Using Duration to Estimate a Bond Price Change

Summary

The document explains the sign convention in the duration approximation for estimating a bond’s price response to a yield change. Under the stated relation, the approximate price change is the negative of bond value times duration times the yield change. A yield decrease is therefore entered as a negative change, and the two negative signs imply an increase in price.

The question concerns a five-year coupon bond and a small yield decline; the worked solution reports the resulting approximate price increase. The answer confirms that the questioner’s signed calculation is correct and says the textbook solution simply cancels the negatives to keep the arithmetic brief. This is a first-order duration estimate, not an exact repricing; the note does not discuss convexity or other effects that can matter for larger yield moves.

Key ideas

• The duration approximation relates price change to bond value, duration, and the negative of the yield change.
• A yield decrease is represented by a negative yield change, producing a positive estimated price effect.
• The textbook solution’s positive arithmetic is consistent with canceling the two negative signs.
• Duration gives an approximate first-order price response and omits convexity effects.

Tags

# Calculate the effect of the change of bond price


# Calculate the effect of the change of bond price












I have the following question from Hull, problem 6.14:

A five-year bond with a yield of 11% (continuously compounded) pays an 8% coupon at the end of each year.

(c) Use the duration to calculate the effect on the bond’s price of a 0.2% decrease in its yield.

The solution is as follows:

Since, with the notation in the chapter $$\begin{align}\Delta B & = -BD\Delta y \qquad (1)\end{align}$$ the effect on the bond’s price of a 0.2% decrease in its yield is: $$\$86.80\times4.256\times 0.002 = 0.74$$

However the bonds price B is \$86.80, and the $\Delta y$ indicates a change, where by notation a decrease is negative, and an increase is positive. So based from this, and using formula (1) shouldn't the equation be:

$$-\$86.80 \times 4.256 \times(-0.002) = 0.74$$

Question:

Is this a mistake in the text book, or is my approach incorrect?

## Answer by Lennart_R (score 2, accepted)

https://quant.stackexchange.com/a/37152

Highly off-topic, but you are correct. The book however is not wrong, the authors just canceled out the negatives to provide short solutions. The solutions tend to be rather brief if i remember correctly.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.