED DURATION A$10 mil B$40 mil.0.47 C$30 mil.0.36 D$20 mil.0.22 total100 mil.1 Portfolio duration = 0.10 x x x x 2 = 5.4

BONDMKT VALUEWEIGHTMODIFIED DURATION A$10 mil B$40 mil.0.47 C$30 mil.0.36 D$20 mil.0.22 total100 mil.1 What if rates increase by 50BP? Portfolio decreases by 0.5 x 5.40 = 2.70% using duration 5.40

CONVEXITY PRICE YIELD CONVEX SHAPE DURATION (linear) Y’ P’ P’’ D+C Y’’ X’ X’’ D+C P’’ D X’’ D

CONVEXITY Cont’d Convexity is a measure of the curvature of the price/yield relationship. Mathematically, convexity is the second derivative of price with respect to yield divided by price. (duration is first)

Consider our 7-year bond 10% coupon priced at 95 with a YTM of 11.06% Modified duration or Duration = 5.31 / ( ) = 4.78 Its convexity is at Using duration and convexity by what % would this bond change by If rates decreased by 200BP? 4.78 x 2 + (½ (31.08) (0.02) 2) x 100) = 10.18%

APPROXIMATING PERCENTAGE PRICE CHANGE USING DUARTION AND CONVEXITY Consider a 25-year 6% bond selling to yield 9%. The modified duration for this bond is and its convexity 183 What is the approximate percentage price change if yield rise by 200 basis points ? Duration Down x 2 = % Convexity (½ (convexity)(  r) 2 ) x 100)= +3.66% Estimated % price change due to duration and convexity = %

You always ADD convexity to duration, never subtract it. Consider a 25-year 6% bond selling to yield 9%. The modified duration for this bond is and its convexity 183 What is the approximate percentage price change if yield decrease by 200 basis points ? Duration up x 2 = % Convexity ½ (convexity)(  r) 2 = +3.66% Estimated % price change due to duration and convexity = %

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