Present Value: How to Do It - New York University
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Present Value: How to Do It
Understanding present value is one of the most important topics in business school. With present value, we can figure out savings and retirement plans, price bonds and stocks, evaluate investment projects and, in general, understand the all important risk-return relationship of investing.
Present value calculations are used for monetary assets and liabilities -- assets and liabilities denominated in dollar terms. Examples are bonds, leases, pensions, and mortgages. In this class, we will concentrate on bonds and leases. However, the intuition and calculations are the same for all types of monetary assets and liabilities.
There is only one formula to remember. Every other formula is just an offshoot of the one formula:
(1) FV(n) = PV(0) (1+r)n ,
where FV(n) is the future value of an investment at time n, PV(0) is the present (today’s) value at time 0, time 0 is today, r is the interest rate, and n is n periods in the future.
For example, suppose I invest $1,000 today at an interest rate of 10%, compounded annually. Using formula (1), the $1,000 will be worth:
Year 1 $1,000 (1.10)1 = $1,100.
Year 2 $1,000 (1.10)2 = $1,210 or $1,100 (1.10)1.
Year 3 $1,000 (1.10)3 = $1,331 or $1,210 (1.10)1.
Note that I am compounding annually. This means that at the end of each year, I receive 10% interest on my entire amount, not just on the original $1,000 interest. The 10% received as interest is then treated as part of the investment. Thus, in year 1, the interest is $100 and the year-end investment (for the beginning of next year) becomes $1,100; in year 2, the interest is $110 (10% of $1,100); and in year 3, the interest is $121 (10% of $1,210).
These calculations are called future values, since it specifies the amount I expect to receive at some future date.
Present value is just the inverse of future value. It translates future values into today’s prices. Using equation (1):
(2) PV(0) = FV(n)/ (1+r)n or = FV(n) (1/ (1+r)n)
where 1/(1+r)n is the discount factor.
Using the example from before, I assumed an interest rate of 10%, compounded annually. Using formula (2), I can calculate present values as:
Investment in Year n Present Value PV(0) Discount Factor
Year 1 $1,100 $1,100 / (1.10)1 = $1,000 0.9091
Year 2 $1,210 $1,210 / (1.10)2 = $1,000 0.8264
Year 3 $1,331 $1,331 / (1.10)3 = $1,000. 0.7513
Note that in all cases, the present value is $1,000. What the above calculations mean is that $1,000 today, if compounded at 10% annually is equivalent to $1,100 in one year. Put differently, I can “price” a financial instrument that gives me $1,100 one year from now as $1,000 today. Similarly, a financial instrument that gives me $1,210 in two years from now can be priced at $1,000 today, assuming an interest rate of 10%, compounded annually.
An annuity is a series of equal cash flows over n periods, in which each cash flow is one period apart. For example, a five year annuity of $1,000 means that 5 payments of $1,000 are made at the end of each of the next five years. The present value of an annuity can be quickly calculated due to the fact that each payment is the same (in this example, $1,000). To calculate the present value of the annuity, I need to know the annuity factor, which is just the sum of the discount factors for the period (in this example, 3 periods).
The present value is $1,000 times the (Annuity Factor, n=3, r=10%). In this case, it is = $1,000 (2.4869) = $2,487. (0.9091+0.8264+0.7513=2.4869 [difference due to rounding]) Annuity factors are printed in tables.
Compounding over smaller intervals
Thus far, I have assumed a compounding rate of once a year. However, financial instruments usually compound over smaller intervals of time, for example, semi-annually, quarterly, daily, and even continuously. To do this:
1. Convert the annual interest rate into the appropriate time interval. Assume a 12% annual interest rate. To convert this to a semi-annual rate, divide the annual rate by 2: 12%/2 = 6%. To convert the annual interest rate to a quarterly rate, divide it by 4: 12%/4 = 3%; monthly compounding is 12%/12 = 1%; daily compounding is 12%/365 = .33%. Thus, the conversion formula is annual rate/number of times compounded during the year.
2. Convert the number of periods into appropriate number. Compounding means that I am collecting interest on my total principal + interest. When we compound yearly, we collect interest once a year. When I compound semi-annually, I collect interest twice a year. When I compound quarterly, I collect interest four times a year. When I compound monthly, I collect interest twelve times a year. Thus, I need a new n, call it n*, equal to n times the number of times we compound per year.
Example: Assume an annual interest rate of 12%, compounded quarterly. What is the future value of $1,000 invested over 3 years?
FV(n) = PV(0) (1+r)n ,
With yearly compounding, n = 3 and r = .12. With quarterly compounding, n* = 3*4 = 12 and r* (the newly converted interest rate) = .12/4 =.03 = 3%.
Thus, the future value is 1,000 (1.03)12 = $1,425.80.
If the compounding period were a year, the future value is 1,000 (1.12)3 = $1,404.93.
The same types of conversions are done with present values and annuities.
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