Present Value distribution of Cash flows

CFA Level III
#1

Hi,

Can someone please explain to me the present value distribution of cash flows in Fixed income portfolio management.

Thanks!

#2

Think about Macaulay duration: you weight the time-to-cash-flow by the present value of that cash flow.

The idea here’s similar:

  • Chop up the future into time periods (usually 6-month periods)
  • Total the present value of all cash flows in each period
  • Divide the present value in each period by the total present value to get the percentage of present value in each period
  • Match the percentage of present value in each period in your portfolio to the percentage of present value in each period in the portfolio (e.g., index) you’re trying to match
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#3

Thank you as always! You are magic!

#4

You’re too kind.

Actually, I’m lucky: my CFA Institute Level III curriculum arrived today.

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#5

Sorry if its seems dumb, i really dont understand what does the sentence in bold means? I saw that in the Kaplan material but its not intuitive on the matching part. Hope someone can explain, thank you,

#6

Suppose that the index portfolio has cash flows in periods 1, 2, 3, and 4:

  • 10% of the PV comes from cash flows that occur in period 1
  • 30% of the PV comes from cash flows that occur in period 2
  • 40% of the PV comes from cash flows that occur in period 3
  • 20% of the PV comes from cash flows that occur in period 4

Your portfolio should have cash flows in periods 1, 2, 3, and 4. Furthermore,

  • 10% of your PV should come from cash flows that occur in period 1
  • 30% of your PV should come from cash flows that occur in period 2
  • 40% of your PV should come from cash flows that occur in period 3
  • 20% of your PV should come from cash flows that occur in period 4

That’s what it means.

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#7

Thanks S2000! your explanation works magic. Just a further bit of clarification, I noticed that the CFAI text uses the duration contribution of each period’s cash flows. This works by considering that since each cash flow is effectively a zero-coupon payment, the time period is the duration of the cash flow, which is then multiplied with the percentage of the total PV to obtain the duration contribution.

To quote CFAI, ‘It is this distribution that the indexer will try to duplicate.’ In this situation, how does this tie in with the PV concept which you’ve kindly clarified?

#8

Recall the definition of Macaulay duration: it’s the present-value-weighted time to receipt of cash flows. Furthermore, modified duration is Macaulay duration divided by (1 + YTM). Therefore, the duration contribution is exactly the PV percentage in each time period multiplied by the (average) time to receipt of cash flows in that period.

#9

We are SO lucky to have S2000magician in the room, he is THE bond guy and his real name is not James!

#10

So, to make sure our portfolio matches that of the benchmark we want our Duration Contributions for each payment interval to match up? Or, do we want the Duration contributions as a % of total duration contributions to matchup. I would assume if there are two bond portfolios with the same starting PV and same duration contributions then thats similar to matching key rate durations and you would be fully immunized, assuming only interest rate risks. Right or wrong?