money market instruments.1
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Money Market Instruments
Corporations and government organizations are continually buying and lending money.Often, they borrow money by issuing bonds, but in many cases, they will raise money
through money market instruments (aka cash equivalents), which consists of short-term, very low risk securities. The money market is the market for buying and sellingshort-term loans and securities. The buyer of the money market instrument is the lender of money and the seller is the borrower of money. Capital markets are the other part of the financial markets, which consists of longer term or riskier securities, such as stocks, bonds, currencies, and derivatives.
While money market instruments are diverse, they have several features in common. Allhave terms of less than 1 year, with most less than 6 months. Many money marketinstruments have terms of 270 days or less, because any instruments with longer maturities would have to be registered with the SEC under the 1933 Act. They are very
low risk securities, and, because of their short terms, they are usually issued at a discount —interest is paid when the holder of the money market instrument is paid par at maturity.Because money market instruments are discounted, their yield is quoted using the bond
equivalent yield, which is the yield that is equivalent to the discount, and allows aninvestor to easily compare yields among different instruments and securities.
Bond Equivalent Yield (BEY)
Sometimes the yields listed for short-term discount instruments have simply beenannualized without compounding the interest. This simplifies the math and can becalculated using a calculator that doesn't have a root or exponential function. This
uncompounded annual interest rate is simply called the annual interest rate todistinguish it from the effective annual yield, but, most often, it is called the bond
equivalent yield (BEY) (aka investment rate yield, equivalent coupon yield). Thesimplified formula appears below:
BEY = Interest Rate per Term x Number of Terms per Year
Below is the formula relating BEY to the face value, price paid for the instrument, anddays left to maturity:
Formula for Calculating Bond Equivalent Yield (BEY)
Interest Rate Per Term Number of Terms per Year
BEY =Face Value - Price Paid ─────────────── Price Paid
xActual Number of Days in Year ─────────────────────── Days Till Maturity
Note that this yield is not compounded, but is the simple interest rate annualized.However, if you only have the simple interest rate of a discount instrument, then this ratecan be converted directly to any compounded rate of interest by using the formula for the
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present and future value of a dollar. (See Calculating the Interest Rate of a DiscountedFinancial Instrument for more info.)
Example — Calculating the Bond Equivalent Yield of a T-Bill
If you buy a 4-week T-bill with a face value of $1,000 for $996.50, what is the bondequivalent yield, assuming it is not a leap year?
($1,000-$996.50)/$996.50 x 365/28 = 4.58% (rounded)
Example—Formula for Finding the Annualized Effective Compounded Rate of Interest for
a Discounted Note
To find the compounded rate of interest for a discounted money market instrument:
1. Divide the par value by the discounted price.2. Raise the result by the number of terms in 1 year, then subtract 1.
So if you bought a 4-week T-bill for $996.50 and receive $1,000 4 weeks later, what isthe effective annual compounded interest rate earned?
Solution:
1. $1,000/$996.50 = 1.0035 (rounded)
Since there are 13 4-week periods in a year, this T-bill rate compounded 13 times wouldequal:
2. (1.0035)13 - 1 = 1.046 - 1 = 4.6% (rounded)
(See how the future value of a dollar is calculated to understand the reasoning better.)
You can use this formula for calculating the yields of any money market instrument soldat a discount.
U.S. Treasury Bills (aka T-bills)
Treasury bills are issued by the federal government and have terms of 28, 91, or 182
days, and are virtually free of credit risk. They are the most actively traded money marketsecurities with very low bid/ask spreads due to their liquidity, and they are also exceptfrom state or municipal taxes. Retail investors can buy T-bills directly from the Treasuryat http://treasurydirect.gov/.
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Federal Funds
The Federal Reserve requires each bank of the Federal Reserve System to maintain aminimum amount of money on deposit at a Federal Reserve bank to insure that the bank has enough reserves to meet customer obligations. Federal funds (aka Fed funds) is the
money that banks deposit at the Federal Reserve bank to maintain the amount of depositsrequired. However, some banks, especially in the large financial centers of major citieslike New York, Chicago, and San Francisco, have greater loan requirements than mostother banks, and often do not have enough to maintain their reserve requirements. Sothese banks borrow from other banks that have an excess amount of money over therequirement. Banks will lend their excess reserves to other banks, or borrow $1 millionand up, if they are short, paying the federal funds rate of interest, usually for 1 day,since most of these loans are overnight loans. Some banks that are always short on moneymay borrow for longer terms—from 1 week to 6 months or, in rare cases, longer—from banks that usually have excess reserves. These longer term Fed funds are called term
Fed funds. The federal funds rate is extremely volatile, and is regulated by the Federal
Reserve to some extent as a means to control the supply and demand of money.
