Business F723

1
Business F723

• Fixed Income Analysis
• Week 4
• Measuring Price Risk

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Basics of Price Risk

• As YTM changes, bond prices change
• Bond prices move in the opposite direction to the
  change in yield
• Not all bonds react the same amount to a given
  change in yield
• For large changes in yield, an increase has a
  higher change than a decrease

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Time to Maturity Effect
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Time to Maturity

• All else held constant the longer the time to
  maturity the larger the price volatility of a
  bond with respect to changing yields
• Intuition if I am paying a premium to lock in an
  above average current yield, I am willing to pay
  more to lock it in for a longer period of time

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Coupon Rate

• Consider two bonds, A and B
• 1,000 face value maturity 10 years
• YTM 9
• Coupon rate A 5 B 10
• Initial Prices PVcoupons PVface
• A 325 415 740
• B 650 415 1,065
• What change in price if YTM ? 8?

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Coupon Rate

• Price A, 8 340 456 796
• Change (796 - 740) / 740 7.6
• Price B, 8 680 456 1136
• Change (1,136 - 1,065) / 1,065 6.7
• In general, the larger the coupon payment, the
  less the change in price with a change in yield.

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Effect of YTM
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Level of YTM

• As the level of interest rates rise, the
  sensitivity of bond prices to changes in the
  yield falls
• Intuition a change from 2 to 2.1 is much more
  significant than a change from 16 to 16.1 as a
  fraction of total return

9
Price Value of a Basis Point

• One measure of price change is the dollar change
  in the price of a bond for a 1 basis point
  increase in the required yield
• Also known as dollar value of an 01
• Stated based on the pricing convention of quotes
  per 100 of face value
• p63 5 year 9 coupon par bond, 3.96

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Yield Value of a Price Change

• Pricing conventions used to quote prices in 32nds
  or 8ths of a point (fraction of a dollar per 100
  of face value)
• This measure converts the minimum price change
  into the effective change to YTM
• 5 year 9 par bond -? 99.875
• New YTM 9.032
• Yield value of an 8th 3.2 basis points

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Macaulays Duration

• First published in 1938
• A bond can be considered to be a package of zero
  coupon bonds
• By taking a weighted average of the maturity of
  those zero coupon bonds, you can approximate the
  price sensitivity of the portfolio that the bond
  represents

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Macaulays Duration

• The average time that you wait for each payment,
  weighted by the percentage of the price that each
  payment represents.
• Captures the effect of maturity, coupon rate and
  yield on interest rate risk.
• The higher the duration the greater the level of
  interest rate risk in an investment.

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Duration Calculation

• Two bonds
• YTM 8
• Maturity 3 years
• Coupon rate
• A 6
• B 10
• Face Value 1,000
• Find the duration.

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Price Elasticity

• Using calculus on the price equation

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Modified Duration

• From the last line of the previous equation, the
  right hand side is
• -1/(1y) x Macaulays duration
• The negative of this is called Modified Duration
• Modified duration Macaulays duration/(1y)
• Often used to approximate percentage price
  changes
• Duration in years D in six month periods/2
• use 6 month rate for (1y) in modified duration

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Alternate Method

• From the annuity formula for the price of a bond
  we can get a formula for modified duration
  instead of calculating weighted average (per 100
  of face value)

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Properties of Duration

• Increases with time to maturity
• Increases as coupon rate decreases to a maximum
  of time to maturity for a zero coupon bond
• Decreases as YTM increases due to face value
  having less weight in portfolio
• Modified Duration is similar, but lower max

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Approximate Price Change

• The change in price for a given change in yield
  can be calculated using modified duration (a.k.a.
  volatility)
• The approximate percentage change in price -
  modified duration x change in yield
• Given MD 7.66, calculate change in price for a
  50 basis point increase in yield
• DP -7.66 x 0.5 -3.83

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Dollar Price Change

• The approximate dollar price change is simply the
  approximate percent price change times the price
• Given the bond on the previous slide, if the
  initial price was 102.5 the decrease in value is
  3.83
• In dollar terms, 3.926

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How Close is This?

