Duration = time horizon under Immunization


Why does CFAI assume duration = time horizon for a liability payable on a single date?

The entire immunization strategy is based on the assumption that duration must equal time horizon until payment.

Meanwhile, they say that “to accomplish the goal of immunization, we use effective duration”.

Which one is being used after all? And why? Thanks!

In other words, are we using Macaulay or Effective Duration? Or both?

Also, can someone explain the reasoning behind the duration of the portfolio matching the duration of the liability.

What is the impact from changing interest rates in a debt I already have? Isn’t it fixed?

Effective duration is best used when a bond has embedded optionality, as it takes into account the effect of these embedded options on cash flow and market value (hence effective - takes into effect).

I believe that the focus in level three is more on bullets with no embedded options, and as such I believe that they are referring to modified duration (someone please correct me if I’m off in my understanding here).

As for your last question. Think of yourself as a pension fund manager, you are managing your asset base against a set of liabilities (pension fund obligations). Your assets can include both equity and fixed income. The fixed income component of your assets has a duration associated with it, and your liability also has a duration. The value of your fixed income assets as well as your fixed income liabilities move with interest rates, hence it is important to know what affect interest rates can have on both sides of your “balance sheet”. I.e. It is important to make sure that the Present Value of Assets (Pension fund assets - composed of Fixed Income and Equity) is greater than the Present Value of Pension Liabilities. This will hopefully create an overall surplus and increase the likelihood of meeting your obligations to pension fund’s beneficiaries (both active and retired members).

For bonds whose cash flows won’t change when their YTM changes, effective duration and modified duration are the same thing.

Thank you for the clarification S2000 :+1: :slightly_smiling_face:

So we’re using modifed=effective duration. Doesn’t the book’s description resemble Macaulay duration? I’m confused there - perhaps I don’t know the relationship of modified and Macaulay well enough.

I think of Macaulay as an actual number of years until cash flows are returned, and modified as the change in price for change in rates.

A liability payable in a single payment is, essentially, a zero coupon bond.

The Macaulay duration of a zero coupon bond is its time to maturity.

The modified duration of a zero coupon bond is its Macaulay duration (time to maturity) divided by (1 + YTM) where YTM means the yield for a single compounding period (usually 6 months). Thus, the modified duration is only slightly less than the time to maturity; using the time to maturity is simpler, and probably sufficiently accurate.

(Personally, I believe that you should use modified or effective duration for both the asset and the liability.)

Thanks guys. But I still have doubts.

  1. Doesn’t the assets (portfolio) and the liabilities move in different directions when rates change? Example: if rates go up, the portfolio value would go down, but if we had a liability linked to interest rates, it would go up. How are we guaranteeing we will be able to pay the obligation in the future by matching the duration of assets with the liability maturity?

  2. We are talking about effective duration or Macaulay duration? I recognize that for a liability payable in one period they are equal, but what if the liability is not due in one period? We would use Effective duration, correct?

  3. I continued on the chapter, and it appears they suggest 4 alternative ways to manage a portfolio against a liability. I thought Classical Immunization took in consideration the liabilities too. What am I missing here? I thought the main flaw of the classical immunization was assuming parallel shifts in yield curve, so not considering yield curve movements.

Thanks guys.

Sounds to me this is your first time through fixed income readings. Well, clearly it is. I struggled too but have a decent grasp now. The logic behind your first question is what gave it away by you not understanding why the PV of liabilities goes down like the PV of assets when rates go up. Read it again. And then again.

Or wait for s2000 to tell you lol. I’m studying just like you so I don’t have time to rehash that entire reading for you on a forum. Your questions are loaded with needed explanation.

Your assets don’t know that they’re assets; they think that they’re bonds.

Your liabilities don’t know that they’re liabilities; they think that they’re bonds.

When interest rates rise, bond prices fall, whether you bought the bond or issued it.

When interest rates fall, bond prices rise, whether you bought the bond or issued it.

Unless the cash flows will change when the yield changes, effective duration and modified duration are equal. And for a liability that comprises a single payment – which is nothing more nor less than a zero coupon bond – modified duration is the time to maturity divided by (1 + YTM), where YTM is the yield for one compounding period (typically 6 months). In short, the difference between effective duration and Macaulay duration is pretty tiny.

All immunization take liabilities into account; that’s the point.

I’m not sure what you’re missing, if anything.

You’re welcome.