The London Interbank Offered Rate (LIBOR) Market
Similar to the Fed funds rate is the LIBOR rate, which is the interbank lending rate that banks charge to each other. Many financial instruments and contracts are based on theLIBOR rate.
Repurchase Agreements (aka Repos, Sale-Repurchase Agreements)
Repurchase agreements (aka repos) are contracts that are sold, often for 1 day or a fewdays, with a minimum denomination of $1,000,000, with the stipulation that they will berepurchased for a price that is higher by the amount of the interest, called the repo rate.
Government security dealers typically use repos to finance the purchase of governmentdebt, especially Treasuries. For instance, if a government bond dealer wanted to buy $1 billion worth of Treasuries, he may submit a winning bid for that amount, but pay only$300,000,000 and finance the rest by promising the U.S. Treasury that he will pay for therest later, after he has customer orders for the rest of the purchase, usually by the next day(an overnight repo). The Treasury will charge daily interest on all issues bid, but not yet paid for. If the bond dealer can sell the Treasuries for more money than he paid, then the
dealer makes a profit; if he sells the Treasuries for less, then the dealer will suffer a loss.
In a reverse repo, the dealer buys the securities with the stipulation that the dealer cansell them back for a higher price—the additional interest. The dealer is, in effect, lendingthe seller money and keeping the securities as collateral.
Term repos have longer maturities of a week to a few months. The market for term reposis larger than the market for overnight repos.
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Sometimes a dealer will have an open repo contract with a lender to provide funds on acontinuing basis, and that allows the dealer the right of substitution —to substitutesecurities of equal or greater value for the loans. Either party may cancel the contract atany time.
Repos are considered safer than Fed funds because they are collateralized, so the reporate ranges from 10 to 200 basis points below the Fed funds rate.
Because repurchase agreements are private agreements between 2 parties, there is nosecondary market for repos, especially considering their very short terms of 1 or moredays.
Bankers' Acceptance
A banker's acceptance is a commercial bank draft requiring the bank to pay the holder of the instrument a specified amount on a specified date, which is typically 90 days from
the date of issue, but can range from 1 to 180 days. The banker's acceptance is issued at adiscount, and paid in full when it becomes due—the difference between the value atmaturity and the value when issued is the interest. If the banker's acceptance is presentedfor payment before the due date, then the amount paid is less by the amount of theinterest that would have been earned if held to maturity.
A banker's acceptance is used for international trade as means of verifying payment. For instance, if an importer wants to import a product from a foreign country, he will oftenget a letter of credit from his bank and send it to the exporter. The letter of credit is adocument issued by a bank that guarantees the payment of the importer's draft for aspecified amount and time. Thus, the exporter can rely on the bank's credit rather than the
importer's. The exporter presents the shipping documents and the letter of credit to hisdomestic bank, which pays for the letter of credit at a discount, because the exporter's bank won't receive the money from the importer's bank until later. The domestic bank then sends a time draft to the importer's bank, which then stamps it "accepted" and, thus,converting the time draft into a banker's acceptance. This negotiable instrument is backed by the importer's promise to pay, the imported goods, and the bank's guarantee of payment.
Commercial Paper
Creditworthy corporations can borrow from banks for the prime rate of interest, but they
may be able to borrow at a lower rate by selling commercial paper to institutionalinvestors and the public—usually banks, pension funds, and other corporations.
Commercial paper are unsecured promissory notes for a specified amount to be paid at aspecified date, and are issued by corporations with excellent credit instead of borrowingmoney from a bank. They are issued at a discount, with minimum denominations of $100,000. The main purchasers are other corporations, insurance companies, commercial
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banks, and mutual funds. Terms range from 1 to 270 days. Commercial paper is the leasttraded money market instrument in the secondary market.
Finance companies sell 2/3 of the total commercial paper, and sell their issues directly tothe public. But corporations that borrow less frequently sell their commercial paper—
called industrial paper —to paper dealers, who then sell them at a markup to other investors. A round lot for a paper dealer is $250,000.
Negotiable Certificates of Deposit
Before 1986, the Federal Reserve Board restricted the amount of interest that banks could pay for savings or other time deposits. Often, corporations would have money availablefor lending, but banks couldn't compete for this money because of the interest raterestriction. Negotiable CDs were a means around the restrictions.
