LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
Three finance drills in the lecture use three different tools. A simulated return that is normal with mean 7 percent and standard deviation 2 percent lands below 5 percent with probability 15.87 percent. A portfolio with weights 0.25, 0.35, and 0.40 on expected returns of 7, 9, and 11 percent has expected return 9.3 percent, computed as a dot product. An annuity due of 200 dollars at the start of each year, for five years, at 6 percent, has future value 1,195 dollars in the answer that was accepted. The lecture's finance rank on that practice site moved up one point, to 51. The rank is bookkeeping. The three numbers are the content.
Why it matters
Monte Carlo work in finance often draws returns from a named distribution and asks for a probability, a portfolio summary, or a cash-flow value. The normal law is fully described by its mean and variance, so a probability is an integral of a known density. Portfolio expected return is linear algebra once the weights and the means are given. An annuity due differs from an ordinary annuity by one extra period of interest, because the payment arrives at the beginning of the period. Mixing those tools up, or using the ordinary-annuity formula for a beginning-of-period payment, is how the drills are failed.
The idea in plain words
Let R be normal with mean 0.07 and standard deviation 0.02, so the variance is 0.02 squared. The density is the usual bell curve, scaled by the reciprocal of sigma and of the square root of two pi, then shaped by a Gaussian exponential in the squared distance from the mean. Its integral from minus infinity to plus infinity is 1. The probability that R is below 5 percent is the integral of that density from minus infinity to 0.05, with mu and sigma set to the given 7 percent and 2 percent. A numerical CDF returns 15.87 percent. The lecture submitted that figure. The integral is not left as a closed expression. The normal CDF does not have an elementary antiderivative. The portfolio vector of weights is a 3 by 1 column, and so is the vector of expected returns. Those shapes do not multiply. Transpose the weights into a row, (0.25, 0.35, 0.40), and multiply by the column (0.07, 0.09, 0.11). The inner dimensions are now 1 by 3 and 3 by 1, and the product is the single number 0.25 times 0.07, plus 0.35 times 0.09, plus 0.40 times 0.11, which equals 9.3 percent. Writing the same sum without matrices is correct and, the lecture argues, a bad habit. A decade of asset returns is the same product with a larger matrix. The drill is the small case of that product.
How to use it
For a normal probability, standardize or hand the integral to a CDF routine. Do not invent a tail number. The 15.87 percent is the routine's value at 5 percent for this mean and this standard deviation. For a portfolio mean, check that weights sum to 1 before you multiply. These do: 0.25 plus 0.35 plus 0.40 is 1. The product is then a weighted average, not a mix of unrelated units. For the annuity, mark whether cash moves at the beginning or the end. Beginning means multiply the ordinary future value by 1.06 in this problem. End means stop before that multiplication.
The lecture had just come from a probability rank it described as 53rd and treated these items as first-level finance. Linear algebra is included on purpose. The transpose is the part candidates skip.
What usually goes wrong
Using 2 percent as the variance instead of the standard deviation. The variance is the square. The density's sigma is 0.02, not 0.0004. Multiplying the two 3 by 1 vectors without a transpose, which the shapes forbid. And applying the ordinary annuity formula to a payment that the problem places at the beginning of the year. The lecture says that single mistake selects the wrong option even when the rest of the algebra is tidy. A calculator result of about 1,195.95 rounded to the listed 1,195. Reporting more precision than the choices offer is not required.
Questions people ask
Why is the normal probability 15.87 percent?
Because the CDF of a normal law with mean 7 percent and standard deviation 2 percent, evaluated at 5 percent, returns that figure. Five percent is one standard deviation below the mean, and the lecture's calculator is what was graded.
What is the matrix product in words?
A quarter of 7 percent, plus 35 percent of 9 percent, plus 40 percent of 11 percent. That sum is 9.3 percent.
Why is the annuity due larger?
Each 200 dollar payment is deposited at the start of the year, so it earns 6 percent for one more year than the matching end-of-year payment. The accepted future value is 1,195 dollars.