Business F723

Fixed Income Analysis

Week 4

Measuring Price Risk

2

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

3

Time to Maturity EffectFor a 10% Coupon Bond

$500

$750

$1,000

$1,250

$1,500

$1,750

$2,000

0% 5% 10% 15% 20%

YTM

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1-year bond 10-year bond par

4

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

5

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%?

6

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.

7

Effect of YTM

YTM Price New Price % Decrease2% $1,541 $1,429 7.28%4% $1,327 $1,234 7.02%6% $1,149 $1,071 6.76%8% $1,000 $935 6.50%

10% $875 $821 6.24%12% $771 $725 5.98%14% $682 $643 5.72%16% $607 $574 5.45%

Effect of a 100 basis point increase on price

for 8% coupon, 10 year bond

8

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¢

10

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; -⅛ = $99.875– New YTM = 9.032%– Yield value of an 8th = 3.2 basis points

11

Macaulay’s 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

12

Macaulay’s 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.

13

Duration Calculation

• Two bonds;– YTM = 8%

– Maturity = 3 years

– Coupon rate• A = 6%

• B = 10%

– Face Value = $1,000

• Find the duration.

Period Payment PV % % x time1 $30 $28.85 3.04% 0.03042 $30 $27.74 2.93% 0.05853 $30 $26.67 2.81% 0.08444 $30 $25.64 2.71% 0.10835 $30 $24.66 2.60% 0.13016 $1,030 $814.02 85.91% 5.1543

$947.58 100.00% 5.5661duration = 2.7831

Period Payment PV % % x time1 $50 $48.08 4.57% 0.04572 $50 $46.23 4.39% 0.08793 $50 $44.45 4.22% 0.12674 $50 $42.74 4.06% 0.16245 $50 $41.10 3.90% 0.19526 $1,050 $829.83 78.85% 4.7310

$1,052.42 100.00% 5.349duration = 2.6745

14

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/(1+y) x Macaulay’s duration

• The negative of this is called Modified Duration

• Modified duration = Macaulay’s duration/(1+y)

• Often used to approximate percentage price changes

• Duration in years = D in six month periods/2• use 6 month rate for (1+y) in modified duration

16

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

18

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

• P% = -7.66 x 0.5% = -3.83%

19

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

20

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

21

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

22

ConvexityFor a 10% Coupon Bond

$500

$750

$1,000

$1,250

$1,500

$1,750

$2,000

0% 5% 10% 15% 20%

YTM

Pri

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1-year bond 10-year bond par

Tangent line for estimated price

23

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

24

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

Coupon rate 9% Period CF t(t+1)CF1/(1+y) (̂t+2) combinedMaturity 5 1 4.5 9.0 0.876297 7.9 YTM 9% 2 4.5 27.0 0.838561 22.6 Price 100 3 4.5 54.0 0.802451 43.3

4 4.5 90.0 0.767896 69.1 5 4.5 135.0 0.734828 99.2 6 4.5 189.0 0.703185 132.9 7 4.5 252.0 0.672904 169.6 8 4.5 324.0 0.643928 208.6 9 4.5 405.0 0.616199 249.6 10 104.5 11,495.0 0.589664 6,778.2

Second derivative = 7,781.0 Convexity measure (half years)= 77.81

Convexity measure (years)= 19.4526

27

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 0.022=3.66%

• Forecast change = -21.24 + 3.66 = -17.58%

• Actual change = -18.03%

28

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

Yield

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Bond A

Bond B

31

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

32

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

33

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

34

Effective Duration

• If a bond has embedded options, that will change the bond’s 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

35

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 bond’s price volatility with respect to changes in the required yield

36

Duration of Floaters

• A floating rate bond usually trades near par since the coupon rate adjusts to changes in interest rates

• Therefore a floater’s 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

37

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

yP

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P+= price if yield up

P0= original price

38

Approximating Convexity

• We can also approximate convexity using a similar method

20

02measureconvexity eapproximat

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P-= price if yield down

P+= price if yield up

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39

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