Cash-Flow Timing Moments and Macaulay Convexity

Summary

The document asks what is represented by a cash-flow-weighted sum of squared payment times and whether it is the same as convexity. The response clarifies that the weights must be present values of the cash flows, normalized by the total present value. Using undiscounted cash flows would not give the standard Macaulay duration expression or its squared-time counterpart.

With present-value weights, the first time moment is Macaulay duration, and the second time moment is Macaulay convexity. The response also distinguishes this quantity from the convexity measure more commonly used in modern bond analysis. It provides a definition-level clarification but does not derive the conversion between conventions or discuss how the measures are used to estimate bond-price sensitivity.

Key ideas

• Macaulay duration weights payment times by each cash flow’s share of total present value.
• The cash-flow weights must be discounted present values, rather than nominal cash amounts.
• The normalized second moment of payment time is Macaulay convexity.
• Macaulay convexity differs somewhat from the convexity convention commonly used in current practice.

Tags

# What is $\sum_{i=1}^n i^2 \frac{CF_i}{CF_{Total}}$


# What is $\sum_{i=1}^n i^2 \frac{CF_i}{CF_{Total}}$












I know that $$\sum_{i=1}^n i \frac{CF_i}{CF_{Total}}$$ is Macaulay Duration, but what is $$\sum_{i=1}^n i^2 \frac{CF_i}{CF_{Total}}$$. I have given it a thought like it is a second moment of something and for a few internet search this has a correlation with Convexity. Is it just convexity if not, what is it ?

## Answer by nbbo2 (score 2)

https://quant.stackexchange.com/a/74369

The first expression is Macaulay Duration only if $CF_i$ is THE PRESENT VALUE of cash flow i, not the cash flow itself. A more explicit formula would be $\sum_i i \frac{PV(CF_i)}{PV_{Total}}$.

The second expression (with the same caveat) is the Macaulay Convexity, which is a little different from the Convexity that is commonly used nowadays.

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