In discrete compounded rates of return, time moves forward in increments, with each increment having a rate of return (ending price / beginning price) equal to 1. Of course, the more frequent the compounding, the higher the rate of return. Take a security that is expected to return 12% annually:
• With annual holding periods, 12% compounded once = (1.12)1 - 1 = 12%.
• With quarterly holding periods, 3% compounded 4 times = (1.03)4 - 1 = 12.55%
• With monthly holding periods, 1% compounded 12 times = (1.01)12 - 1 = 12.68%
• With daily holding periods, (12/365) compounded 365 times = 12.7475%
• With hourly holding periods, (12/(365*24) compounded (365*24) times = 12.7496%
With greater frequency of compounding (i.e. as holding periods become smaller and smaller) the effective rate gradually increases but in smaller and smaller amounts. Extending this further, we can reduce holding periods so that they are sliced smaller and smaller so they approach zero, at which point we have the continuously compounded rate of return. Discrete compounding relates to measurable holding periods and a finite number of holding periods. Continuous compounding relates to holding periods so small they cannot be measured, with frequency of compounding so large it goes to infinity.

The continuous rate associated with a holding period is found by taking the natural log of 1 + holding-period return) Say the holding period is one year and holding-period return is 12%:

ln (1.12) = 11.33% (approx.)

In other words, if 11.33% were continuously compounded, its effective rate of return would be about 12%.

Earlier we found that 12% compounded hourly comes to about 12.7496%. In fact, e (the transcendental number) raised to the 0.12 power yields 12.7497% (approximately).

As we've stated previously, actual calculations of natural logs are not likely for answering a question as they give an unfair advantage to those with higher function calculators. At the same time, an exam problem can test knowledge of a relationship without requiring the calculation. For example, a question could ask:

Q. A portfolio returned 5% over one year, if continuously compounded, this is equivalent to ____?

A. ln 5
B. ln 1.05
C. e5
D. e1.05

The answer would be B based on the definition of continuous compounding. A financial function calculator or spreadsheet could yield the actual percentage of 4.879%, but wouldn't be necessary to answer the question correctly on the exam.

Monte Carlo Simulation
A Monte Carlo Simulation refers to a computer-generated series of trials where the probabilities for both risk and reward are tested repeatedly in an effort to help define these parameters. These simulations are characterized by large numbers of trials - typically hundreds or even thousands of iterations, which is why it's typically described as "computer generated". Also know that Monte Carlo simulations rely on random numbers to generate a series of samples.

Monte Carlo simulations are used in a number of applications, often as a complement to other risk-assessment techniques in an effort to further define potential risk. For example, a pension-benefit administrator in charge of managing assets and liabilities for a large plan may use computer software with Monte Carlo simulation to help understand any potential downside risk over time, and how changes in investment policy (e.g. higher or lower allocations to certain asset classes, or the introduction of a new manager) may affect the plan. While traditional analysis focuses on returns, variances and correlations between assets, a Monte Carlo simulation can help introduce other pertinent economic variables (e.g. interest rates, GDP growth and foreign exchange rates) into the simulation.

Monte Carlo simulations are also important in pricing derivative securities for which there are no existing analytical methods. European- and Asian-style options are priced with Monte Carlo methods, as are certain mortgage-backed securities for which the embedded options (e.g. prepayment assumptions) are very complex.

A general outline for developing a Monte Carlo simulation involves the following steps (please note that we are oversimplifying a process that is often highly technical):
1. Identify all variables about which we are interested, the time horizon of the analysis and the distribution of all risk factors associated with each variable.
2. Draw K random numbers using a spreadsheet generator. Each random variable would then be standardized so we have Z1, Z2, Z3... ZK.
3. Simulate the possible values of the random variable by calculating its observed value with Z1, Z2, Z3... ZK.
4. Following a large number of iterations, estimate each variable and quantity of interest to complete one trial. Go back and complete additional trials to develop more accurate estimates.
Historical Simulation
Historical simulation,
or back simulation, follows a similar process for large numbers of iterations, with historical simulation drawing from the previous record of that variable (e.g. past returns for a mutual fund). While both of these methods are very useful in developing a more meaningful and in-depth analysis of a complex system, it's important to recognize that they are basically statistical estimates; that is, they are not as analytical as (for example) the use of a correlation matrix to understand portfolio returns. Such simulations tend to work best when the input risk parameters are well defined.
Sampling and Estimation

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