Negotiable certificates of deposit (aka jumbo certificates of deposits, jumbo CDs) are
tradable certificates issued by commercial banks as unsecured time deposits. Termsrange from a minimum of 14 days to 1 year or more. Most have terms of 1 to 3 months, but some can have maturities of 3 to 5 years, or longer. They have a minimumdenomination of $100,000, but usually are issued in denominations of $1,000,000 or more. Most CDs have a fixed rate of interest, although there are some that pay a variablerate of interest. CDs are actively traded in the secondary market in round lots of $5,000,000.
Broker's Loans and Call Loans
Broker's loans are loans from commercial banks to brokers so that the broker's
customers can finance stock purchases. The broker uses the stocks, held in street name,for collateral for the loans.
Time notes are loans that must be paid by a specific date for a specified interest rate,with terms of 6 months or less. A demand note (aka call loan) is a loan that is payableon demand the next day at 1 day's interest. If the note is not demanded, then the term isextended by another day, and so on, up to 90 days. The interest rate for each day varieswith the prevailing interest rate.
Eurodollars, Eurocurrency
Eurodollars usually refers to U.S. dollars deposited in banks outside of the UnitedStates. Eurocurrency is a more general term that can refer to any currency that isdeposited in banks whose domestic currency is different from the deposited currency, andit can involve any country, including the Far East and the Cayman Islands. Eurodollars or Eurocurrency does not necessarily involve either Europe or the Euro. Multi-nationalcorporations deposit their domestic currency in foreign banks because they can often get better terms trading their currency with the locals than by exchanging domestic currencyfor foreign currency at a bank. The interest paid on these deposits is usually equal to the
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London Interbank Offer Rate (LIBOR ), which is slightly higher than the yield for 3-month Treasuries.
This is a sample of key money rates that were published in the Wall Street Journal onApril 20, 2007. Note that because many of these rates are negotiated between private
parties, and the rates fluctuate throughout the day, these rates are mostly averages that donot necessarily reflect individual transactions. Note also how the yields of the variousmoney market instruments compare with the key interest rates.
onds - Table of Contents
Bond Fundamentals
Interest Rates
The Present Value and Future Value of Money
The Present Value and Future Value of an Annuity
Bonds - An Introduction
Bond Indentures
Bond Yields
Bond Pricing
Bond Risks
Bond Ratings and Credit Risk
Volatility of Bond Prices in the Secondary Market
Primary Bond Market
Term Structure of Interest Rates
Interest Rate Risk
Duration and Convexity
Bond Income Taxation
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Types of Bonds
International Bonds
Convertible Bonds
Preferred Stock
Fixed Rate Capital Securities (FRCS)
Money Market Instruments
Money Market Instruments ◄ Current Document
Certificates of Deposit
Commercial Paper
Repurchase Agreements (Repos)
Federal Funds
Bankers Acceptances
Medium-Term Notes
Government Securities
Municipal Bonds
United States Savings Bonds
Treasury Securities: Bills, Notes, Bonds, TIPS, and STRIPS
Primary and Secondary Markets for Treasury Securities
United States Treasury Auctions
Corporate Bonds
Corporate Bonds - An Introduction
Corporate Bond Types: Mortgage Bonds, Collateral Trust Bonds, Equipment TrustCertificates (ETCs), Debentures, and Guaranteed Bonds
High-Yield Bonds (Junk Bonds)
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Asset-Backed Securities
Asset-Backed Securities (ABS)
Asset-Backed Securities: Structure
Prepayment Models
Asset-Backed Securities: An Overview of Credit Ratings
Asset-Backed Securities: Credit Analysis
Special Purpose Entity (SPE)
Credit Enhancements
Credit Card Asset-Backed Securities
Credit Card Asset-Backed Securities: Structure and Cash Flow Allocation
Auto Asset-Backed Securities
Auto Lease Asset-Backed Securities
Mortgage-Backed Securities
Collaterized Mortgage Obligations
Collateralized Debt Obligations (CDOs)
Covered Bonds
Derivatives
Derivatives - An Overview
Death Bonds
Interest Rate Swaps
Credit Default Swaps
Overnight Index Swaps
Derivatives Resources
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Bond Resources
Bond Blog
Bond Formulas
Interest and Interest Rates
Interest rates are the rate of growth of money per unit of time. It is one of the mostfundamental factors in investments, since so many financial assets depend on its value. Itis used to determine the present and future value of money and annuities. Many securitieseither pay interest or the payoff depends on the interest rate. Whether a business willinvest in capital or issue securities depends on the interest rate. Hence, the interest rateallocates economic resources more efficiently. Governments control their economies byadjusting key interest rates through monetary and fiscal policies.
