Bond Price Sensitivity

1
Bond Price Sensitivity

• September 25, 2001

2
Price-Yield Curve

At low yields, prices rise at an increasing rate
as yields fall
price
At higher yields, prices fall at a decreasing
rate as yields rise
yield
3
Bond Price Volatility

• (1) Price and yield goes in opposite directions
• (2) For small (/-) yield changes, the percentage
  price (/-) change for a bond are nearly equal
• The curvature of the line doesnt have much
  impact
• (3) For large (/-) yield changes the percentage
  (/-) changes for a specific bond are not equal
• The curvature of the line begins to matter
• (4) For a given yield change, the percentage
  price increase when rates fall is greater than
  the percentage price decrease when rates rise,
  i.e. convexity
• Why do we care about price volatility?
• Want to know the risk exposure, and to hedge it

4
Bond Price Volatility

• (5) For a given term to maturity and initial
  yield, the price volatility is greater, the lower
  the coupon rate.
• (6) For a given coupon rate and initial yield,
  the longer the term to maturity, the greater the
  price volatility.
• (7) For a given change in yield, price volatility
  is greater when yield levels in the market are
  low.
• What should an investor do if he thinks that the
  interest rate will fall soon?
• Should buy bonds with longer maturity and lower
  coupon, other things equal.
• What should a bond portfolio manager do if he is
  worried that the interest rate is rising due to
  inflation pressure?
• He should switch his portfolio to bonds with
  shorter maturity.

5
Measures of Price Volatility

• Properties (5) and (6) show that maturity and
  coupon rate influence bond prices simultaneously.
• How then can we compare bonds having different
  maturities and coupon rates?
• E.g., how do you choose between a 20 year bond
  with 8 coupon and a 15 year bond with 12
  coupon?
• Duration a single measure of volatility
  combining both the effects of coupon and maturity.

6
Price Value of a Basis Point

• The change in price if the required yield changes
  by 1 basis point.
• Indicates dollar price volatility, not percentage
  price volatility.
• Is usually expressed as an absolute value.
• Is it the same for a yield increase as for a
  yield decrease?

7
Yield Value of a Price Change

• What is the change in yield corresponding to a
  specified price change?
• Treasury notes and bonds yield value of a 32nd.
• Corporate and municipal bonds yield value of an
  8th.

8
Macaulays Duration

• Weighted average of the time to receipt of cash
  flows.

9
Duration

• What is the duration of a 2 year bond paying 10
  annually when the market interest rate is 12,
  semi-annually compounded?

10
Duration

• Modified duration duration /(1Y)
• Dollar duration durationP
• Modified duration tells you what the proportional
  price change will be for a small change in yield.

11
Duration

• What is the modified duration of a 5-year zero
  coupon note with face value 1,000,000? The
  semi-annual yield is Yield/2y0.03.
• The current price is 1,000,000/(10.03)10
  744,094. Suppose right after buying the 5-year
  zero, its yield goes up from 6 to 6.01 (1 basis
  point), i.e. y0.0601/20.03005.
• new price 1M/(10.03005)10 743733
• old price 744094
• proportional change in price -361/744094
  -0.000485 -9.709 0.00005

12
Duration

• So how do we interpret MD 4?
• for 1 basis point change in the yield, the
  proportional price change is 4 basis points.
• The approximation will get worse the larger the
  yield change.
• Example This time yield goes from 0.06 to 0.08.
• new price 1,000,000/(10.04)10 675564
• old price 744094
• proportional price change -68530/744094
  -0.092
• approx by MD -9.7090.01 -0.097
• approximation error 0.005 (or 3712)

13
Convexity
price
Error in estimating price based only on duration
Error in estimating price based only on duration
y
yield
14
Summarize Formulae

• What we have so far

15
Price Approximation, again

• The same example as before, except now the yield
  jumps from 0.06 to 0.08
• new price 675564
• old price 744094
• change in price -68530/744094-0.0921

16
Price Approximation, again (cont)

• Approximate using duration and convexity
• -0.09710.010.5103.69(0.01)2
  -0.0921
• -MD0.010.5convexity (0.01)2
• approximation error lt 1 basis point !

17
Computations for Coupon Bonds

• The exercise for coupon bonds is more complicated
  than for zero coupon bonds, but the formulae stay
  the same. The following are simplified formulae

18
Example

• A 10 1/8, 8 year coupon note, with a face value
  of 1,000,000, is priced to yield 12.5.
• What is the bond price?
• The duration is 91,316,841 and the modified
  duration is 10.353. The convexity is 1.269
  109 and the convexity is 143.866. Suppose the
  bonds yield immediately falls to 11.5.
• Use the duration and convexity information to
  approximate the bond price change as a result of
  the change in yield.

19
Some Properties of Duration

• Duration of a zero-coupon bond equals maturity.
• Holding time to maturity constant, duration is
  higher when coupons are lower.
• Holding coupon constant, duration increases with
  time to maturity.
• Holding other factors constant, duration is
  higher when yield to maturity is lower.
• Bottom line we have found a single measure that
  can summarize the bond price volatility.
• But as we have shown, duration can be very useful
  for estimating future price changes.

20
Duration of a Portfolio

• The duration of a portfolio is simply the
  weighted average of the duration of the bonds in
  the portfolios.
• Example A portfolio manager holds the following
  bonds
• Bond market value duration
• A 10 million 4
• B 40 million 7
• C 30 million 6
• D 20 million 2
• What is the portfolios duration?
• Total value 10403020100
• (10/100)4 (40/100)7 (30/100)6 (20/100)2
  5.4

21
Value of Convexity

• Consider Bond A and B, where B has greater
  convexity. In general you would like to hold
  bonds with greater convexity, other things equal.
  Why?
• If you hold bond B, if rates fall, your gain will
  be accelerated and if rates rise your loss will
  be slowed.
• This will be priced into the securities.
• If investors expect increased volatility, higher
  convexity bonds will increase in price relative
  to low convexity bonds.

22
Convexity
price

• Bond A has less convexity than B. Which bond
  would you short to hedge your current portfolio?
• B is always preferable to A for bond holder. So
  A is preferable if we are shorting.
• Implication hedge using instruments of lower
  convexity than your target bond.

B
A
yield
23
Properties of Convexity

• As yields increase, convexity decreases.
• Examine the price-yield curve - as yields
  increase, the P-Y curve flattens out.
• Conclude convexity is more important at higher
  interest rates.
• It is a safety factor
• Because bond price falls at a decreasing rate,
  but also increases at a decreasing rate!

24
Properties of Convexity (cont)

• For a given yield and maturity, the lower the
  coupon rate, the greater the convexity.
• So zeroes have greater convexity than coupon
  bonds of the same maturity.
• For a given yield and modified duration, the
  lower the coupon rate, the smaller the convexity.
• So zeroes have smaller convexity than coupon
  bonds of the same duration.