Interest rate risk, Duration and Convexity

Interest rate risk: the risk that an investment will lose value based on a change in interest rates

Interest rates are influenced by many macroeconomic forces:
monetary policy, inflation, the strength of the economy

When interest rates go up, things get more expensive: debts get more expensive, commodities get more expensive. Interest rate is the cost of money.

Higher duration means higher sensitivity the bond has to interest rates.

Duration is the "weighted average maturity for all future bond cash flows"

It estimates the impact a 1% change in interest rates could have on a bond's price.

The bond's coupon: Low coupon bonds tend to have higher durations and are more interest rate sensitive than equivalent higher coupon bonds.

Note: lower yields also tend to equate to higher duration

The bond's maturity: Long maturity bonds tend to have higher durations and are more sensitive than equivalent short maturity bonds

A defensive investor, or someone who expects interest rates to rise, will aim to reduce duration

The reverse is also true.

In bond investing duration is therefore key to understanding and managing risk and reward.

Duration:

When interest rates rise, the price of the bond that we are holding is going to fall.

What is MacD? It is the weighted average maturity of the cashflows from a bond. Looking at these bonds, how long is it going to take us to receive these cashflows? The one with the higher Macauley duration is going to be more sensitive to interest rate changes.

The formula for Macauley Duration: PV of the ith cash flow/ Price of the bond.

The numerator is talking about the time weighted present value of the cash flows.

Managing duration

Modified duration: sensitivity of the bond's price to yield.

Modified duration= Macauley Duration/ (1+ yk/k)

if modified duration is -0.0443, then the price decreases by 4.43% when there is a change in 1% yield.

Modified duration is the derivative of the bond's price with respect to the yield.

Modified duration= mac duration/ (1+r)

Ex) Use a three year bond that pays 5 % annual coupons with a YTM of 7%. calculate its modified duration.

Modified duration = Mac Duration /(1+r)
Price= 94.75

MacDur= 2.855

Mod Dur= 2.855/ (1+0.07) = 2.668

If YTM increases by 1%, the price is going to go down by 2.668%.

A fixed income security that pays quarterly coupons has a current price of 95.63 based on a 5.14% annual yield-to-maturity. If the yield-to-maturity increases 10 basis points, the price will decrease to 94.96.
If the yield-to-maturity decreases 10 basis points, the price will increase it 96.35.
What is the approximate modified duration and the approximate Macaulay duration.

Macaulay duration: weighted average time to cash flow receipt

Effective duration: measures sensitivity of bond price relative to benchmark yield

Convexity

percentage change in the bond (y-axis)
percentage change in YTM (x-axis)

Yield and price are inversely related

Duration rule says that:
1. there is a direct relationship between bond price and yield
2. said differently, the price-yield relationship is linear
allow equal changes in yield on the positive and negative side to study the response of Bond Price

Curvature of the price-yield relationship is called Convexity.

Yield and price are inversely related.
In reality the price and yield relationship is not linear relationship.

It is more of a curved line.
the reality it is convex
how the bond is changing to the changes in the yield
People like convexity.