### The Dilemma of Duration

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Before going to the concept of Duration and related terms lets see what is YTM = Yield To Maturity

#### The Yield to Maturity for a Coupon Paying Bond has been defined as the effective rate of interest at which the discounted value of proceeds of a bond equals the price.

Now let's Come to Duration

#### Background: An Investor is concerned whether his assets in a portfolio are sufficient enough to fund the liabilities or not. So for portfolio with investments in Fixed interest securities investor is more concerned in knowing the change in its portfolio due to change in its interest rate.

Now for simplicity, suppose interest rate will remain same throughout the term of a security.

#### also known as Volatility. It measures sensitivity in cashflows due to change in interest rate. Let X be the present value of Payments at rate YTM. So Effective duration is the change in the X  with respect to change in "i"(i.e. YTM) .

Duration:

• also known as Macaulay Duration and Discounted Mean Term.
• It is a measure of a bond's sensitivity to interest rate changes. Technically, duration is the weighed average number of years the investor must hold a bond until the Present Value of the bond’s cash flows equals the amount paid for the bond.
• Macaulay duration is simply the weighted average life of the cash flows .The weight is the present value of that cash flow divided by the total present value and then you're simply multiplying that by the time when the cash flow is receives that's why it's the weighted average life.

Duration is measured in years. Generally, the higher the duration of a bond or a bond fund (meaning the longer you need to wait for the payment of coupons and return of principal), the more its price will drop as interest rates rise.

For example, if a bond has a duration of five years and interest rates increase by 1%, the bond's price will decline by approximately 5%. Conversely, if a bond has a duration of five years and interest rates fall by 1%, the bond's price will increase by approximately 5%.

So we have seen that Duration = Macaulay Duration = DMT
and Effective Duration = Volatility

Modified Duration: used to estimate percentage price change when interest changes by one percent.

To find the modified duration, You have to take the Macauley duration and divide it by 1 + (yield-to-maturity / number of coupon periods per year).

Notes to Remember:
• First, as maturity increases, duration increases and the bond becomes more volatile.
• Second as a bond's coupon increases, its duration decreases and the bond becomes less volatile.
• Third, as interest rates increase, duration decreases and the bond's sensitivity to further interest rate increases goes down.
• Fourth, The duration of "n" year zero coupon bond is "n". Because there is only one payment that must be the time of that cashflow

Other Notes:

YIELD DURATION: Measures bond price sensitivity to its Yield to Maturity
Examples: Macaulay Duration , Modified Duration.

CURVE DURATION: Measures Sensitivity of a Security to a benchmark yield Curve.
Example: Effective Duration

Effective Duration*(1+i)= DMT

The difference between modified duration and effective duration is that modified duration assumes that the cash flows won’t change, while effective duration allows that the cash flows might change.  The cash flows could change because:
1. The bond has embedded options (call option, put option, conversion option, prepayment option)
2. The bond has a floating interest rate

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