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Bond Duration

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Linear measure of the sensitivity of a bond's price to fluctuations in interest rates.

Measured in units of time; always less-than-equal to the bond’s maturity because the value of more distant cash flows is more sensitive to the interest rate.

“Duration" generally means Macaulay duration.

For small interest rate changes, duration is the approximate percentage change in the value of the bond for a 1% increase in market interest rates.

The time-weighted average present value term to payment of the cash flows on a bond.

The proportional change in a bond’s price is proportional to duration through the yield-to-maturity

A 10-year bond with a duration of 7 would fall approximately 7% in value if interests rates increased by 1%.

The higher the coupon rate of a bond, the shorter the duration.

Duration is always less than or equal to the overall life (to maturity) of the bond.

A zero coupon bond will have duration equal to the maturity.

Duration x Bond Price: the change in price in dollars, not in percentage, and has units of Dollar-Years (Dollars times Years).

The dollar variation in a bond's price for small variations in the yield.

For small interest rate changes, duration is the approximate percentage change in the value of the bond for a 1% increase in market interest rates.

Uses zero-coupon bond prices as discount factors

Uses a sloping yield curve, in contrast to the algebra based on a constant value of r - a flat yield.

Macaulay duration is still widely used.

In case of continuously compounded yield the Macaulay duration coincides with the opposite of the partial derivative of the price of the bond with respect to the yield.

Modified Duration – where n=cash flows per year.

and

What will happen to the price of a 30 year 8% bond priced to yield 9% (i.e. $897.27) with D* of 11.37 - if interest rates increase to 9.1%?

- Rule 1: the duration of a zero coupon bond is equal to its time-to-maturity.
- Rule 2: holding time-to-maturity and YTM constant, duration is higher when the coupon rate is lower.
- Rule 3: holding coupon constant, duration increases with time-to-maturity. Duration always increases with maturity for bonds selling at par or at a premium.
- Rule 4: cateris parabus, the duration of coupon bonds are higher when its YTM is lower.
- Rule 5: duration of a perpetuity is [(1+r)/r].

- Bond prices do not change linearly, rather the relationship between bond prices and interest rates is convex.
- Convexity is a measure of the curvature of the price change w.r.t. interest rate changes, or the second derivative of the price function w.r.t. relevant interest rates.
- Convexity is also a measure of the spread of future cash flows.
- Duration gives the discounted mean term; convexity is used to calculate the discounted standard deviation of return.

Duration versus Convexity

Price

Yield