## Friday, October 28, 2011

### Duration

CFA Level 1 - Fixed Income Investments

Duration is an estimated measure of the price sensitivity of a bond to a change in interest rates. It can be stated as a percentage or in dollar amounts. It can be helpful to "shock" or analyze what will happen to a bond when market rates increase or decrease.

Example:
Let's assume that the calculation yields a duration of 6.14, this means that if interest rates change, the value of the bond will change by 6.14%. If      there is a 50 basis point change, the value will change by 3.07% and for a 25 basis point change would equal a 1.53% change.

Calculating Duration
The easiest way to calculate duration or the percentage price change is to average the percentage price change coming from an equal increase and decrease in interest rates measured in basis points. To compute duration one needs to develop a valuation model to determine the new prices. The duration value is only as good as the valuation model.

Example:
Stone & Co. bonds are selling at 95, yielding 5.25%Let's assume that yields increase by 25bps, causing the price to decline to 93.

Therefore, the price changes by 2.1%. Then, take 2.1% and divide it by 25bps equaling a .084% change. This represents a 1 basis points move.

Now let's assume that yields decrease by 25bps, causing the price to increaseto 98. The price change is now 3.06%.Then, take 3.06% and divide it by 25bps equaling a .1224% change. This represents a 1 basis points move.

As a final step, just average the two percentage price changes for a 1 basis points move in rates.

Here's how the calculation should look:
Duration = (.084 + .1224)/2 = .1032 = price change of 10.32%  (.1032*100).

Formula 15.4

 Duration = Price if yield decline - Price if yield increase / 2 * (initial price) *change in yield in decimals

As such:98-93/ 2*95*.0025 = 10.52

Approximate Percentage Price Change of a Bond Given a Change in Duration
Let's continue with the above duration of 10.52. This would equal a percentage price change of 10.52 % for a change of 100 basis points in either direction. If the basis points change were 50, then the percentage price change would be 5.26% (10.52/2). If it were a 25bps change, the value would be 2.63% (10.52 / 4).

Approximate New Price of a Bond Given the Duration and New Yield Level
Let's return once again to working with a duration of 10.52. This time, we'll add a total market value of the Stone & Co bonds of \$10,000,000.

Assume that the rates change by 100 bps. This would cause the value of the bonds to change by \$ 1,052,000 (\$10,000.000 *.1052). This is also known as dollar duration.The price will then range from \$11,052,000 to \$8,948,000.

If rates increase by 50 basis points, however, the dollar change would be \$526,000 giving the bonds a price range of \$ 10,526,000 to \$ 9,474,000.

Duration and Yield-Curve Risk for a Portfolio of Bonds
Portfolios have different exposures to how the yield curve shifts. These differences represent yield-curve risk. Because a portfolio tends to have different maturities, if there is not a parallel movement of rates or an equal amount of change in the yield curve across all maturities of the yield, the durations for the different maturities will not react in the same manner. Therefore, the simple procedure discussed above concerning duration will not be able to be applied to the entire portfolio, but will have to be applied over the different maturities as well as to the amount of rate changes in those maturities.

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