<p>To make sense of your question, your <span class="math-container">$U$</span> should be
<span class="math-container">$$
U = u(C_1) + \beta (1-p)E_1 [ u(C_2) ] + \beta p E_1[u(Y_2 - T_2)].
$$</span>
(This definition of <span class="math-container">$U$</span> assumes that the default event is independent of other sources of uncertainty in the economy.)</p>

<p>If the bond defaults, saving <span class="math-container">$S_1$</span> from the first period is wiped out.
If there is uncertainty in period-2 production/endowment/transfer/etc, then period-2 consumption is still random in the event of default.</p>

<p>Assuming that agent only chooses consumption <span class="math-container">$(C_1)$</span> (therefore <span class="math-container">$C_2$</span> in the event of no default), then real interest rate <span class="math-container">$r$</span> is determined by
<span class="math-container">$$
1 = E_1[\beta \frac{u'(C_2)}{u'(C_1)} (1-p) (1 + r)].
$$</span>
This is just an extension of the standard Euler equation for interest rate
<span class="math-container">$$
1 = E_1[\beta \frac{u'(C_2)}{u'(C_1)} (1 + r)]
$$</span>
when <span class="math-container">$p = 0$</span>.</p>

<p>If equilibrium consumption remains the same (e.g. equals exogenous endowment), then higher probability of default <span class="math-container">$p$</span> leads to higher interest rate <span class="math-container">$r$</span>, as you would expect. High probability of default incentivizes agent to consume today instead of saving for tomorrow. To keep him from borrowing and clear the bond market, equilibrium interest rate must be higher. 
Equivalently, higher default risk means agent demands higher liquidity premium on bonds.</p>

<p>If <span class="math-container">$p$</span> and <span class="math-container">$r$</span> are small, <span class="math-container">$(1-p)(1+r) \approx e^{r-p}$</span>, which implies that <span class="math-container">$r$</span> moves one-to-one with respect to <span class="math-container">$p$</span>.</p>

CC BY-SA 4.0
2020-06-05T14:44:53.527Z
2020-06-06T16:49:40.577Z
2020-06-06T16:49:40.577Z
Michael
<p>So I want to calculate the effect of default risk on the required interest of bonds (not on the price of bonds as that is normalized to one). </p>

<p>I thought of using the consumption capital asset model and calculating the following.</p>

<p><span class="math-container">$$ U = u(C_1) + \beta (1-p)E_tu(C_2) + \beta p u(C_2) $$</span></p>

<p>Where the following applies:</p>

<p><span class="math-container">$$ p = no \; default ,\; (1-p) = default $$</span> </p>

<p>and the following budget constraints apply:</p>

<p><span class="math-container">$$C_1 = Y_1 - T_1 - S_1$$</span>
<span class="math-container">$$C_2 = Y_2 - T_2 + (1+r)S_1$$</span></p>

<p>I got to the following result using Lagrange, such that the consumer maximizes utility by deciding on how much to invest:</p>

<p><span class="math-container">$$1 = \frac{\beta E_t[U'(C_2)p(1+r)]}{(u'(C_2)}$$</span></p>

<p>However, I want to get it ultimately in an equation that uses covariances but I do not know how. </p>

<interest-rate><default>
Effect of default risk on the interest rate of bonds
Mart
# Effect of default risk on the interest rate of bonds

(Asked by: username_0 on 2020-06-05)

So I want to calculate the effect of default risk on the required interest of bonds (not on the price of bonds as that is normalized to one).

I thought of using the consumption capital asset model and calculating the following.

$$ U = u(C\_1) + \beta (1-p)E\_tu(C\_2) + \beta p u(C\_2) $$

Where the following applies:

$$ p = no \; default ,\; (1-p) = default $$

and the following budget constraints apply:

$$C\_1 = Y\_1 - T\_1 - S\_1$$
$$C\_2 = Y\_2 - T\_2 + (1+r)S\_1$$

I got to the following result using Lagrange, such that the consumer maximizes utility by deciding on how much to invest:

$$1 = \frac{\beta E\_t[U'(C\_2)p(1+r)]}{(u'(C\_2)}$$

However, I want to get it ultimately in an equation that uses covariances but I do not know how.

**Tags:** interest-rate, default

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## 1 answers

### Answer by username_1 on 2020-06-05. Score: 2 | Accepted answer

To make sense of your question, your $U$ should be
$$
U = u(C\_1) + \beta (1-p)E\_1 [ u(C\_2) ] + \beta p E\_1[u(Y\_2 - T\_2)].
$$
(This definition of $U$ assumes that the default event is independent of other sources of uncertainty in the economy.)

If the bond defaults, saving $S\_1$ from the first period is wiped out.
If there is uncertainty in period-2 production/endowment/transfer/etc, then period-2 consumption is still random in the event of default.

Assuming that agent only chooses consumption $(C\_1)$ (therefore $C\_2$ in the event of no default), then real interest rate $r$ is determined by
$$
1 = E\_1[\beta \frac{u'(C\_2)}{u'(C\_1)} (1-p) (1 + r)].
$$
This is just an extension of the standard Euler equation for interest rate
$$
1 = E\_1[\beta \frac{u'(C\_2)}{u'(C\_1)} (1 + r)]
$$
when $p = 0$.

If equilibrium consumption remains the same (e.g. equals exogenous endowment), then higher probability of default $p$ leads to higher interest rate $r$, as you would expect. High probability of default incentivizes agent to consume today instead of saving for tomorrow. To keep him from borrowing and clear the bond market, equilibrium interest rate must be higher.
Equivalently, higher default risk means agent demands higher liquidity premium on bonds.

If $p$ and $r$ are small, $(1-p)(1+r) \approx e^{r-p}$, which implies that $r$ moves one-to-one with respect to $p$.


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