# Pricing bond backed by collateral

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?

• 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. – noob2 Jun 22 '19 at 16:26