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JOB MARKET ADVICE FOR MY STUDENTS University of New Orleans Tulane University Securities Business & Brokerage Firms Economic Analysis, Industry Analysis, Company Analysis How to set personal and professional goals.
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One of the “main ideas” of finance is that Money has a time value. “A dollar today is worth more than the promise of a dollar tomorrow.” The “promise” of the dollar tomorrow takes away the possibility of the use of that dollar today. The promise of the dollar tomorrow involves the risk that the dollar will not be delivered. Given the time and risk, you expect compensation for NOT having the money TODAY. The time value of money problems are used extensively in finance. ALL assets (investments, real estate, business ventures) are valued using time value techniques. The 6 equations of Time Value:
1. Future value of a single sum. Here is a time-value-related article that you may enjoy reading. It is about the horrible savings rate in the US and how we will pay dearly for our short-term thinking and the failure to save. Click here: America's Savings Rate Article.
The simplest example of the future value of a single sum is $100 on deposit for one year at 6% interest.
We could easily calculate the annual earnings to this deposit by
multiplying 0.06 (the decimal version of 6%) by $100.
We would reach an answer of $6 [.06 x $100]. The formula for the Future Value of a Single Sum is: P ( 1 + I )n Where P is the Principal. In our example, “P” equals $100. “I” equals the rate of interest, or the rate of return. But, “I” is actually more complex. It is: the interest rate divided by the number of compounding periods per year. In our example, the interest rate is 6%, the compounding period is 1; interest was paid at the end of the year, it was paid once. “n” is the time component of the equation. “n” equals: the number of compounding periods per year x the number of years. In our [simple] example, the number of compounding periods per year was “1” and the number of years was “1.” Therefore, the exponent was 1x1=1. The exponent of “1” needed not be written. A second example, which is not so simple, is: $2715.16 is on deposit for 6 ½ years at 8 ¼% interest compounded weekly. What is the amount in the account at the end of the term? The solution would appear:
$2715.16 ( 1+ .0825/52)6.5x52 = $4639.82.
The idea of present value is particularly significant in finance because so many financial decisions, personally or in business, evaluate a project, or an investment that has expectations of some return that is realized in the future. The challenge is to value that project or investment TODAY. Investments that are “throwing off” cash flows, such as a certificate of deposit or bond that pays periodic interest, those cash payments can be valued today, even if they are to be received later than today. The Present Value is an amount known in the future that is valued today [in the present]. The first formula that we learned for the Future Value, is used for the Present Value. The new formula is: P (1 + I )-n . The formula is the same as the Future Value with one clear difference, the exponent is negative. Remember that the exponent is the time variable in these equations. If the exponent is positive, as in Future Value calculations, time is positive. In Present Value, the exponent is negative, time is negative, and we are bringing a future, known amount, back through time to today. An example of the Present Value of a Single Sum: An investor wants to have $5000 in 4 years. If money is worth 8%, compounded monthly, how much would they have to deposit TODAY. $5000 (1 + .08/12)-4x12 = $3634.60 The answer is the amount that has to be deposited today, in an account with the interest rate specified, to reach the $5000 goal. The present value calculation takes the guess out of the goal. ********************************************************* The number of applications to the Future/Present value of an annuity is countless. Financial applications are what we are focused on, but any number that is growing, like populations, inflation, etc. can use this formula to predict, estimate or calculate future numbers. ********************************************************* ANNUITIES We have been looking at the first examples of time value in the simplest form, the present and future values of single sums. As we would begin a savings plan for ourselves, we could invest money that we already have, such as, money that we have in the bank. That would be a single sum calculation. As a practical matter, people save money in similar frequency to their paycheck. This regular “stream of payments” is known in finance as an Annuity. Annuities are best more easily calculated if a consistent amount is saved/invested in consistent intervals. For example, $50 could be saved monthly for 5 years at a rate of 8%.
[ P * ((1 +
i )n –1) As powerful as it is for calculating relatively lengthy problems, this formula was formed from the Future Value of a Single Sum formula that we first used. There is one assumption – the cash flow must equal to the compounding frequency. If the payment or deposit is monthly, the money must be compounded monthly. The actual compounding frequency may be greater than the one used in the calculation; this would make our answer to the problem more conservative than the actual. [For example, you could be making monthly deposits into a savings account at a local bank that pays interest compounded daily. Our calculation of that problem would produce a slightly smaller result than the actual. However the calculation of that problem would be far more cumbersome than it would be worth.] Here is an example: $150 is deposited into an account monthly that earns 8% interest for 5 years. What is the value in the account at the end of the term? 150 [( 1 + .08/12 )12x5 –1 ] / (.08/12) = $11,021.53 Another example: A person deposits $250 monthly into an account earning 7 ¾% {compounded monthly} for 42 years. What is the value of the account at the end of the term? 250 [( 1 + .0775/12 )12x42 -1] / (.0775/12) = $954,183.98.
[ P * (1- (1 + i )-n) ] / i The present value of an annuity can be used to obtain a single [present] value of a stream of payments. With a slight modification of the future value of an annuity formula, we can solve problems like this example: A person signs a lease with an apartment complex for 3 years, paying $550 per month in rent. With money worth 6% [compounded monthly] what is the present value of the 36 payments? 550 [ 1- ( 1 + .06/12 )-12x3 ] / (.06/12) = $18,079.06.
Take the concepts explained above and think of a compounding interval being infinitely fast. This is continuous compounding. There is an easy formula to handle what may be thought of as a complex problem: Pert Where "P" is principal.
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