Interest is the cost of money, and like the price of virtually everything else, it isdetermined by supply and demand. In the United States and most other developedcountries, the government has a major influence on the interest rate, adjusting it higher tocool the economy and adjusting it lower to stimulate it. The government can also increasethe money supply by printing money, or through monetary and fiscal policies. Another source of supply is the savings of people, businesses, and other organizations. The maindemand for money is for loans by people and businesses. Demand can also be affected bythe monetary policies of the government.
Charging interest on a loan is sometimes called usury, although in more recent times, ithas acquired a negative connotation of excessively high or illegal interest rates being
charged. In fact, when the usury rate is limited by law, the rate is referred to as a usuryceiling. However, at least 2 states in the United States do not have usury limits: Delawareand South Dakota, which is why many credit card issuers are located in those states.
The concept of interest has a long history. Aristotle thought interest was evil, andaccording to the Koran, God condemned the charging of interest. The earliest knownexamples of interest were in ancient Mesopotamia, beginning in the 3rd millennium B.C.,when an interest rate of either 20% or 33% was charged depending on whether the loanwas paid in silver or barley. However, the interest rate did not depend on the amount of time. [No doubt this simplified calculations that required using a sexagesimal (base 60)numbering system and pressing wedge-shaped (cuneiform) styles into wet clay tablets.]
Nominal and Real Interest Rates
The nominal interest rate is the stated interest rate. If a bank pays 5% annually on asavings account, then 5% is the nominal interest rate. So if you deposit $100 for 1 year,you will receive $5 in interest. However, that $5 will probably be worth less at the end of the year than it would have been at the beginning. This is because inflation lowers the
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value of money. As goods, services, and assets, such as real estate, rise in price, it takesmore money to buy them.
The real interest rate is the nominal rate of interest minus inflation, which can beexpressed approximately by the following formula:
Real Interest Rate = Nominal Interest Rate – Inflation Rate = Growth of PurchasingPower.
For low rates of inflation, the above equation is fairly accurate. However, the actualgrowth of your purchasing power is equal to the nominal interest rate divided by theinflation rate:
Formula Relating the Real Interest Rate,
Nominal Interest Rate, and Inflation Rate
1 + R =
1 + N
───────── 1 + I
R = Real Interest Rate
N = Nominal Interest RateI = Inflation Rate
The above equation can be solved for the real interest rate.
Solving for the Real Interest Rate
(1+R)(1+I) = 1+N Multiply both sides by (1+I).1+I+R+RI = 1+N Multiply out left-hand side to get terms.R+RI = R(1+I) = N-I Subtract 1 and I from both sides.R = (N-I)/(1+I) Divide both sides by 1 + I.
Real Interest Rate Formula
R = N - I ──── 1 + I
R = Real Interest Rate N = Nominal Interest RateI = Inflation Rate
Because people invest to earn more purchasing power, they will only invest or lendmoney that pays more than the expected inflation rate. In this case, the nominal rateequals the real interest rate plus the expected inflation.
Nominal Interest Rate Equilibrium
Although there are many different interest rates, their differences result mainly from risk, but they all move up or down along with the prevailing rates. Thus, these rates can beabstracted as a single interest rate—the prevailing interest rate. Generally, as interestrates increase, saving increases and borrowing decreases, and vice versa. If investments pay higher interest, then more people, businesses, and other organizations will invest toearn more money. If interest rates decline, then the motivation to invest declines also, but borrowing increases, which increases demand for money. There is a point where supplyequals demand—this is the observed or nominal interest rate.
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Irving Fisher looked at interest rate equilibrium as the desire for a specific real rate of return plus the expected inflation rate:
Nominal Interest Rate = Real Interest Rate + Expected Inflation Rate.
If the expected inflation rate was high, then people would demand a higher nominal ratefor their investments; for why would anyone invest if they did not expect a real return?Although no one can really know what future interest rates will be, the nominal interestrate can be somewhat indicative of the expected interest rates.
The Taxation of Nominal Interest Rates
Most earned interest, or any positive return from investments, is taxed. However, taxescurrently apply to the nominal rate of return, not the real rate—thus, the tax rate on thereal rate of return is greater than the published tax rate.
Real Rate of Return = Nominal Interest Rate x (1 – Your Tax Bracket) - Inflation Rate
Example — Calculating the Real Interest Rate after taxes
If you earned 5% nominal interest on your money with 3% inflation, and you are in the25% tax bracket, what is your real interest rate after taxes?
Solution:
Using the above formula:
Real Rate of Return = 5% x .75 - 3%. = .75%
As you can see from the above, if you are in a high tax bracket, you will have to earnsignificantly more than 5% to earn a decent real return. If you are in the 35% bracket,given the above nominal interest rate and inflation rate, your real interest rate would be 0!You can see why the wealthy invest in tax-free municipal bonds.
