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# Reinvestment rate w.r.t Term structure of yield curve

#### Arka Bose

##### Active Member
Hi @David Harper CFA FRM and all,

This is not related to FRM, but still concept applies, so I am asking here. In the CFA L3 textbook, this para is given:

"In general, for an upward-sloping yield curve, the immunization target rate of return will be less than the yield to maturity because of the lower reinvestment return. Conversely, a negative or downward-sloping yield curve will result in an immunization target rate of return greater than the yield to maturity because of the higher reinvestment return."

In the glossary: "the immunization target rate of return is defined as “the assured rate of return of an immunized portfolio, equal to the total return of the portfolio assuming no change in the term structure.”

Can anyone tell me why in an upward sloping term structure, the reinvestment rate is lower?
I have tried finding similar in Bruce Tuckman's book but no luck.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @arkabose Really interesting! I had to stare at it for a bit but I think I get it. My understanding is reflected in this tiny XLS https://www.dropbox.com/s/t8p6m9268dkfk5j/0208-immunize.xlsx?dl=0 (snapshot below)
The immunization target return is (basically) a certain realized yield even if the rate curve shifts; to immunize, the portfolio's duration must be rebalanced so it keeps matching the maturity of the immunized liability (this may not be in Tuckman, but it's in Fabozzi for sure); e.g., if the liability is $X due in 3 years, a portfolio with (modified) duration of 3.0 years will balance reinvestment risk with interest rate (duration) risk. Below is 3-year bond with 2.0% annual coupon and three different spot rate curves: flat at 2.0%, upward sloping (1, 2, 3%) and downward sloping. In the flat rate case, the bond's yield of 2.0% will equal its realized yield because the coupons will be reinvested at 2.0%. My "Future" column shows the realized cash flows (and the realized return/CAGR). Key idea: yield (YTM) is only realized if the coupons reinvest at the same yield. So the second scenario shows how, if the curve is upward sloping and further we have an unchanged term structure, the realized yield will be less than the YTM (2.960 2.947 vs 2.973). Why? because of the unchanged term structure: go forward one year, and the first$2.00 coupon is not invested at 3.0% (nearer to the yield) but at 2.0% because the bond matures in 2.0 years. Go forward two years, and the next coupon reinvests at only 1.0%. Hence the meaning of "because of the lower reinvestment return" which means, put another way, an unchanged and upward-sloping term structure implies coupons are reinvested at progressively lower rates.

Put even more simply, a portfolio's yield will (mostly, depending on coupon rate) be near to its maturity (e.g., a 3 year bond has a mod duration of > 2.6 years) but the guaranteed, realized return (i.e., immunization target) is going to be lower, in this scenario, because the coupons will be reinvested at less than the yield (if coupons reinvest at less than yield, realized yield is less than YTM). I hope that helps, i did this quickly so i can't say it's impervious, I'll take any feedback, thanks!

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#### Arka Bose

##### Active Member
Thanks a lot David! The unchanged term struture is the main thing to look at i got it now!

#### Arka Bose

##### Active Member
I have one question though. Lets look at case 2, the cash flow at the end of second year \$50, in your excel, it seems u have taken 3% rate. Shouldnt the reinvestment rate be 1% as only 1 year is remaining and we are having unchanged TS? Similar thing u have done in case 3

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@arkabose Yes agreed, my mistake copying down, good catch. I think you got it! Fixed in XLS and above (added the reinvestment rate column). With that correction, the directional outcome is the same but more pronounced, yay