Duration is the change in the value of a bond from a % change in interest rates. There are three variations of Duration.

#### Macaulay Duration

Macaulay Duration, commonly known as Duration, is the time the investor takes to recover an invested money in the bond through coupons and principal repayment. Frederick Macaulay gave this concept. Duration refers to the weighted average time before repayment or when cash flow is received. Duration attempts to measure how long it takes, in years, for an investor to recoup the bond’s price from the bond’s total cash flows. Macaulay Duration can only be extended on instruments with fixed cash flows.

Macaulay Duration= {C/(1+y)^t}*1 + {C/(1+y)^2}*2 +—– {C+M/(1+y)^n}*n

where:

- C=periodic coupon payment
- y=periodic yield
- M=the bond’s maturity value
- n=duration of bond in periods

**Example**

Assume an investor buys a 6% annual payment bond with 3 years to maturity. The bond has a yield to maturity of 8% and is currently priced at 95 per 100 of par. Calculate Macaulay Duration.

Macaulay Duration = [{6/(1.08^1)}*1]+[{6/(1.08)^2}*2] + [{(6+100)/(1.08)^3}*3] = 2.82

#### Modified Duration

Modified Duration measures the price sensitivity of a bond to interest rate movements. Modified Duration (ModDur) extends Macaulay duration and helps measure a bond’s sensitivity to changes in interest rates.

Modified Duration measures the percentage change in price for a unit change in yield and refers to the price sensitivity. Modified Duration depends on yields, whether or not the instruments have fixed cash flow or not. This is useful to measure the sensitivity of a bond’s market price to a finite interest rate, such as yield movement.

Modified Duration = Macaulay Duration /{1 +(y/k)}

Where:

y = yield to maturity

k = number of coupon payments per year

**Example:**

A semi-annual 5-year bond with a 4% coupon yielding 6% and a Macaulay duration of 4.50. The modified Duration of the bond will be

= 4.50/(1+(0.06/2))

= 4.37.

#### Duration and Bond Characteristics

Duration reflects a bond’s particular term to maturity, coupon, and yield. They are all related in the following ways.

- Modified Duration will always be less than the term to maturity when a bond has coupons because the calculation gives weight to these payments.
- A bond with no coupon payments, such as a stripped coupon, will have a Macaulay duration equal to its term to maturity.
- The size of a coupon and modified Duration are inversely related. For example, as a coupon becomes smaller, modified duration increases because less of the total cash flow of a bond comes from interest payments.
- Term to maturity and modified Duration are positively related.
- Market yield and modified Duration are inversely related. The higher the market yield, the lower the Duration becomes.

### Duration and Price Change

Using the following formula, the modified duration can be used to calculate an approximate percent

price change of an individual bond from a given yield to any other yield:

#### Approximate Percentage Price Change = – (Modified Duration) * (Y) *100

**Where:**

(Y) is the change in yield expressed as a decimal

For example, on a 5-year bond with a 4% coupon and a modified duration of 4.37, yields increase from 6% to 6.50%. The approximate price change is:

-(4.37)(0.0050)(100) = -2.19%

The formula to find the approximate dollar change is similar to the formula approximating the

percent change.

#### Approximate Price Change = – (Modified Duration) * (Y) * MV

MV is the market value of the bond or portfolio. For example, an investor holds INR 50,000 worth of 5-year 4% bonds when it yields 4%, and its modified duration is 4.80. The approximate price change in the bond when yields fall 50 basis points would be: -(4.80)* (-0.0050)* (50,000) = INR 1,200.

### Convexity and Price Change

All duration measures only approximate the price change after a fluctuation in yields.

- For significant decreases in Yield, any measure of duration will understate the actual price change.
- For significant increases in Yield, any measure of duration will overstate the actual price change.

To improve the estimate of the price change, a non-callable bond’s convexity adjustment is added to the duration calculation. Duration plots out as a straight line on the price/yield graph.

Convexity is a measure of how much a bond’s price/yield curve deviates from the linear approximation of that curve.

The approximate percentage price change due to Convexity is equal to the following:

#### Percent Price Change due to Convexity =Approximate Percentage Price Change due to Convexity * (Convexity/2) * dY *dY *100

**Where;**

dY is the change in Yield.

For instance, assume a 5-year bond with a 4% semi-annual coupon trades to yield 6% (from the above example). Its Convexity is 36.36. Yields increased by 50 basis points. The approximate percent price change from Convexity is equal to the following:

Approximate Percentage Price Change due to Convexity = (36.36/2) (0.0050) (0.0050) (100)

= +0.045%.

The total price change equals -2.19% + 0.045% = -2.15%.

#### Convexity and Bond’s Characteristics

Like duration, Convexity reflects the properties of a non-callable bond’s coupon, maturity, and Yield.

- Coupon amount and Convexity are inversely related. Therefore, Convexity will increase with a lower coupon, assuming all else is constant.
- Term to maturity and Convexity are positively related. Convexity will increase the longer the time to maturity becomes, assuming all else is constant.
- Yield and Convexity are inversely related. Convexity increases with a decreasing yield.
- Graphically this implies the price/yield curve is more convex at the lower yield end of the curve.
- Convexity is always positive.

Callable bonds suffer from price compression below a certain yield level. Therefore, the limit in price appreciation creates a negative convexity for callable bonds below this yield threshold. The negative Convexity compensates for overestimating the price by the duration at low yield levels to ensure the price estimation is close to the actual price.