LOS f: Calculate and interpret the duration and dollar duration of a bond.

Duration is directly related to maturity and inversely related to the coupon rate and yield to maturity (YTM). Duration is approximately equal to the point in years where the investor receives half of the present value of the bond's cash flows. Therefore, the later the cash flows are received, the greater the duration. 

The longer the time to maturity, the greater the duration (and vice versa). A longer-term bond pays its cash flows later than a shorter-term bond, increasing the duration. The lower the coupon rate, the greater the duration (and vice versa). A lower coupon bond pays lower annual cash flows than a higher-coupon bond and thus has less influence on duration. The lower the YTM, the higher the duration. This is because the bond's price (or present value) is inversely related to interest rates. When market yields fall, the value (or cash flow) of a bond increases without increasing the time to maturity.

Because of convexity, it will be approximately a 7.5% change in price, not an actual 7.5% change in price. The readings are very explicit about this distinction.

The sensitivity of a bond’s price to changes in yield is known as a bond’s effective duration. Macaulay’s duration is calculated by the timing of cash flows weighted by the proportionate value of each flow’s present value.

If bonds are identical except for maturity and coupon, the one with the longest maturity and lowest coupon will have the greatest volatility. 

The relationship of maturity to volatility is direct - the longer the time to maturity, the greater the volatility. A longer-term bond pays its cash flows later than a shorter-term bond, increasing the volatility. This is because a bond’s price is determined by discounting the value of the cash flows. A longer-term bond pays its cash flows later than a shorter-term bond, and longer-term cash flows are discounted more heavily. 

The relationship of coupon to volatility is indirect - the lower the coupon rate, the greater the volatility. This is because a bond’s price is determined by discounting the value of the cash flows. A lower coupon bond pays less cash flows over the bond's life and more at maturity than a higher coupon bond. As noted above, longer-term cash flows are discounted more heavily.

This question is asking: given a change in yield, which of the bonds will exhibit the greatest price change? Of the four choices, the bond with the longest maturity and lowest coupon will have the greatest price volatility. 

All else equal, the bond with the longer term to maturity is more sensitive to changes in interest rates. Cash flows that are further into the future are discounted more than near-term cash flows. Here, this means that one of the 20-year bonds will have the highest volatility. Similar reasoning applies to the coupon rate. A lower coupon bond delivers more of its total cash flow (the bond's par value) at maturity than a higher coupon bond. All else equal, a bond with a lower coupon than another will exhibit greater price volatility. Here, this means that of the 20-year bonds, the one with the 10% coupon rate will exhibit greater price volatility than the bond with the 15% coupon.

If bonds are identical except for maturity, and coupon, the one with the shortest maturity and highest coupon will have the shortest duration. The rationale for this is similar to that for price volatility. Duration is approximately equal to the point in years where the investor receives half of the present value of the bond's cash flows. Therefore, the earlier the cash flows are received, the shorter the duration.

The relationship of maturity to duration is direct - the shorter the time to maturity, the shorter the duration. A shorter-term bond pays its cash flows earlier than a longer-term bond, decreasing the duration. Here, one of the 10-year bonds will have the shortest duration.

The relationship of coupon to duration is indirect - the higher the coupon rate, the shorter the duration. A higher coupon bond pays higher annual cash flows than a lower coupon bond and thus has more influence on duration. Here, the 10-year bond with the highest coupon (8.00%) will have the shortest duration. Note: In addition to having the highest price volatility, zero-coupon bonds have the longest duration (at approximately equal to maturity). This is because zero coupon bonds pay all cash flows in one lump sum at maturity.

Portfolio managers are very interested in a bond’s sensitivity to changes in interest rates. Bonds can be different in terms of maturity and coupon level, while both characteristics impact the change in the bond’s price given changes in interest rates. Duration is a measure that can assesses the time element of bonds in terms of both coupon and term to maturity.

Bonds with larger coupons have a smaller duration, all other things the same. A zero coupon bond has an effective duration equal to its maturity. Duration measures the approximate change in price given a change in interest rates. Therefore, the duration of the bond does not change with the yield to maturity of the bond.

For an option-free zero coupon bond, effective and modified duration will be almost identical and both will be approximately equal to the bond's years to maturity.

Duration decreases as market yields rise.

For a zero-coupon bond duration is approximately equal to the number of years to maturity. Here, there are 4 years until maturity, so the effective duration is approximately equal to 4 years. We use the term approximately because this ignores the curvature of the price/yield curve.

Duration is inversely related to yield to maturity (YTM). The higher the YTM, the lower the duration. This is because the change in the bond's price (or present value) is inversely related to changes in interest rates. When market yields rise, the value (or cash flow) of a bond decreases without decreasing the time to maturity.

Duration is also a function of volatility (risk).  Higher volatility (risk) = higher duration.  A higher coupon bond has a lower duration relative to a similar bond with a lower coupon because the bond holder is getting more of their cash value sooner (because of the higher coupon).  This lowers the overall risk of the bond resulting in a lower duration.

For coupon paying bonds, duration is less than maturity.

Duration is approximately equal to the point in years where the investor receives half of the present value of the bond's cash flows. Since zero-coupon bonds only have one cash flow at maturity, the duration is approximately equal to maturity. Any coupon amount will shorten duration because some cash flow is received prior to maturity.

The statement that a bond's percentage change in price and dollar change in price are both tied to the underlying price volatility is correct.

The effective duration formula result is for a 1.00% change in interest rates (100 basis points equals 1.00%, or 0.01 in decimal form). The denominator is multiplied by 2.

The duration of a floating rate bond is equal to the time until the next coupon payment takes place. As the coupon rate changes semi-annually with the level of the interest rate, a floating rate bond has the same duration as a pure discount bond with time to maturity equal to the time to the next coupon payment of the floating rate bond.

The duration of a zero coupon bond is approximately equal to its time to maturity.

All else constant, a bond with a longer maturity will be more sensitive to changes in interest rates. All else constant, a bond with a lower coupon will have greater interest rate risk.

Duration = [1001 ? 999] / [2 × 1000 × 0.0001] = 10.