Duration

Duration is a measure of the slope of the price-yield function, steeper at low interest rate and flatter at high interest rate for non-callable bonds. It represents the percentage change in price for a 100 basis point change in yield.

Effective Duration
Duration is the approximate percentage change in price for a 100 basis point change in rates.
Effective Duration = (V_- V+) / (2 x Vo x dy in decimal).

Note:
· go down or up by same no. of basis points

Modified Duration
Modified duration is the approximate percentage change in a bond’s price for a 100 basis points change in yield, assuming that the bond’s expected cash flow does not change when the yield changes. This works for option-free bonds such as Treasuries but not with option-embedded bonds because the cash flows may change due to a call or prepayment.

Macaulay duration
Macaulay’s duration is the weighted average number of years remaining to receive the present value of a bond.

It gives the analysis a short cut to measure modified duration. But because modified duration is flawed by not incorporating the change in cash flows due to an embedded option, so are Macaulay durations.

Modified duration = Macaulay’s Duration/ (1 + yield/k)
Macaulay duration = Modified duration x (1 + BEY/2).

Notes:
· The duration of zero coupon bond = its maturity;
· Duration of a floater coupon bond = the time to the next reset date

When is Effective Duration a Better Measure?

When a bond has an embedded option, the cash flows can change when interest rates change because of prepayments and the exercise of calls and puts. Effective duration takes into consideration the changes in cash flows and values that can occur from these embedded options.

Duration of a portfolio
Duration of a portfolio equals the weighted average of the durations of the bonds in the portfolio.