Bond Duration & Convexity Risk
Estimate interest rate risk using Modified Duration and Convexity. Enter values for instant results with step-by-step formulas.
Formula
Mod Duration = Mac Duration / (1 + y/f)\nPrice %ฮ โ -ModDur ร ฮy + ยฝ ร Convexity ร (ฮy)ยฒ
Duration measures the linear sensitivity of a bond's price to interest rate changes. Convexity measures the curvature (non-linear) sensitivity. Together, they provide a highly accurate estimate of price risk.
Worked Examples
Example 1: 5-Year Bond
Problem:5% Coupon, 6% YTM, 5 Years.
Solution:Price < Par (Discount). Duration ~4.3 years.
Result:If rates rise 1%, price drops ~4.1%.
Frequently Asked Questions
What is Modified Duration?
It adjusts Macaulay Duration to estimate the percentage change in price for a 1% change in yield.
Why is Convexity important?
For large yield changes (>1%), Duration becomes inaccurate. Convexity corrects this error.
Does a higher coupon reduce duration?
Yes. Higher coupons mean you get cash back sooner, lowering the weighted average time (Duration).
What is Negative Convexity?
Occurs in callable bonds or mortgages (MBS). As rates fall, prices don't rise as much because the borrower might refinance/call the bond.
Is Duration measured in years?
Macaulay Duration is in Years. Modified Duration is technically a % sensitivity, but often quoted as a unitless number.
What is 'Effective Duration'?
A measure used for bonds with embedded options (calls/puts), where cash flows change as rates change. Bond Duration & Convexity Risk assumes fixed cash flows.
How does frequency affect duration?
More frequent payments (monthly vs annual) slightly reduce duration as cash is received sooner.
What is a bond ladder and when should I use one?
A bond ladder staggers bond maturities across several years โ for example, buying bonds maturing in 1, 2, 3, 4, and 5 years with equal portions of capital. As each bond matures, you reinvest the proceeds into a new long-dated bond, maintaining the ladder. This structure reduces interest rate risk: if rates rise, you reinvest maturing bonds at higher yields rather than locking all capital into one long-term bond at the wrong time. It also provides predictable cash flows at regular intervals, useful for retirees managing income. Bond ladders are most effective for investors who hold to maturity and want capital preservation with some yield.
Background & Theory
Duration vs Maturity
A 10-year Zero Coupon bond has a duration of 10 years. A 10-year 10% Coupon bond has a duration of ~7 years. Why? Because you get much of your money back earlier via coupons. Duration is the "effective" maturity.
Why Convexity is Good
Positive convexity is beneficial for bondholders. It means that as yields fall, prices rise more than duration predicts. As yields rise, prices fall less than duration predicts. It is a cushion.
Practical Application
- Hedging: Pension funds match the duration of their assets (bonds) to their liabilities (future payouts) to immunize against rate changes.
- Trading: If you expect rates to fall, buy High Duration bonds (maximize gain). If rates rise, move to Low Duration (minimize loss).
History
Frederick Macaulay
In 1938, Frederick Macaulay introduced the concept of "Duration" to measure the weighted average time to receive cash flows. Before this, investors looked only at Maturity, which ignored the timing of coupon payments.
The 1970s Inflation
Bond math became critical in the 1970s and 80s when interest rates became volatile. Simple yield calculations weren't enough; traders needed to hedge "Duration Risk".
Convexity Importance
As derivatives and complex trading grew, Duration (a linear approximation) proved insufficient for large rate moves. Convexity was added to account for the curve in the Price-Yield relationship, becoming essential for managing large portfolios.