Simple Interest
Simple interest, often called the nominal annual percentage rate (APR ), isuncompounded interest, which is calculated by multiplying the principal times the
interest rate. The earned interest is not added to the principal, so the amount of interestearned is always the same for a given interest rate.
Interest = Principal x Interest Rate
A good example of simple interest is the interest earned by bonds. Most bonds pay acoupon rate, which is simply the stated interest rate of the bond when it is first issued.When the interest is earned, it is sent to the bondholder—it is not added to the bond's
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principal and the interest earns no additional interest unless the bondholder reinvests theinterest in another investment, such as a savings account.
Compounding Interest
Compounded interest is calculated using the principal plus previously earned interest.For instance, if you deposit $100 in a savings account that pays 6% interest, compoundedsemiannually, then this means that you are actually earning 3% every 6 months, so that atthe end of 6 months, you would have $103. But in the next 6 months, there would be$103 earning interest instead of just $100, so $103 x 3% = $3.09. Add this to the 1st $3already earned will yield a total of $6.09 for the 1st year, which is 9 cents more than if theinterest rate was simple interest. This would be equivalent to a simple interest rate of 6.09% per year. Because money earns interest, it has a future value that is greater than its present value by the amount of the interest earned—this is referred to as the future valueof money or the future value of a dollar. The future value can be expressed as:
Future Value = Principal x (1 + Interest Rate per Compounding Period) Number of Compounding
Periods
or
Future Value of a Dollar (FVD)
FV = P(1+i)n
FV = Future ValueP = Principali = interest rate per compounding periodn = number of compounding periods
Using the above example: $100 x (1+.03)2 = $106.09. Interest rates are often used tocompare investments, but not all investments have the same compounding period, or itmay not be compounded at all, as is the case for a zero coupon bond, which pays nointerest. The interest is earned by buying the bond at a discount and receiving face valueat maturity. However, an effective compounded interest rate can be found even for adiscounted bond, because it is possible to convert compounding interest rates into other rates with different periods of compounding. Most investments that pay interestnormalize the interest rate to an annual rate—the APR. Thus, using the above example, asavings deposit that pays 6% compounded semiannually is approximately equivalent to
6.09% compounded annually. By normalizing interest rates to an effective annual percentage rate, different investments can be easily compared.
Rule of 70 — Quick Method to Find Doubling Time
The Rule of 70 is a simple method to find how quickly a principal that is earning acompounding interest rate will take to double: divide 70 by the interest rate for thecompounding period.
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Time to Double = 70 / Interest Rate
Examples: How long will a savings account paying 5% compounded annually take todouble? 70/5 = 14 years. As a check, using part of the formula for future value listedabove, (1.05)14 ≈ 1.98, so the Rule of 70 is a close approximation. Note, however, that the
Rule of 70 approximation becomes less accurate for higher interest rates. For instance, if the interest rate is 14%, then 70/14 = 5, but (1.14)5 ≈ 1.93
Continuous Compounding Interest
Many portfolio simulations and pricing models for derivatives use a continuouslycompounded interest rate formula.
If a savings account paid a nominal interest rate of 6%, that was compoundedsemiannually, the real compounded rate can be found using the following formula:
1. Formula For Finding the Compounded Interest Rate from a Nominal InterestRate
C = ( 1 +R ─ n
)
n
- 1 = ( 1 +.06 ─ 2
)
2- 1 = 0.0609 =6.09%
C = Compounded Interest RateR = Nominal Interest Raten = number of compounding periods
To find the daily compounded rate for a nominal annual interest rate of 6%, divide theinterest rate by 365, and raise the quantity in parentheses to the 365th power. We note thatas n increases, the increase in the 1st term becomes less and less, reaching a limit as nincreases to infinity. This limit is the natural logarithm base e:
2. Formula For Finding the Natural Logarithm Base e
As n → ∞, ( 1 +1 ─ n
)n
→ e = 2.718281828... n = number of compounding periods
As a corollary of the above equation, we arrive at formula 3:
As n → ∞, ( 1 +R ─
n
)n
→ eR R = Rate of growth or interest rate
n = number of compounding periods
Thus, by substituting the result of formula 3 into formula 1, we see that:
Continuous Compounded Interest Rate = eR - 1
Example — Calculating the Continuously Compounded Interest Rate or the Effective
Annual Percentage Rate
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If a bank advertises a savings account that pays a 6% nominal interest rate that iscompounded continuously, what is the effective annual percentage rate?
