Duration and Convexity Calculator

Duration and Convexity Formulas:

\[ \text{Duration} = \frac{\sum \left( \frac{t \times C}{(1 + r)^t} \right)}{\text{Price}} \] \[ \text{Convexity} = \frac{\sum \left( \frac{t(t+1) C}{(1 + r)^{t+2}} \right)}{\text{Price}} \]

Duration:

Convexity:

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1. What are Duration and Convexity?

Duration measures the sensitivity of a bond's price to interest rate changes, representing the weighted average time to receive cash flows. Convexity measures how the duration changes as interest rates change, capturing the curvature in the price-yield relationship.

2. How Does the Calculator Work?

The calculator uses these formulas:

\[ \text{Duration} = \frac{\sum \left( \frac{t \times C}{(1 + r)^t} \right)}{\text{Price}} \] \[ \text{Convexity} = \frac{\sum \left( \frac{t(t+1) C}{(1 + r)^{t+2}} \right)}{\text{Price}} \]

Where:

\( t \) — Time period (years)
\( C \) — Cash flow at time t (USD)
\( r \) — Yield to maturity (decimal)
\( \text{Price} \) — Current bond price (USD)

Explanation: Duration is the present-value-weighted average time to receive cash flows. Convexity accounts for the fact that duration changes as yields change.

3. Importance of Duration and Convexity

Details: Duration helps estimate price sensitivity to interest rate changes. Convexity improves this estimate, especially for large yield changes. Together they help manage interest rate risk in bond portfolios.

4. Using the Calculator

Tips: Enter bond price in USD, yield rate as a decimal (e.g., 0.05 for 5%), and cash flows as period,amount pairs (one per line). All values must be valid (price > 0, rate ≥ 0).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between Macaulay and modified duration?
A: This calculator computes Macaulay duration. Modified duration (for % price change) equals Macaulay duration divided by (1 + yield).

Q2: What are typical duration values?
A: Short-term bonds might have 1-3 years duration, long-term bonds 10-30 years. Zero-coupon bonds have duration equal to maturity.

Q3: How does convexity affect bond prices?
A: Positive convexity means price increases more when rates fall than it decreases when rates rise (good for investors).

Q4: Are there limitations to these measures?
A: They assume parallel yield curve shifts and don't account for embedded options in callable/putable bonds.

Q5: How are these used in portfolio management?
A: Portfolio duration/convexity help immunize against interest rate risk or position for expected rate changes.