(2) Bond Price Volatility

When is a portfolio said to be duration neutral?

A portfolio is duration natural when the duration for both the portfolio and index is approximately they same.

How does the convex relationship of price and yield impact volatility? What are the four price volatility characteristics of option-free bonds?

As the required yield rises, the price of the option-free bond declines. The convex shapes drives the the four price volatility characteristics of bonds. (1) Prices of all option-free bonds move in the opposite direction of change in required yield, but the percentage price change is not the same for all bonds. (2) For small yield changes, the percentage price change for a given bond is roughly the same, whether the required yield increases or decreases. (3) For large changes in the yield, the percentage price change is not the same for an increase as it is for a decrease in the yield. (4) For a given large change in basis points, the percentage price increase is greater than the percentage price decrease.

Define convexity and explain how modified duration and convexity are used to approximate the bond's percentage change in price, given a change in interest rates.

Convexity measures the curvature of the bond's price-yield curve. Such curvature means that the duration rule for bond price change (which is based only on the slope of the curve at the original yield) is only an approximation. Adding a term to account for the convexity of the bond increases the accuracy of the approximation. That convexity adjustment is the last term in the following equation:

How is modified duration and Macaulay duration linked?

Investors refer to the ratio of Macaulay duration to (1+y) as the modified duration.

What two characteristics determines a bonds price volatility?

Maturity: For a given coupon rate and initial yield, the longer the term to maturity, the greater the price volatility. Coupon: For a given term to maturity and initial yield, the lower the coupon rate, the greater the price volatility. To increase (decrease) a portfolio's price volatility because they expect interest rates to fall (rise), should hold bonds with long (short) maturities.

What is modified duration? Why is modified duration preferred to maturity?

The modified duration is related to the approximate percentage change in price for a given change in yield to maturity. For option-free coupon bonds, modified duration is a better measure of the bond's sensitivity to changes in interest rates. Maturity considers only the final cash flow, while modified duration includes other factors, such as the size and timing of coupon payments, and the level of interest rates (yield to maturity).

What is convexity?

Convexity measures the relationship between bond prices and bond yields. It is used as a tool to manage risk and measure the amount of exposure on a portfolio of bonds. As the convexity of a bond portfolio increases, the systematic risk to that portfolio increases, making it more sensitive it to overall fluctuations in interest rates. As a general rule, the higher the coupon rate, the lower the convexity of a bond.

What is an important property of duration?

Coupon bond: The modified and Macaulay duration of a coupon bond are less than the maturity. Zero-coupon bond: Macaulay duration of a zero-coupon bond equals its maturity. Modified duration of a zero-coupon bond is less than its maturity.

How can the dollar price change be approximated for modified duration?

For small changes in the required yield, the equation works in estimating the change in price. For large changes in the required yield, dollar duration or modified duration cannot approximate the price reaction. Duration will overestimate (underestimate) the price change when the required yield rises (falls), thereby underestimating (overestimating) the new price.

How can "price value of a basis point" measure bond price volatility?

The price value of a basis point, is the change in the price of the bond if the required yield changes by 1 basis point.

Why does the 6% coupon bond have a longer duration than the 9% coupon bond?

A lower coupon rate implies a greater modified duration and a greater price volatility. Bond coupon decreases from 8% to 4%: Modified duration increases as the coupon decreases. Bond maturity decreases from 15 years to 7 years: Modified duration decreases as maturity decreases.

What are the three properties of convexity?

Property (1): The following property holds for all option-free bonds. As the required yield increases (decreases), the convexity of a bond decreases (increases). This property is referred to as positive convexity. An implication of positive convexity is that the duration of an option-free bond moves in the right direction as market yields change. In other words: Rise in yields: If market yields rise, the price of a bond will fall. The price decline is slowed down by a decline in the duration of the bond as market yields rise. Fall in yields: If market yields fall, duration increases so that percentage price change accelerates. Property (2): For a given yield and maturity, the lower the coupon, the greater the convexity of a bond. Thus, a zero-coupon bond has the highest convexity. Property (3): For a given yield and modified duration, the lower the coupon, the smaller the convexity. Zero-coupon bonds have the lowest convexity for a given modified duration.

How can the percentage price change be approximated for modified duration?

Suppose that the yield on any bond changes by 100 basis points, by substituting 0.01 into the equation (dy). Thus, modified duration can be interpreted as the approximate percentage change in price for a 100-basis-point change in yield.

How can "duration" measure bond price volatility?

The Macaulay duration is one measure of the approximate change in price for a small change in yield.

How can "yield value of a price change" measure bond price volatility?

The change in the yield for a specified price change is estimated by calculating the bond's yield to maturity if the bond's price is decreased by X dollars. Then the difference between the initial yield and the new yield is the yield value of an X dollar price change. The smaller this value, the greater the dollar price volatility, because it would take a smaller change in yield to produce a price change of X dollars.

What is the price-yield relationship for an option-free bond?

The price of an option-free bond changes in the opposite direction to the required yield, as the price of a bond is equal to the present value of its expected cash flows. The shape of the price-yield relationship for any option-free bond is convex. Decrease bond price: An increase in the required yield decreases the present value of its expected cash flows and therefore decreases the bond's price. Increase bond price: A decrease in the required yield increases the present value of its expected cash flows and therefore increases the bond's price.