Solution:
Using the above formula:Continuously Compounded Interest Rate = e.06 - 1 = 1.061837 - 1��� 6.1837%
Although it sounds like you'll make a lot of money by having it continuouslycompounded, it's not much more than the daily compounded rate of:
6% Compounded Daily = (1 + .06/365)365 ≈ 0.061831 ≈ 6.1831%
Note that since 1 + Growth Factor = e(Growth Factor), we can simplify the 1st formula relatingreal interest rates, nominal interest rates, and inflation rates by the following equation:
Formula Relating the Real Interest Rate, Nominal Interest Rate,
and Inflation Rate Using Continuously Compounded Rates
eR =e N
─── eI
= e(N-I)R = Real Interest Rate N = Nominal Interest RateI = Inflation Rate
Taking the natural logarithm of both sides, simplies the above equation even further:
R = N - I
Thus, for continuously compounded rates, the approximation formula for relating the realinterest rate to the nominal interest rate and inflation rate becomes exact.
The Present Value and Future Value of
Money
< prev: Interest Rates How would you like to make more than 100% interest compoundedannually, virtually guaranteed, and with very little risk? This is not a misprint, and it isnot a lure to sell you something. I have nothing to sell. Read on. When you learn aboutthe present value of a dollar and the future value of a dollar, you can see things that mightnot be so obvious at first.
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Money makes money. And the money that money makes makes more money. — Benjamin Franklin
Money has a time value because it can be invested to make more money. Thus, a dollar received in the future has lesser value than a dollar received today. Conversely, a dollar
received today is more valuable than a dollar received in the future because it can beinvested to make more money. Formulas for the present value and future value of moneyquantify this time value, so that different investments can be compared. If a saver deposits $100 in a savings account today, and it pays 5% interest, what will it be worth 5years from now, or 10 years from now? If an investor buys stock for $25, then sells it 3years later for $45, what was its rate of return? A business has money and many ways tospend or invest it. What is the best use of that money?
The present value and future value of money, and the related concepts of the presentvalue and future value of an annuity, allow an individual or business to quantify andminimize its opportunity costs in the use of money. Opportunity cost, in terms of the use
of money, is the benefit forfeited by using the money in a particular way. For instance, if I spend $100 instead of depositing it in a bank that pays 5% interest, I forego the interestthat I would have earned in the savings account by spending it instead of saving it, and if I would have saved it, then I forfeit the benefit of what I purchased. Of course, it might be possible to buy some stock, instead, that may double or triple, incurring an evengreater opportunity cost. However, the future value of a stock is unpredictable, and thetrue opportunity cost of anything is really not knowable. However, the opportunity costcan be compared among specific investments where the rate of return is dependent on aninterest rate that is either known or can be reasonable estimated by using the formulas for the present value and future value of money. Or a reasonable interest rate can be assumedsimply to compare different investments.
The Future Value of a Dollar
The future value of a dollar is considered first because the formula is a little simpler.
The future value of a dollar is simply what the dollar, or any amount of money, will be
worth if it earns interest for a specific period of time.
If $100 is deposited in a savings account that pays 5% interest annually, with interest paidat the end of the year, then after the 1st year, $5 of interest will be added to the $100 of
principal for a total of $105. In the 2nd year, interest will be earned not only on the principal of $100, but also on the $5 of interest earned. Thus, at the end of the 2nd year,there will be 5 more dollars of interest earned from the principal added to the account, plus 25¢ earned from the previous year's interest of $5. Thus, at the end of the 2nd year there will be $105 + $5 + $.25 = $110.25 total in the account. This is an example of compounding interest, interest that is paid on interest previously earned. This processcan be continued for any number of years.
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Expressing this as an equation, if P = principal and i = interest rate per year, then theamount of money in the account after the 1st year can be expressed by the equation P (1
+ i) = P + i*P = $100 + .05 * 100 = $100 + $5 = $105 . To find the amount after the 2ndyear multiply 105 by the same factor— (1 + i). This equation can be expressed in terms of the 1st equation: P (1 + i) (1 + i), which reduces to P (1 + i)2. This equation can be
extended to P (1 + i)
n
, with the superscript n equal to the number of years. Thus, theamount of money in the account after 3 years is P (1 + i)3. For this example, 100 (1 + .
05)3 = 100 (1.05)3 = 100 * 1.157625 = $115.76, rounded to 2 decimal places.
