Fixed Income Analysis Week 4 Measuring Price Risk Com 4FJ3 Fixed Income Analysis Week 4 Measuring Price Risk

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

Time to Maturity Effect

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

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%?

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.

Effect of YTM

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

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¢

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

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

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.

Duration Calculation Two bonds; Find the duration. YTM = 8% Maturity = 3 years Coupon rate A = 6% B = 10% Face Value = $1,000 Find the duration.

Price Elasticity Using calculus on the price equation

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

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)

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

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%

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

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

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

Convexity Tangent line for estimated price

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

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

Convexity Measure The convexity measure is the second derivative of the price divided by the price

Convexity Example

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%

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

Note on Convexity Different writers compute the convexity measure differently One method moves the ½ into the measure

Value of Convexity Price Bond A Bond B Yield

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

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

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

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

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

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

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

Approximating Convexity We can also approximate convexity using a similar method P-= price if yield down P+= price if yield up P0= original price

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