Financial Risk Manager Part 1

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A zero-coupon bond with a notional value of $156 and price x has the following probability density function:

f(x) = \frac{x^2}{144000} \quad \text{for } 0 \leq x \leq 200

Determine the probability that the price of the bond is between $126 and $130 inclusively.

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NNikitesh

Last updated: February 2, 2026 at 10:22

0.4267

0.3621

0.4551

0.4088

Explanation:

Explanation

For a continuous probability density function $f(x)$ , the probability that $x$ lies between $a$ and $b$ is given by:

P(a \leq x \leq b) = \int_a^b f(x) \, dx

Given:

Step-by-step calculation:

\begin{aligned} P(126 \leq x \leq 130) &= \int_{126}^{130} \frac{x^2}{144000} \, dx \\ &= \frac{1}{144000} \int_{126}^{130} x^2 \, dx \\ &= \frac{1}{144000} \left[ \frac{x^3}{3} \right]_{126}^{130} \\ &= \frac{1}{144000} \left( \frac{130^3}{3} - \frac{126^3}{3} \right) \\ &= \frac{1}{144000} \cdot \frac{1}{3} \left( 130^3 - 126^3 \right) \\ &= \frac{1}{432000} \left( 2,197,000 - 2,000,376 \right) \\ &= \frac{1}{432000} \left( 196,624 \right) \\ &= 0.4551 \end{aligned}

Verification of calculations:

Therefore, the probability that the bond price is between $126 and $130 is 0.4551, which corresponds to option C.

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