• For small yield changes, the approximation is
  reasonable, p. 70 example, for a 1 basis point
  increase on a 25 year 6 coupon bond with an
  initial yield of 9, the forecast change is
  -0.0747 actual is -0.0746
• For large changes it is not as good
• Reason duration is a linear approximation of the
  price/yield relationship

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Portfolio Duration

• Since duration is simply a weighted average of
  the time to the coupon payments and face value,
  portfolio duration is simply the weighted average
  of the durations of the individual bonds
• Portfolio managers look at the contribution to
  portfolio duration to assess their interest rate
  risk of a single bond issue

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Convexity
Tangent line for estimated price
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Convexity

• Due to the shape of the yield curve, the
  predicted price will always be lower than the
  actual price
• How close the approximation is depends on how
  convex the price/yield relationship is for a
  given bond

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Measuring Convexity

• Convexity is based on the rate of change of slope
  in the price/yield relationship
• That means that we need the second derivative of
  the price of a bond
• This is the dollar convexity

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Convexity Measure

• The convexity measure is the second derivative of
  the price divided by the price

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Convexity Example
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Price Change Example

• Given 25 year, 6 bond yielding 9
• Required yield increases to 11
• Mod. Duration 10.62
• change due to duration -10.62 x 2-21.24
• Convexity in years 178
• change due to convexity 1/2 x 178 x 0.0223.66
• Forecast change -21.24 3.66 -17.58
• Actual change -18.03

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Alternate Calculation

• We could also take the second derivative of the
  annuity based price formula
• Divide by price for convexity measure
• Divide by m2 to convert to years

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Note on Convexity

• Different writers compute the convexity measure
  differently
• One method moves the ½ into the measure

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Value of Convexity
Price
Bond A
Bond B
Yield
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Value of Convexity

• Two bonds offering the same duration and yield,
  but with different convexity
• Bond A will outperform bond B if the required
  yield changes
• Bond A should have a higher price
• Increase in value of A over B should be related
  to the volatility of interest rates

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Positive Convexity

• As required yields increase convexity will
  decrease
• As yields increase the slope of the tangent line
  will become flatter
• Implication
• as yield increases, prices fall and duration
  falls
• as yield decreases, prices rise and duration rises

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Properties of Convexity

• For a given yield and maturity, the lower the
  coupon rate, the higher the convexity
• For a given yield and modified duration, the
  higher the coupon rate, the higher the convexity
• Although coupon rate has an impact on the
  convexity it has a bigger impact on duration

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Effective Duration

• If a bond has embedded options, that will change
  the bonds price sensitivity to changes in
  required yields
• The value of a call option on the bond decreases
  as yields increase, and increases as yields
  decrease
• Effective duration can be calculated to account
  for the fact that expected cash flows may change
  in yields change

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Duration vs. Time

• With plain vanilla bonds, duration can be seen as
  a measure of time
• With more complex instruments, this link is
  broken
• Modified duration is a measure of the bonds
  price volatility with respect to changes in the
  required yield

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Duration of Floaters

• A floating rate bond usually trades near par
  since the coupon rate adjusts to changes in
  interest rates
• Therefore a floaters duration is near zero
• An inverse floater has a high duration (possibly
  greater than its maturity) since, when interest
  rates go up its coupon payments go down,
  exaggerating the impact of a change in yields
• A double floater could have a negative duration

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Approximating Duration

• Instead of using duration to approximate price
  changes, we can use price changes to approximate
  duration
• Potentially useful for complex instruments as a
  measure of price volatility

P- price if yield down P price if yield up P0
original price
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Approximating Convexity

• We can also approximate convexity using a similar
  method

P- price if yield down P price if yield up P0
original price
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Changing Yield Curve

• What happens if the shape of the yield curve
  changes?
• It is possible that prices on 30 year bonds could
  change while short term rates are stable
• Duration calculations can change to key rate
  durations, duration vectors, partial durations,
  etc.
• Key rate durations are illustrated in the text
  they are calculated using the approximation
  formula