What Is The Time Value Of Money?
(click to watch video tutorial on this topic)The principle of time value of money - the notion that a given sum of money is more valuable the sooner it is received, due to its capacity to earn interest - is the foundation for numerous applications in investment finance.
Central to the time value principle is the concept of interest rates. A borrower who receives money today for consumption must pay back the principal plus an interest rate that compensates the lender. Interest rates are set in the marketplace and allow for equivalent relationships to be determined by forces of supply and demand. In other words, in an environment where the market-determined rate is 10%, we would say that borrowing (or lending) $1,000 today is equivalent to paying back (or receiving) $1,100 a year from now. Here it is stated another way: enough borrowers are out there who demand $1,000 today and are willing to pay back $1,100 in a year, and enough investors are out there willing to supply $1,000 now and who will require $1,100 in a year, so that market equivalence on rates is reached.
The Five Components Of Interest Rates
Think of the total interest rate as a sum of
five smaller parts, with each part determined by its own set of factors.
1. Real Risk-Free Rate - This assumes no risk or
uncertainty, simply reflecting differences in timing: the preference to spend
now/pay back later versus lend now/collect later.
2. Expected Inflation - The market expects
aggregate prices to rise, and the currency's purchasing power is reduced by a
rate known as the inflation rate. Inflation makes real dollars less valuable in
the future and is factored into determining the nominal interest rate (from the
economics material: nominal rate = real rate + inflation rate).
3. Default-Risk Premium - What is the chance that
the borrower won't make payments on time, or will be unable to pay what is
owed? This component will be high or low depending on the creditworthiness of
the person or entity involved.
4. Liquidity Premium- Some investments are
highly liquid, meaning they are easily exchanged for cash (U.S. Treasury debt,
for example). Other securities are less liquid, and there may be a certain loss
expected if it's an issue that trades infrequently. Holding other factors
equal, a less liquid security must compensate the holder by offering a higher
interest rate.
5. Maturity Premium - All else being equal, a
bond obligation will be more sensitive to interest rate fluctuations the longer
to maturity it is.
Time Value Of Money Calculations
Here we will discuss the effective annual rate, time value of
money problems, PV of a perpetuity, an ordinary annuity, annuity due, a single
cash flow and a series of uneven cash flows. For each, you should know how to
both interpret the problem and solve the problems on your approved calculator.
The Effective Annual Rate
We'll
start by defining the terms, and then presenting the formula.
The stated annual rate, or quoted rate, is the interest rate on an investment if an institution were to pay interest only once a year. In practice, institutions compound interest more frequently, either quarterly, monthly, daily and even continuously. However, stating a rate for those small periods would involve quoting in small fractions and wouldn't be meaningful or allow easy comparisons to other investment vehicles; as a result, there is a need for a standard convention for quoting rates on an annual basis.
The stated annual rate, or quoted rate, is the interest rate on an investment if an institution were to pay interest only once a year. In practice, institutions compound interest more frequently, either quarterly, monthly, daily and even continuously. However, stating a rate for those small periods would involve quoting in small fractions and wouldn't be meaningful or allow easy comparisons to other investment vehicles; as a result, there is a need for a standard convention for quoting rates on an annual basis.
The effective annual yield
represents the actual rate of return, reflecting all of the compounding periods
during the year. The effective annual yield (or EAR) can be computed given the
stated rate and the frequency of compounding. We'll discuss how to make this
computation next.
Formula 2.1
Effective annual rate (EAR) = (1 + Periodic interest rate)m - 1 Where: m = number of compounding periods in one year, and periodic interest rate = (stated interest rate) / m |
Example: Effective Annual Rate
Suppose we are given a stated interest rate of 9%, compounded monthly, here is what we get for EAR:
EAR = (1 + (0.09/12))12 - 1 = (1.0075) 12 - 1 = (1.093807) - 1 = 0.093807 or 9.38%
Keep in mind that the effective annual rate will always be higher than the stated rate if there is more than one compounding period (m > 1 in our formula), and the more frequent the compounding, the higher the EAR.
Solving Time Value of Money Problems
Approach these problems by first converting both the rate r and the time period N to the same units as the compounding frequency. In other words, if the problem specifies quarterly compounding (i.e. four compounding periods in a year), with time given in years and interest rate is an annual figure, start by dividing the rate by 4, and multiplying the time N by 4. Then, use the resulting r and N in the standard PV and FV formulas.
Suppose we are given a stated interest rate of 9%, compounded monthly, here is what we get for EAR:
EAR = (1 + (0.09/12))12 - 1 = (1.0075) 12 - 1 = (1.093807) - 1 = 0.093807 or 9.38%
Keep in mind that the effective annual rate will always be higher than the stated rate if there is more than one compounding period (m > 1 in our formula), and the more frequent the compounding, the higher the EAR.
Solving Time Value of Money Problems
Approach these problems by first converting both the rate r and the time period N to the same units as the compounding frequency. In other words, if the problem specifies quarterly compounding (i.e. four compounding periods in a year), with time given in years and interest rate is an annual figure, start by dividing the rate by 4, and multiplying the time N by 4. Then, use the resulting r and N in the standard PV and FV formulas.
Example: Compounding Periods
Assume that the future value of $10,000 five years from now is at
8%, but assuming quarterly compounding, we have quarterly r = 8%/4 = 0.02, and
periods N = 4*5 = 20 quarters.
FV = PV * (1 + r)N = ($10,000)*(1.02)20 = ($10,000)*(1.485947) = $14,859.47
Assuming monthly compounding, where r = 8%/12 = 0.0066667, and N = 12*5 = 60.
FV = PV * (1 + r)N = ($10,000)*(1.0066667)60 = ($10,000)*(1.489846) = $14,898.46
Compare these results to the figure we calculated earlier with annual compounding ($14,693.28) to see the benefits of additional compounding periods.
Source: www.investopedia.com
FV = PV * (1 + r)N = ($10,000)*(1.02)20 = ($10,000)*(1.485947) = $14,859.47
Assuming monthly compounding, where r = 8%/12 = 0.0066667, and N = 12*5 = 60.
FV = PV * (1 + r)N = ($10,000)*(1.0066667)60 = ($10,000)*(1.489846) = $14,898.46
Compare these results to the figure we calculated earlier with annual compounding ($14,693.28) to see the benefits of additional compounding periods.
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