净利差

Here for the Duration: Understanding the Duration Ratio
04 JAN 2011   by Ben T

People looking to earn interest income benefit from a higher interest rates; people looking to borrow money do not. KESDEE’s learning module covering basic market risk has a course on Interest Rate Risk which details and explains the duration ratio.

Two common duration measures

There are two basic duration measures for bonds without embedded options, Macaulay and modified duration. They are near enough in practice, but technically they have different meanings (under the special case of continuous compounding, they are identical!). In risk, we tend to employ the modified duration because modified duration is the more accurate “sensitivity” measure: a first approximation estimate of the bond’s price change given a change in the interest rate (yield).

But when somebody, for example, refers to duration of 5.0 years, the use of “years” betrays Macaulay duration. Notice this has a time dimension and is therefore more intuitive for some. And contrary to many beliefs, Macaulay duration is denoted in time units. Specifically, Macaulay duration is the weighted-average time to maturity of the bond’s cash flows. Weighted by what? The weights are the present values of the cash flows divided by the bond price (since the bond price is the sum of the present value of cash flows, such weights will sum to 1.0 by definition!).

Similar to the KESDEE definition, if the bond pays an annual coupon (to keep this example simple), we can express the Macaulay duration by the following:



Where (T) = time period or year, C(t) is the cash flow, (y) is the yield, and (P) is the current market price of the bond.

Duration of a 10 year bond

For example, assume a $1,000 par 10-year bond that pays an annual coupon of 4.0% and has a yield of 6.0%:



The denominator is the easier piece, it is the sum of the discounted cash flows, which should equal the bond price (in this case, $852,90). The numerator multiplies the present value of each cash flow by its time period; e.g., year 2 * $35.60 PVCF = $71.20. Then each of these ten products are summed; e.g., in this case, the sum is $7,062.44. This number itself has no natural interpretation. But when we divide by the bond price, we are effectively producing a weighted average of the time to maturity of the bond’s cash flows!

Duration ratio

The duration ratio is the ratio of the portfolio’s asset duration to its liability duration. Duration ratios come in three varieties: equal to one, less than one and more than one.

If a cash flow has a duration ratio that is equal to one, its net sensitivity is match and fluctuations in interest rates have no effect on the cash flow. The cash flow is not sensitive… perhaps it needs to get in touch with its feelings.

If a cash flow’s duration ratio is less than one, its net sensitivity favors assets and an increase in interest rates will have a positive impact on it. A cash flow with a duration ratio that is greater than one has a net sensitivity which favors liabilities and would benefit most from a decrease in interest rates.

For example, assume both assets and liabilities have a market value of $1,000. The duration of assets is 7.0 and the duration of liabilities of 5.0. This implies dollar durations, respectively, $7,000 and $5,000. The duration ratio is therefore is equal to $7,000/$5,000 or 1.40.



So, if the interest rate increases by 1.0%, then we estimate assets will lose $70 in value and liabilities will lose only $50 in value. (Two technical caveats: 1. Notice we employed the Macaulay duration as if it were the modified duration; this is imprecise but near enough. Better would be to use 7/1.06 for asset duration and 5/1.06 for liability duration. 2. Duration is only a first order approximation that ignores convexity; we want to always bear in mind that it is not precise).

From: http://www.bionicturtle.com/care ... -the-duration-ratio