Quantitative Methods - Discrete and Continuous Compounding
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
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.
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
A general outline for developing a
- 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.
- Draw K random numbers using a spreadsheet generator. Each random variable would then be standardized so we have Z1, Z2, Z3... ZK.
- Simulate the possible values of the random variable by calculating its observed value with Z1, Z2, Z3... ZK.
- 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, 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.
Personal FinanceMany advisors display similar skillsets that can make distinguishing between them difficult. The following guidelines can help you better understand their qualifications and services.
InvestingManagers, in developing their investment process, need to determine some “general rules” that make it meaningful. We offer six.
ProfessionalsRead an in-depth comparison about working as a Financial Analyst vs. working as an Investment Banker, two highly prestigious business careers.
ProfessionalsThe right advisor credentials can make all the difference, but wading through some 100 certifications can be a challenge. Here's some help.
Investing BasicsAsset managers must accurately represent all of a clients assets in the client portfolio. This can be tricky for unique and hard-to-value assets.
Personal FinanceCleats, pads and profits: we take a look at the top 10 most valuable sports teams in the world.
ProfessionalsThe Chinese economy's ongoing slowdown is having a major impact on iron ore demand.
Personal FinanceA strong balance sheet sets a company apart and boosts investor confidence. How healthy is Costco based on an analysis of its balance sheets from the last two years?
Investing BasicsBrokers and registered investment advisors have some key differences. Here's what you need to know.
ProfessionalsDCF and Comparables models are widely used in equity valuation. We explain the pros and cons of each method.
Professionals who help individuals manage their finances by providing ...
Formerly known as the Association for Investment Management and ...
A professional designation given by the CFA Institute (formerly ...
A financial professional who studies various industries and companies, ...
The differences between a Chartered Financial Analyst (CFA) and a Certified Financial Planner (CFP) are many, but comes down ... Read Full Answer >>
According to the CFA Institute, a person who holds a CFA charter is not a chartered financial analyst. The CFA Institute ... Read Full Answer >>
The types of positions that a Chartered Financial Analyst (CFA) is likely to hold include any position that deals with large ... Read Full Answer >>
Prepaid expenses benefit both businesses and individuals. Prepaid expenses are the types of expenses that are bought or paid ... Read Full Answer >>
If you are looking specifically for an investment banking position, an MBA may be marginally preferable over the CFA. The ... Read Full Answer >>
You may still pass the Chartered Financial Analysis (CFA) Level I even if you fare poorly in the ethics section, but don't ... Read Full Answer >>