# Immunization Rate

What exactly does the immunization rate represent? The minimum return of the assets to meet the future liabilites?

Also - on page 41 of CFAI volume 4, the text describes when the YTM drops to 375% the porftolio value is now \$541.36, how is this amount being derived?

asked and answered the exact same question just yesterday. search for 541.36 on the site and you shall be provided the answer…

For that value you would use— FV 500 mill. IY- 1.875 (3.75/2) N- 20 (10 year note semi annual) Pmt- 11.875 (500*.0475/2)

541.36

Is this continigent immunization stuff that critical? I keep getting turned around with it with what time period to use for which and other things. Does anyone have tips or a step by step or should I not even worry about it?

It’s critical only if it appears on the exam in June.

what is the immunization rate?

The rate you require to meet your obligation.

If you have a €2,000,000 liability due in 3 years and you have €1,800,000 today, then your (effective annual) immunization rate is 3.5744%, because

€1,800,000 × 1.035744³ = €2,000,000.

I know the objective of all this is to build a bond portfolio that produces an assured rate of return that is immune from (parallel) interest rate fluctuations in order to match the liability.The curriculum defines the immunization target rate of return as the total return of the portfolio assuming no change in the term structure… I understand that YTM is the single discount rate used to discount all CFs such that it is equal to the bond price… but what exactly is the immunization target rate of return and what is the purpose of determining this rate?

The curriculum also states that for an upward-sloping yield curve, the immunization rate will be less than the YTM because of lower reinvestment return… but wouldn’t the rate be greater since an upward curve imply that coupons will be reinvested at subsequently higher (upward) rates relative to YTM where the yield curve is flat? The reinvestment rate (interest on interest) is the only component of total return that is subject to changes in the yield curve, is this correct?

I am clearly confused as even phrasing this question was a challenge… any help would be much appreciated.

Thank you…

If you look at my post (two above this one), you’ll see that the immunization rate is the minimum rate you need to earn to meet your objective. In that example, if we can earn exactly 3.5744% per year for three years, we’re fine. If we can earn more than 3.5744%, we’re on velvet: we’ll pay off our liability and have enough left over to throw a party.

The purpose of determining that 3.5744% is the immunization rate is to have a basis against which to compare our actual return. If we’re earning 3.5%, we should be worried; if we’re earning 4%, we should be delighted.

As for the yield curve: suppose that you have a par yield curve with the following rates:

• 1-year: 2.0%
• 2-year: 3.0%
• 3-year: 3.8%
• 4-year: 4.4%
• 5-year: 4.8%
• 6-year: 5.0%

You buy a 6-year, annual-pay bond with a coupon of 5% (so it’s selling at par). For you to earn 5% on your investment, you have to reinvest the coupons at 5%. If the par curve doesn’t change over the next year, then you’ll get a coupon payment, which you’ll invest for 5 years (the remaining time to maturity on your bond). However, the 5-year par rate is only 4.8%, not 5.0%. The next year, if the par curve doesn’t change, you’ll reinvest the coupon payment at only 4.4%. And so on.

The upshot: your realized yield will be less than 5%.

(Here’s a question: what, exactly, will it be?)

(I inputted a table but it did not show correctly)

Principal = 10,000

Coupon = 6 years x 500 = \$3000

Reinvestment Return (Total = \$90)

Yr 1 - 4.8% x 500 = 24

Yr 2 - 4.4% x 500 = 22

Yr 3 - 3.8% x 500 = 19

Yr 4 - 3.0% x 500 = 15

Yr 5 - 2.0% x 500 = 10

My total would be 10,000 + 3000 + 90 = \$13090 and my realized return would be (13090/10000)^1/6 = 4.59%

Is this correct (or do I need to account for interest on interest)? and if so, 4.59% would be the immunization rate?

Thank you so much for your help, s2000!!

You need to account for interest on interest.

The first coupon is reinvested at 4.8% for 5 years.

The second coupon is reinvested at 4.4% for 4 years.

And so on.

I got 4.9015%.

The point is that with an upward-sloping yield curve, your realized return is less than the 5% YTM.

This doesn’t have anything to do with the immunization rate; that rate is determined by how much you have to invest today compared to what you need in the future.

(13325.71/10000)^1/6 = 4.9015%

Thank you for your invaluable help!

You’re quite welcome.

I’ll be writing some articles on this stuff pretty soon.

Really look forward to them as your notes on FEH123 have helped me tremendously

That’s good to hear.

so why does the return have to be less than the immunization rate in a continent immunization plan? Wouldn’t you think it would have to be more if you want to continue to actively manage?

I’m not sure that I follow this.

There are three rates of interest in contingent immunization:

• Required rate of return
• YTM of the portfolio
• Actual (total) return on the portfolio

As long as the actual return on the portfolio exceeds the required return, you can continue active management; as soon as the actual return drops to or below the required return, you must immunize.

With an upward-sloping yield curve, the actual return on the portfolio will be less than the YTM. That doesn’t really matter.

(As an aside, if you’ve seen my avatar, you know that I would rather be riding as well.)

So are the required return and immunization rate the same?

Yes.