Modified duration ^ Macaulay duration, briefly again

The nice pivot diagram on http://en.wikipedia.org/wiki/Bond_duration is for Macaulay duration — dollar-weighted average maturity. Zero bond has duration equal to its maturity. (I think many textbooks use this diagram because it’s a good approximation to MD.)

The all-important sensitivity to yield is …. MD i.e. modified duration. Dv01 is related to MD (not Macaulay) — http://bigblog.tanbin.com/2012/05/bond-duration-absolute-1-relative-x.html

MD is the useful measure. It turned out that MD is different from Macaulay duration by a small factor.

quantitative feel of bond duration – map absolute 1% -> relative x%

In the simplest illustration of modified duration, if a bond has modified duration == 5 years, then a 100bps yield change translates to 5% dollar price (valuation) change.

Note that 100 bps is an Absolute 1% change in yield, whereas the 5% is a Relative 5% change in valuation. If original valuation == $90 [1], then 100 bps =>> $4.5 change.

After we clear this little confusion, we can look at dv01. Simply set the absolute yield change to 1 bp. The valuation change would be a Relative 0.05% i.e. $0.045. The pattern is

Duration == 5 years => dv01 == 0.05% Relative change
Duration == 6 years => dv01 == 0.06% Relative change
Duration == 7 years => dv01 == 0.07% Relative change

Note 0.05% Relative change means 0.05% times Original price, not Par price.  Original price can be very different from par price, esp. for zero bonds.

[1] 90 in bond price quote means 90% of par value. For simplicity we would assume par is $100, though smallest unit is $1000 in practice.

(See P10 of YieldBook publication on Duration.)

question on our Reo pricing engine — effective duration

Hi Jerry,

I recently worked on eq derivative pricing. I realized traders need to know their sensitivities to a lot of variables. That made me start thinking about “your” pricing engine — If a bond trader has 100 open positions, she also need to know her aggregate sensitivity to interest rate (more precisely the yield curve).

To address this sensitivity, I know Reo displays dv01 at position level (and rolls up to account/sector levels), but how about effective duration?

If we do display duration on a real time basis, then is it calculated using dv01 or is there option-adjusted-spread factored in for those callable bond positions?

Thanks.
Tanbin

dv01 ^ duration – software algorithm

Q: Do dv01 and duration present the same level of software complexity? Note most bonds I deal with have embedded options.

I feel answer is no. dv01 is “simulated” with a small (25 bps?) bump to yield… Eff Duration involves complex OAS. See the Yield Book publication on Durations. In AutoReo, eff duration is computed in a separate batch-driven risk system. No real time update.

By contrast, eq option (FX option probably similar) positions need to have their delta and other sensitivities updated more frequently.

bond duration vs option sensitivity greeks

Just as an option’s market value has sensitivities to underlier price, vol, time to expiration… collectively known as the greeks, a fixed-income portfolio valuation has a much-watched sensitivity to yield, known as Modified-Duration.

Interestingly, The most complex part of bond duration is … option adjusted spread for bonds with embedded options!

A 2nd common type of bond-with-options are caps/floors. Is any Greek widely used for these? Yes since these instruments are mathematically equivalent to calls/puts on an underlying bond.

I feel caps/floors are less popular than swaption, whose market is quite large.

bond duration (and KRD) – learning notes 2

Jargon warning: yield is best written in bps/year, like 545bps/year. If you say 5.45% it gets ambiguous in some contexts such as modified duration. “1% rise in yield” could mean 2 things

– 5.45% —-> 6.45%,
– 5.45% –x-> 5.50% is a misunderstanding

This is not academic; this is real. Portfolio sensitivity to yield fluctuations is a key concern of banks on Wall St or Main St. It’s all about x bps change in yield. (From now on, always use bps to describe yield; avoid percentage.)

DV01 is dollar value of a “basis point”, free of any ambiguity.

DV01 and modified duration are 2 of the most widely used bond math numbers. Both are derived from bond cash flow.

Mac duration — definition — weighted average of wait time for the cash flows.
Mac duration — usage — not much in real world trading

Modified duration — definition — Mac duration modified “slightly”, by a tiny factor. REDUCED by (1+r)
Modified duration — usage — more useful than Mac duration. It measures price sensitivity to a yield shift, on a given bond.

For a simple example of a bond with modified duration of 5 years. 100 bps yield change results in a 5% dollar price change.

Key Rate Duration is an natural (and intuitive) extension of the duration concept, useful in MBS etc.

discount curve – cheatsheet

Based on P245 of [[complete guide to capital markets]]

Discount curve is designed to tells us how to discount to NPV $1 received x days from today, where x can be 1 to 365 * 30. If the curve value for Day 365 is 0.80, then SPOT rate is 25% or 2500bps — 80 cents invested today becomes $1. Note the value on the curve is not bps or spot rate, but a discount factor value from 0 to 1.

Q: how do I get forward rate between any 2 dates?
A: P246. Simple formula.

Discount curve is “built” from and must be consistent with
+ ED CD rates (spot rates) of 1,7,14..,90 days. As loans, these loan term always starts today.
+ ED futures rates (forward rates). Loan term always last 3 months, but start on the 4 IMM dates of this year, next year, next next year ….
(Note ED futures rates are determined on the market; ED CD rates are announced by BBA.)

Discount curve needs to cover every single date. First couple of months are covered by latest announced ED CD rates, interpolating when necessary. After we pass 90, all subsequent dates (up to 30 years) are covered by ED futures rates observed on CME. Being forward rates, these aren’t directly usable as those CD rates, but still very simple — If the 3-month forward rate 3/19/2008 – 6/19/2008 is 200bps, and discount factor for 3/19 is 0.9, then 6/19 discount factor is (0.9 / 1.02)