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I'm new to quantitative finance, and trying to derive an interest rate for a collateralized bond.

Imagine there are two parties, Alice and Bob. Alice wants to lend $X$ units of an asset to Bob. The loan matures in one year, at which point Bob returns $X \cdot (1+R)$ units to Alice.

The asset has an expected return $r$ and variance $\sigma^2$.

Now, it is possible that Bob defaults and does not return the asset. Because of this, he has to post some collateral with Alice of a different asset. This other asset has return $r'$ and variance $\sigma^{2\prime}$. Bob posts $C$ units of this asset as collateral.

My question is, is there a formula to determine $R$ as a function of $X,r,\sigma^2,r',\sigma^{2\prime}$ and $C$? If not, what other information would I need?

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    $\begingroup$ In practice for something to be used as collateral it has to be nearly risk-free i.e. $\sigma_C^2 \approx 0$. The lender usually does not accept risky collateral, that is the point of collateralization: to provide something of known value. $\endgroup$ – noob2 Jun 22 at 16:26

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