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I thought I had a pretty good grasp on how to calculate this but I'm getting questioned on it and just want to be sure I'm not getting it mixed up. In my notation you enter into the repo contract at $t=t_0$ and exchange a cash nominal, C, for bonds where C is determined by $C=B_{t_0} (1-HC)$ where HC is the haircut, e.g. 2%. The maturity of the repo is at $t_2$

The way I view it is that the present value of the repo at $t_1$ would be the value of the cash leg up to $t_2$ minus the cost of closing out the repo by entering into an opposite repo agreement from $t_1$ to $t_2$. The value of the cashleg of the reverse repo at that point would be $B_{t_1} (1-HC)$ and you get interest at the prevailing repo rate at the time that may be different from that of the first repo. I have put the calculations below. Is this correct? I am asking because I am being told by industry professionals that the value of the bond at the time doesn't affect the present value, only the risk. As a side note, if I am doing this correctly should you also discount this value from $t_2$ to $t_1$ to get the PV?

\begin{equation} PV = C_{t_2} - B_{t_1} (1 - HC) \exp{(r_2(t_2-t_1))} \end{equation} \begin{equation} C_{t_2} = C_{t_0} \exp{(r_1(t_2-t_0))} = B_{t_0} (1-HC) \exp{(r_1 (t_2-t_0))} \end{equation}

C = Cash leg value

B = Bond leg value

HC = haircut

PV = present value

$r_i$ = prevailing repo rate at time $t_i$

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As you have been advised, the value of the bond at $t_1$ is not relevant. This is because in any repo, the amount of bonds posted changes on a daily basis to maintain the haircut at the correct level. The amount of cash lent in the repo does not change. Hence , what matters is how the repo rate has changed from $t_0$ to $t_1$. For example , if the initial repo was at Fed Funds+ 25 and now it is Fed Funds + 20, the value of the repo is now the PV of 5bp over the remaining life of the repo.

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  • $\begingroup$ "This is because in any repo, the amount of bonds posted changes on a daily basis to maintain the haircut at the correct level." I did not know this. So this in effect means that in my calculations above B_{t_1] (1−HC) should in fact just be C_{t_1}, as the B_{t_1} will be the value of more (less) bonds in order to make it so? Is this always the case or can a repo be constructed as I have done above? When looking at valuations of repos and comparing the two calculations I find that mine is the one that is almost always more in line. $\endgroup$
    – Oscar
    Commented May 7, 2020 at 9:23
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    $\begingroup$ What I’ve described is standard practice - I’m not sure if your alternative structure has been executed at some time. $\endgroup$
    – dm63
    Commented May 8, 2020 at 22:54
  • $\begingroup$ Thank you, after looking into it some more I think you're right. I think in practice it is the case that additional collateral is posted as cash though, not extra bonds? The math should be unchanged either way though, right? $\endgroup$
    – Oscar
    Commented May 9, 2020 at 4:46
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    $\begingroup$ Yes, the additional collateral could be cash or bonds , but it doesn’t change the math $\endgroup$
    – dm63
    Commented May 9, 2020 at 10:59

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