# YTM on a leveraged bond position, bond not trading at par

I have to build a tool for my clients so they can see the YTM for a bond taking into account the leverage we offer them. I will keep it simple for the purpose of this discussion.

Rate Charged on Loan= 0

Equity Amount=100,000

Borrowed Amount=100,000

Bond YTM 5% trading at par, dirty price=clean price=100

The YTM of the position in this case is simply 5*(200/100)=10%

The problem I am facing is when the bond is not trading at PAR, this simple calculation no longer holds, I confirm this by laying out the cash flows of the bond along with the leverage (double the coupons, at maturity make a loss/gain which gets magnified with leverage). While doing an IRR on the cashflows is the correct YTM of the position, I do not want to do this approach as it complicates the understanding for the clients. Is there a mathematical way i can arrive at the solution?

Bonus question, what is exactly causes this issue when bond is not trading at PAR, I know it is the nature of the IRR but I just cant put an explanation together…

Thanks

The NPV of your cash flows has to equal zero with IRR. It does not double because you pay 5k less at origination and dont get that money until the end and IRR is a money weighted return. If you take the PV of each cash flow & sum them up using your IRR discount rate the NPV will equal zero. If you do the same but use your “unlevered” rate *2 your NPV will be negative because this is no longer the balancing figure needed. If you look at the difference between the values for each cash flow you will start to under the nature of IRR. It is a plug/balancing figure everyone has agreed is important. The fact that it doubles when at par is a special case. Im sure S2k can probably give a much better theoretical mathmatical answer.

Sorry if this isnt a fantastic response, i can add more later.

thanks, been a few busy days. I will read in details. Just wanted to take the time out to that you.