Which of the following explain why financial instruments, such as bonds or loans, carry a default risk? Check all that apply.

A natural disaster

An unexpectedly large budget surplus

A firm that issued bonds expanded too quickly and its revenue fell

A firm that issued bonds expanded too quickly and its revenue rose above the expected level

Let $I{R}_{\text{m}\text{a}\text{r}\text{k}\text{e}\text{t}\text{,}i,t}$ be the market interest rate the borrower i pays at time $t$; let $I_{\text{r}\text{i}\text{s}\text{k}-\text{f}\text{r}\text{e}\text{e}\text{,}t}$ be the interest rate a borrower who has no default risk would pay

at time $t,$ and let $DR{P}_{i,t}$ be the default risk premium borrower i must pay at time $t.$

Which of the following represents the default risk premium $($DRP$)?$

$DR{P}_{i,t}=I{R}_{\text{m}\text{a}\text{r}\text{k}\text{e}\text{t}\text{,}i,t}$

$DR{P}_{i,t}=\frac{I{R}_{\text{m}\text{a}\text{r}\text{k}\text{e}\text{t}\text{,},t}-I{R}_{\text{r}\text{i}\text{s}\text{k}-\text{f}\text{r}\text{e}\text{e},t}}{I{R}_{\text{r}\text{i}\text{s}\text{k}\text{k}-\text{f}\text{r}\text{e}\text{,},}}$

$DR{P}_{i,t}=I{R}_{\text{m}\text{a}\text{r}\text{k}\text{e}\text{t},i,t}-I{R}_{\text{r}\text{i}\text{s}\text{k}-\text{f}\text{r}\text{e}\text{e}\text{,}t}$

$DR{P}_{i,t}=I{R}_{\text{m}\text{a}\text{r}\text{k}\text{e}\text{t},i,t}\frac{x}{x}$I $R_{\text{r}\text{i}\text{s}\text{k}-\text{f}\text{r}\text{e}\text{e}\text{,},t}$

Suppose that in December $2016,$ Redondo & Sons Corporation issues a bond that pays $7\mathrm{.}5\%.$ At the same time, the US Treasury, when it issues debt,

pays an interest rate of $3\%.$

Based on the formula for the default risk premium,

must pay a DRP of

because the rate on

is considered the risk$-$free rate.