IRR vs YTM and portfolio duration vs average duration in an upward sloping yield curve

“In an upward-sloping yield curve, portfolio duration and IRR will be higher-than-average duration and YTM of the bonds because portfolio statistics reflect all cash flows (and return) to be received and the longer maturity bonds will impact the portfolio for a longer time.”

This sentence is in the KS notes and and struggling to understand it.

In a nutshell, when the yield curve isn’t flat, the duration of the portfolio won’t be the weighted average of the durations of the constituent bonds and the YTM of the portfolio won’t be the weighted average of the YTMs of the constituent bonds. In short, what they taught you at Level I about duration and YTM for a portfolio was all a lie. And, most likely, a conspiracy.

You have to aggregate the cash flows at each payment date and work it out from first principles.

Note that you won’t have to do the calculations on the exam. You simply have to know that when the yield curve slopes upward the weighted average duration and weighted average YTM will be too low, and when it slopes downward it’ll be too high.

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By the way, it would be an interesting exercise for you to try in Excel: put together 3 or 4 bonds, and an upward sloping yield curve, calculate the YTM and the modified duration of each bond, calculate the price-weighted averages of those, then calculate the YTM and modified duration of the portfolio. Then try it with a flat yield curve and a downward sloping yield curve.

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Thank you S2000. That’s a good idea.

Here is another way to add more intuition to it.

Most of the time, the yield curve has an upward slope, right?

In Page 85, there is this very classic example in which they used 3 bonds to immunize the liabilities. The weighted average yield is significantly lower (almost 15%) than the real yield, calculated using the cashflow as S2000magician mentioned. “The difference arises because of the steepness in the yield curve. The key point is that the goal of the immunization strategy is to achieve a rate of return close to high (3.76%), not low (3.3)”

The rate calculated using the cashflow always has the same trend as the yield curve direction.

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Hi! So portfolio IRR and duration > weighted-avg IRR and Duration for an upward YC. What about portfolio convexity vs weighted-average convexity for an upward-sloping YC? Will the portfolio convexity also be higher than weighted-average convexity?