59 Yield-Based Bond Duration Measures and Properties

• Interest rate risk, also known as price volatility, is greater for bonds with longer maturities and lower coupons

Bond Volatility

• A bond with a low coupon and long maturity is most likely to show the greatest volatility due to interest rate changes

Money Duration

• Money duration is expressed in currency units

Price Value of a Basis Point (PVBP)

• PVBP is equivalent to the initial price less the price if yield changed by 1 bps
• For an 18-year, 8% annual pay bond with a par value of $1,000 and a yield of 9%, the PVBP is approximately $0.82
• The calculation for PVBP = initial price - price if yield changed by 1 bps

Interest Rate Sensitivity

• A zero-coupon bond has the highest interest rate sensitivity among five-year bonds
• A zero-coupon bond's Macaulay duration equals its time to maturity

Modified Duration of a Bond

• A bond's price decreases while its yield increases when interest rates increase
• As the bond price decreases, the yield increases
• As the yield increases, price volatility (duration) falls for a 30-year bond selling at a discount

Approximate Modified Duration

• For a 30-year semi-annual coupon bond issued at par with market rates at 6.75%, a yield decline of 30 basis points increases the price to $1,039.59
• A yield increase of 30 basis points decreases the price to $962.77
• The approximate modified duration is 12.80

Duration and Yield Changes

• Duration alone underestimates the price increase when yield decreases and overestimates the price decrease when yield increases, for large changes in yield
• This is a result of duration's linear estimate not accounting for the convexity (curvature) in the price/yield relationship

Interest Rate Risk

• Holding all other factors constant, a coupon bond's interest rate risk is higher when its yield to maturity is lower
• Higher coupon rate and higher current yield result in lower interest rate risk

Money Duration Calculation

• Whittaker's $12 million position in Dewey bonds has an annual modified duration of 8.0
• The money duration of the position is $96.0 million
• Money duration is calculated as annual modified duration multiplied by portfolio value

Bond Duration

• The bond with the longest maturity and lowest coupon has the greatest duration, assuming bonds are otherwise identical
• Longer cash flow times increase duration

Interest Rate Sensitivity

• Given bonds identical except for term to maturity, the longer-term bond is more sensitive to changes in interest rates

Option-Free Bonds

• The 10-year 4% coupon bond has the highest interest rate sensitivity of the option-free bonds when comparing 10-year bonds to 5-year bond

Approximate Modified Duration

• A 25-year, $1,000 par semiannual-pay bond with a 7.5% coupon and a 9.25% YTM has an approximate modified duration of 10.03 based on a yield change of 50 basis points

Price Change Estimation

• A non-callable bond with 10 years to maturity, a 5.5% annual coupon, and a $1,000 par value has a YTM of 4.7%
• If rates rise by 75 basis points, the estimated price change using duration is approximately -$61.10

Price Value of a Basis Point

• The price value of a basis point (PVBP) for a 7-year, 10% semiannual pay bond with a par value of $1,000 and yield of 6% is closest to $0.64
• The calculation for PVBP = initial price - price if yield changed by 1 bps

Interest Rate Risk

• An increase in the maturity of a coupon bond will typically increase its interest rate risk

Coupon Rate

• A decrease in the coupon rate of a coupon bond will typically increase its interest rate risk

Bond Principal Payments

• Lower coupons cause greater relative weight to be placed on the principal repayment

Bond Maturity

• The probability that interest rates will change increases as the maturity of a bond increases

Bond Price Value

• The price value of a $1,000 par value, 6-year, 4.2% semiannual coupon bond is $958.97
• The bond's price value of a basis point is closest to $0.50

Approximate Modified Duration Calculation

• An option-free 20-year 7% annual-pay par bond has an approximate modified duration of 10.6, based on a 25 basis point change in yield

Approximate Modified Duration

• An annual-pay bond with a current price of 102.50 per 100 of face value drops to 100 if its YTM increases by 0.5% and rises to 105.5 if its YTM decreases by 0.5%
• The approximate modified duration of the bond is 5.37

Bond Price

• A bond's price will decrease by 4.21% for a 1% increase in yield to maturity, or increase by 4.45% for a 1% decrease in YTM
• If the bond is currently trading at par value, the bond's approximate modified duration is closest to 4.33

Modified Duration

• Modified duration measures the price sensitivity of a bond to changes in interest rates
• Duration = (V- – V+) / [2V0(change in required yield)] where: V- = estimated price if yield decreases by a given amount, V+ = estimated price if yield increases by a given amount, V0 = initial observed bond price

Money Duration

• A $100,000 par value bond with a full price of $99,300, a Macaulay duration of 6.5, and an annual modified duration of 6.1 possesses a money duration per $100 par value is closest to $606
• Money duration per $100 par value = annual modified duration × full price per $100 par value

Bond Durations

• When coupon rates increase bond duration decreases, because investors are receiving more cash flow sooner
• When maturities increase, duration also increases because the payment are spread out over a longer period of time

Interest Rate Risk

• The inclusion of a call feature is least likely to increase the interest rate risk of a bond

Bond Price Volatility

• Putable bonds will have less price volatility at higher yields

Yield Curve Shift

• A par value government bond maturing in five years is likely to experience the smallest percent change in price given an upward parallel shift in the yield curve

Call Features

• A call provision decreases the bond's duration
• The potential for price appreciation is reduced
• Investors are exposed to additional reinvestment rate risk

Callable Bonds

• Cash flows associated with callable bonds become unpredictable due to the call provision
• A call provision introduces prepayment risk that should be factored in the calculation

Largest Change In Value

• A zero-coupon bond will experience the largest change in value given an instantaneous parallel upward shift of 100 basis points

Zero-Coupon Bonds

• Because the only cash flow made is the principal payment at maturity, duration is equal to time to maturity
• Zero-coupon bonds possess the highest interest rate sensitivity

Modified Duration

• Modified duration is lower than Macaulay duration unless Yield to Maturity equals zero
• Modified duration = Macaulay duration / (1 + YTM)

Bond Duration

• A bond with a 10-year maturity and a 10% coupon rate has the shortest duration

Yield to Maturity

• A bond with a yield to maturity of 8.0% is priced at 96.00. If its yield increases to 8.3% its price will decrease to 94.06; if its yield decreases to 7.7% its price will increase to 98.47
• The modified duration of the bond is closest to 7.66
• Change in the yield = yield change is 30 basis points

Modified Duration of a Bond

• Given an option-free 5-year 6% annual-pay bond selling for $979.22 per $1,000 of par value and having a Macaulay duration of 4.4587, the modified duration is closest to 4.187
• Yield To Maturity = 6.5%
• Modified duration = Macaulay duration / (1 + YTM)

Coupon Rates

• Bonds with higher coupons have lower interest rate risk