Future Value of a Dollar (FVD)
FV = P(1+i)n
FV = Future Value of a Dollar P = Principali = interest rate per year n = number of years
Using a calculator to determine future value:
If you have a calculator that has the exponential function—usually designated by the yx
key—then this equation is easy to solve. Add the interest rate in decimal form to 1, then press yx, then enter 3, then press the = key. Take this product, the interest factor, andmultiply it by the principal. So for our example, enter 1.05, then press yx, then enter 3,then press = to arrive at the interest factor 1.157625. Multiply this by 100 to get $115.76,the amount of money in the account after 3 years. Because exponentiation has priorityover multiplication, you can also enter it this way: 100 X 1.05 yx 3 = $115.76.
Comp0unding Interest
In all formulas that compute either the present value or future value of money or annuities, there is an interest rate that is compounded at certain intervals of time. Thisinterval of time is assumed to be 1 year, but, if it is less than 1 year, as it frequently is,then there are 2 adjustments that must be made to the formulas:
1. The number of time periods must be changed to represent the number of timesthat interest is compounded. The number of years must be multiplied by thenumber of compounding periods within a year.
2. The interest rate itself must be changed to reflect the interest rate per time period.The annual interest rate must be divided by the number of compoundings in a year.
Note also that most of the solutions to these formulas are rounded.
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Example — Adjusting a Formula for Non-annual Compounding of
Interest
If you put $100 in a savings account that pays 5% interest annually, but is compoundeddaily, how much will be in the account after 10 years?
Solution: This is finding the future value of a savings account, but since this account iscompounded daily, the formula will have to be adjusted as follows:
FV = P(1+i ─ c
)n*c
FV = Future Value of Savings AccountP = Principali = interest rate per year n = number of yearsc = number of compounding periods in a year
Thus, we find the solution by plugging the values into the formula:
FV= 100 * (1+.05 ─── 365
)10*365=$164.87
Note that with compounding interest, doubling either the interest rate or the amount of time more than doubles the amount of interest earned. For instance, $100 earning 5%interest that is paid yearly would equal $62.89 of earned interest after 10 years; after 20years, earned interest would equal $165.33.
Thus, the future value of a dollar is the value that it will have after a specific time
earning a specific interest rate.
The Present Value of a Dollar
Suppose you buy a zero coupon bond that matures in 10 years, then pays $1,000. Howmuch is that future payment of $1,000 worth today at a 5% interest rate? In other words,if the prevailing interest rate is 5%, how much should you pay for a zero coupon bondthat is sold at a discount to its par value?
In determining the future value of money, we know how much money we are starting
with, and we want to know how much it will be worth at some point in the future at aspecific interest rate. When we know how much a future payment will be, then we wantto determine what its value is today at a given interest rate.
The present value is the current value of a payment that will be received in the future.Discounting is the process of determining the present value of a payment from a knownfuture payment, or future value. This is the reverse of determining the future value of a
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payment, because in this case, we already know the future value. It is found by dividingthe future value by the same interest factor, (1 + i)n, used to determine future value.
The Present Value of a Dollar (PVD)
PVD =FVD
───── (1+i)n
PVD = Present Value of a Dollar
FVD = Future Value of a Dollar i = interest rate per time periodn = number of time periods
Example — Calculating the Worth of a Zero Coupon Bond
How much would a zero coupon bond sell today, that pays $1,000 in 10 years, assumingan interest rate of 5% that is compounded and paid annually?
Solution: The zero coupon bond pays $1,000 in 10 years, so that is its future value in 10years. If the prevailing interest rate is 5%, then to find the present value of the zero:
PVD =
1,000 ──────── (1+.05)10 = $613.91
Using a calculator to determine present value:
Enter 1,000, press the divide key, ÷ enter 1.05, then press the exponential key, yx, thenenter 10, then the = key. The calculator should do the exponentiation 1st, becauseexponentiation has priority over division, then the division to arrive at the correct answer of $613.91, rounded to 2 places.
Summary: 1,000 ÷ 1.05 yx 10 = $613.91.
Calculating the Interest Rate of a
Discounted Financial Instrument
To find the present value, we need to know the future value and the interest rate; to findthe future value, we need to know the present value and the interest rate. But sometimes, both the present value and the future value are known, but not the interest rate. A good
example of this problem is the zero coupon bond. A zero coupon bond pays no interestduring its term, but is bought at a discount to its par value. Thus, in this case, the purchase price is known, which is its present value, and its future value is the par value of the bond, usually $1,000, which is paid when the bond matures. But what is theequivalent interest rate? As we will see below, even though a zero pays no interest, itstill has an equivalent interest rate, which can be calculated and compared to other investments. How would it compare to a savings account that pays 5% interestcompounded annually, for instance?
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To find the equivalent interest rate, i, we transpose the equation for the future value of money to equal i. The equation for future value is:
Present Value * (1 + i)n = Future Value
First, divide both sides by the Present Value:(1 + i)n = Future Payment/Present Value.
