<p>Set the inequality <span class="math-container">$P&gt;Par$</span>, substitute the bond price equation, and then manipulate the inequality to find your result:</p>

<p><span class="math-container">$$P&gt;Par$$</span>
<span class="math-container">$$\frac{c}{y}\left(1-\frac{1}{(1+y)^T}\right)+\left(\frac{Par}{(1+y)^T}\right)&gt;Par$$</span>
<span class="math-container">$$\frac{c}{y}\left(1-\frac{1}{(1+y)^T}\right)&gt;Par\left(1-\frac{1}{(1+y)^T}\right)$$</span>
<span class="math-container">$$\frac{c}{y}&gt;Par$$</span></p>

<p>Let <span class="math-container">$c=r_cPar$</span> so that <span class="math-container">$r_c$</span> is the coupon rate (this is the fraction of the Par value that the coupon represents. Then the last equality is equivalent to</p>

<p><span class="math-container">$$r_c&gt;y$$</span></p>

<p>Therefore, the first inequality is true if and only if all of the following inequalities are. In particular, the last inequality gives the result you are looking for.</p>

CC BY-SA 4.0
2019-04-18T23:40:48.180Z
2019-04-19T00:07:46.653Z
2019-04-19T00:07:46.653Z
Regio
<p>I know that if the coupon rate on a bond is larger than the yield-to-maturity, then the price must be higher than the par value. Yet I have the bond price equation <span class="math-container">$P=\frac{c}{y}\left(1-\frac{1}{(1+y)^T}\right)+\frac{Par}{(1+y)^T}$</span>, and I wonder how I manipulate this equation to show that what I mentioned above is correct. Grateful for any tips to solve this. </p>

<mathematical-economics><finance><financial-economics>
Financial economics: Bond Price
Dogge
# Financial economics: Bond Price

(Asked by: username_0 on 2019-04-03)

I know that if the coupon rate on a bond is larger than the yield-to-maturity, then the price must be higher than the par value. Yet I have the bond price equation $P=\frac{c}{y}\left(1-\frac{1}{(1+y)^T}\right)+\frac{Par}{(1+y)^T}$, and I wonder how I manipulate this equation to show that what I mentioned above is correct. Grateful for any tips to solve this.

**Tags:** mathematical-economics, finance, financial-economics

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## 1 answers

### Answer by username_1 on 2019-04-18. Score: 1

Set the inequality $P>Par$, substitute the bond price equation, and then manipulate the inequality to find your result:

$$P>Par$$
$$\frac{c}{y}\left(1-\frac{1}{(1+y)^T}\right)+\left(\frac{Par}{(1+y)^T}\right)>Par$$
$$\frac{c}{y}\left(1-\frac{1}{(1+y)^T}\right)>Par\left(1-\frac{1}{(1+y)^T}\right)$$
$$\frac{c}{y}>Par$$

Let $c=r\_cPar$ so that $r\_c$ is the coupon rate (this is the fraction of the Par value that the coupon represents. Then the last equality is equivalent to

$$r\_c>y$$

Therefore, the first inequality is true if and only if all of the following inequalities are. In particular, the last inequality gives the result you are looking for.


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