# How does the yield of a floater change when the discount/required margin changes?

On this site: https://ebrary.net/14293/economics/actual_floater, it says that the yield of a floater is deteremined like this:

That yield is determined by assuming the coupon rate on the floater is swapped to a synthetic fixed rate and then solving for the internal rate of return.

But how is actually the synthetic bond created? Lets say that the disount/required margin of a floater increases by on basis point. Then the price of the floater decreases. How would one then find an equivalent fixed rate security?

I assume that market value must be the same for the synthetic bond and the floater. So in the synthetic security we can either increase the yield, or the yield can stay the same and we can decrease the coupon payments, or a comination?

Is there any rule to how the yield of a floater changes when the discount margin changes?

This is not how most people calculate the yield of a floater.

The way most people calculate the yield of a floater is:

1 for each remaining unset coupon, project the values of the index that will be used (such as 3Mo LIBOR, daily SOFR, SONIA, ESTR, etc - see Forecast 3m LIBOR USD. Budget purpose for example); and project the coupons. For example, if a floater has a coupon that in 1 year is reset from 3 months LIBOR plus 100 basis points ("quoted margin"), and if the projection curve predicts that 3 months LIBOR will be 30 basis points when this coupon is reset, then you project that the coupon will be 30+100 = 130 basis points.

2 Just as you would for a fixed-coupon instrument, solve for the internal rate of return (IRR) of the cash flows where you pay the dirty price on setlement date and receiver the set coupon(s), the unset coupons projected in step 1, and principal repayments.

Various spreads that make sense for fixed-coupon bonds (Z-spread, OAS, etc) work just as well for floaters, discounting the future cash flows by your bond funding cost. You can also calculate the discount margin (DM), which is similar to these spreads.

(Sometimes people calculate "flat yield" and "flat discount margin" by taking the current value of the index and assuming that it will remain constant, so you don't need to project it from the curve in step 1. This is the default behavior of the Bloomberg Terminal - a setting which you may want to change.)

The yield of a floater changes every time the projection curve changes, so, unlike fixed-coupon bonds, floaters are almost never quoted on yield. Floaters are usually quoted on price or on DM. Backing out price from DM (the inverse of the price to DM calculation) is depends somewhat on the projection curve assumptions.

If you want to reprice the bond under various risk scenarios, then you should not assume that your discount curve (funding) and the projection curve (used to predict unset coupons) are the same. But the two curves will be correlated. In particular, for a dv01 scenario, you may want to perturb both discount and projection curves by the same 1 basis point up and down.

If the DM moves (e.g. because the bond issuer's credit changes), then you back out the new bond price (the inverse of the price to DM calculation), and use the new price to calculate the new yield, Z-spread, and other spreads, which will all change by about the same amount as the perturbed DM.

As you see, there's no synthetic fixed-coupon bond in this picture. You could make one up, but I don't see how that would benefit anyone in understanding the value of the floater.

• Thank you!, I just want to say that I am studying this and not working with it, so I don't use a Bloomberg terminal etc.. I have some quick questions for you. 1. Could you say a little about point 1, how are the index projected? 2. Can you please say a little about what you mean about "If you want to reprice the bond under various risk scenarios, then you should not assume that your discount curve (funding) and the projection curve (used to predict unset coupons) are the same." Is the discount curve the yield curve, and the projection curve the values of the future projected index values? – arnis Dec 27 '20 at 16:28
• 3. The relationship between the marketvalue, the DM and the yield is seen here: ebrary.net/14292/economics/… in equation 7.7. In theory we assume we don't have the new price, and so both the yield and the market value is not known, then I don't see how we can do what you describe in your second to last paragraph? – arnis Dec 27 '20 at 16:28
• OK, I will edit my answer to add more details. I must admit, I don't like like what's written on tge pages you linked. – Dimitri Vulis Dec 27 '20 at 16:45
• “Backing out price from DM (the inverse of the price to DM calculation) is depends somewhat on the projection curve assumptions”. This is a little misleading. Discount margin is always calculated using flat yield and no projection curve is involved. There are some forward DM measures (I think POINT was z-DM or something) but they aren’t used for quoting purposes. – Bond wiz Dec 28 '20 at 3:19
• @Bondwiz Thanks! Few things are "always" anything. :) If someone quotes frn dm to you, it's better to ask them whether they mean constant "assumed rate", or uses projection curve, rather than to assume. I've seen both - consistent within one guy's runs, but not necessarily within runs generated by different people on the same desk. It doesn't make a huge different anyway. – Dimitri Vulis Dec 28 '20 at 4:11