## Macaulay Duration, Modified Duration and Effective Duration

ashok.kothavle
Posts: 4
Joined: Sun Jul 13, 2014 4:10 am

### Macaulay Duration, Modified Duration and Effective Duration

Dear Forum,

Need some clarification regarding duration.

(A) Macaulay Duration

Macaulay duration measures the average time that would be taken to receive the cashflows from the invested bond. It is the weighted average term to maturity of the cash flows from a bond. The weight of each cash flow is determined by dividing the present value of the cash flow by the price.

Formula: -

Macaulay Duration= ∑ ( 1/P × Ci / [ e ^ ((YTM^c×ti ) ) ] × ti )

Where

n = number of cashflows
P = Current Price of the Bond
Ci = Amount for ith cashflow
YTMc = Continuously Compounded YTM
ti = time (in years) until cashflow Ci becomes due from today

Assumption

1. Macaulay Duration measures interest rate risk only for bonds where cashflows do not change with change in the yield (i.e. for plain vanilla bonds and not for bonds with embedded options)

2. Macaulay Duration assumes yield curve is flat and so cashflows are reinvested at constant YTM rate over the bond period.

3. Macaulay Duration does not consider the fact that duration does not remain constant and duration changes with level of YTM rates.

Questions

1. Can Macaulay Duration be computed for floating rate bonds because the cashflows change with change in the yield on the coupon reset date.

2. Is Macaulay Duration also computed as negative of (percentage change in the price of the bond with 1% change in YTMc)?

i.e. Macaulay Duration = -1 × ( % ∆P ) / ( ∆YTM^c of 1% )

(B) Modified Duration

Macaulay Duration does not consider the fact that duration does not remain constant and duration changes with level of YTM rates. To correct this error, modified duration is computed.

Modified Duration=(Macaulay Duration)/((1+ YTM/m) )

Where m = number of compounding periods in a year
YTM = Annualized YTM rate which is not continuously compounded
Modified Duration is used when YTM is not a continuously compounded rate.

Assumptions

1. Modified Duration measures interest rate risk only for bonds where cashflows do not change with change in the yield (i.e. for plain vanilla bonds and not for bonds with embedded options)

2. Modified Duration assumes yield curve is flat and so cashflows are reinvested at constant YTM rate over the bond period.

3. Modified Duration changes with changes in the level of YTM rates.

Questions

1. What is the mathematical as well as practical importance of YTM rate being continuously compounded?

2. Can Modified Duration be computed for floating rate bonds because the cashflows change with change in the yield on the reset date.

3. Is Modified Duration also computed as negative of (percentage change in the price of the bond with 1% change in YTM)?

i.e. Modified Duration= -1 ×( %∆P ) / ( ∆YTM of 1% )

4. Is the only difference between the above equation of modified duration and that of macaulay duration relate to the YTM rate being used (annual vs continuously compounded)?

(C) Effective Duration

Effective duration takes into account that expected cash flows will fluctuate as interest rates change. This becomes useful to compute duration for options embedded bonds

Effective Duration=(P(-∆i) - P(+∆i)) / (2 × P0× ∆i)

Assumptions

1. As YTM could not be calculated for options embedded bonds, change in interest rate is used instead of change in YTM

Questions

1. Change in interest rates is used instead of change in YTM. So is it assumed that the complete interest rate term structure has a 1% parallel shift?

Regards
Ashok D Kothavle

edupristine
Finance Junkie
Posts: 722
Joined: Wed Apr 09, 2014 6:28 am

### Macaulay Duration, Modified Duration and Effective Duration

MACAULAY DURATION

1) No, Macaulay duration cannot be calculated for floating rate bonds (with non fixed cash flows). It is used only for fixed cash flow bonds.

2) The idea behind Macaulay duration is to find out the weighted average time until a bond holder would receive the cash flows from a bond.
The concept of percentage change in price wrt to 1% change in yield relates to Modified Duration.

MODIFIED DURATION

1) When yields are continuously compounded, then Macaulay Duration is numerically equal to Modified Duration.

But when the yields are periodically compounded, it means there is a little difference between the two as shown below:

Modified Duration = Macaulay Duration/ (1+y/k) where "y" is the yield to maturity of an asset periodically compounded
And "k" is the compounding frequency per year.

2) Yes, Modified Duration can be calculated for floating rate bonds. This means if the cash flows change at reset dates as per the floating rate formula, modified duration can be used.

3) Yes, the negative sign in modified duration formula simply implies that the relationship between prices and yield change is inverse.

4) As already discussed in point 2 of MACAULAY DURATION above, Macaulay Duration = -1 × (% ∆P) / (∆YTM of 1 %) formula is not valid. This formula is essentially for Modified Duration.

EFFECTIVE DURATION
1) Yes, the assumption of 1% parallel shift is correct. If however, you want to calculate the price change for an arbitrary shift in Yield,

%ΔP ≈ − (Effective Duration) ×Δy

ashok.kothavle
Posts: 4
Joined: Sun Jul 13, 2014 4:10 am
Thanks a lot for the guidance