Questions:
What is the difference between a bond's duration and convexity? How do you use these values to estimate the price change from a change in yields?

James is a bond portfolio manager at Kentech. He is worried about the impact of changes in the yields on the prices of his bonds. 

One of his bonds was issued by a company in Boston and has a modified duration of 31.11 and its convexity is 300.90. 
Moreover, his second bond is in an emerging market in Asia. Its modified duration is 59.21 and its convexity is 350.26.
The third bond in his portfolio is also in the U.S., however it is a high-risk bond. Its modified duration is 60.23 and its convexity is 360.9.

In the next six months, the expected change in the U.S. is 26 basis points. Whereas, the expected change in yields in the emerging market in Asia is 53 basis points due to high volatility in the market. 

Which bond is the riskiest based on the expected changes in price in the next 6 months?

Reference answers:
# Definitions

## Modified Duration

A mathematical formula that measures how much a bond's value changes in response to a 1% change in interest rates

## Convexity

Convexity is the curvature in the relationship between bond prices and interest rates.

## Difference

Bond-modified duration and convexity measure the sensitivity of a bond's price to the change in yields. The difference is that modified duration measures the linear relationship and convexity measures the degree of curve in the relationship between bond prices and change in yields. 

Although modified duration can provide a good measure, measuring convexity can enhance the accuracy of the change in price as a result of a change in yield.

# Formula 
The change in the price of the bonds can be found using the following 

Percentage change in price = -(Annual Modified Duration * Change in yield) + [1/2 * Annual Convexity * (Change in yield)^2]

Bond 1: 

Percentage change in price = -(Annual Modified Duration * Change in yield) + [1/2 * Annual Convexity * (Change in yield)^2]
Percentage change in price = - (31.11 * -0.0026) + [1/2 * 300.90 * (0.0026)^2]
Percentage change in price =  0.1826 
Percentage change in price = 18.26 %

Bond 2: 

Percentage change in price = -(Annual Modified Duration * Change in yeild) + [1/2 * Annual Convexity * (Change in yeild)^2]
Percentage change in price =  -(59.21 * -0.0053) + [1/2 * 350.26 * (0.0053)^2]
Percentage change in price = 0.3187
Percentage change in price = 31.87%

Bond 3: 

Percentage change in price = -(Annual Modified Duration * Change in yield) + [1/2 * Annual Convexity * (Change in yield)^2]
Percentage change in price =  -(60.23 * -0.0026) + [1/2 * 360.9 * (0.0026)^2]
Percentage change in price = .01578
Percentage change in price = 15.78%

# Result

The expected changes in the prices of the bonds are 18.26% for bond 1, 31.87% for bond 2 and 15.78% for bond 3. 

Bond 2 has the highest expected change in price for the expected change in yields.

This is because bond 2 is in an emerging market with volatile interest rates which has been confirmed by the calculations.