Take the nth root of both sides:
1 + i = (Future Payment/Present Value)1/n.
Then subtract 1 from both sides, to arrive at i, the interest rate for the discount:
The Interest Rate of a Discount (IRD)
i = (FV ───── PV )
1
―
n
- 1i = Interest Rate of Discount per time periodn = number of time periodsFV = Future ValuePV = Present Value
or
Example — Calculating the Interest Rate of a Zero Coupon Bond
What interest rate is a zero coupon bond paying, that costs $600 and pays $1,000 in 10years, assuming an interest rate that compounds annually?
Solution: Future Value = $1,000 par value, Present Value = $600 purchase price, n = 10years
(1,000/600)1/10 - 1 = 5.24%
Using a scientific calculator: 1000 ÷ 600 = x√y 10 - 1 = .0524 = 5.24%
Note that if the interest is compounded at different intervals, such as quarterly or daily,then the interest rate i and the number of compounding periods must be adjusted. But if compounding of interest is not specified, as with the zero coupon bond, what value do weuse? We use the value that allows us to compare it to another investment. We can specifythat the interest rate be compounded daily instead of annually, which will result in alower interest rate. We would do this to compare the zero, for instance, to a savingsaccount that is paying interest compounded daily. So how would our zero coupon bondexample compare with a savings account that paid 5%, compounded daily? In other
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words, if we put $600 in the savings account instead of buying the bond, would we havemore or less than $1,000 after 10 years?
We can solve this problem in 2 ways. We can solve the problem either by calculating thefuture value of $600 earning 5% compounded daily, or we can calculate the equivalent
interest rate for the zero coupon bond when compounded daily. In either case, we need toknow how many compounding periods there are. Since there is 365 days to a year, thereis 3,650 compounding periods in 10 years. However, because the interest rate is listed as5% per year, compounded daily, we need to find that .05/365 = .000136986 = the dailyinterest rate. Substituting these values into the IRD formula, the future value of thesavings account is:
Future Value = 600 * (1 + .000136986)3650 = $989.20
We can already see that the zero coupon bond pays better, but let's see what the interestrate of the bond would be if compounded daily, like the savings account.
(1,000/600)1/3650 - 1 = .000139962 (daily interest rate) * 365 = .0511 = 5.11% annualinterest rate.
Example — Calculating the Interest Rate of a Fluorescent Bulb
What did you say? Fluorescent bulbs don't pay interest! Let's see.
You have a light bulb in your house, that's on quite a bit, and it's a 100 watt bulb. Youtypically go through 13 bulbs in 5 years, for a total of 10,000 hours of light. But, what if you buy a 23-watt fluorescent bulb, instead, which gives off almost the same amount of
light, and it last 13 times longer than a typical 100-watt bulb. And let's say that your electric rate is 10¢ a kilowatt hour. Let's also assume that you pay about 33% of your income in federal, state, and local taxes.
100-watt bulbs generally cost about .25 per bulb, so 13 will cost $3.25. A 23-wattfluorescent bulb can be bought for nearly the same price as the 13 incandescent bulbs.Yes, fluorescent bulbs are more expensive than incandescents, but wait! The amount of electricity consumed by a series of 100-watt bulbs in 10,000 hours is 1,000 kilowatts (100x 10,000 = 1,000,000 watts or 1,000 kilowatts, since a kilowatt is 1,000 watts). At 10¢ akilowatt hour, that's $100 worth of electricity needed for incandescent bulbs.
The fluorescent bulb consumes 230 kilowatts of electricity for the same amount of light(23 x 10,000 = 230,000 watts = 230 kilowatts). At 10¢ a kilowatt hour, that's $23 worthof electricity for a savings of $77 over the incandescents. So, in 5 years, for this one lightsocket, you would save $77 over a period of 5 years. But, to save $77 is the same thing asearning $77 tax-free! So you paid $3.25 to earn $77 tax-free over a period of 5 years. Sowhat is the equivalent interest rate that is compounded annually of this?
Solution: Future Value = $77 saved, Present Value = $3.25 purchase price, n = 5 years
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(77/3.25)1/5 - 1 = 88.33%
Using a calculator: 77 ÷ 3.25 = x√y 5 - 1 = .883308... = 88.33%
That's a pretty impressive rate of return—a tax-free rate of return—and not only that, it's
virtually guaranteed! Your only small risk is that the bulb breaks or turns out to bedefective, in which case, you can probably return it to the store for another one. Note thatthe $77 is real money, even though you only saved it and didn't really earn it. It's real,however, because if you had bought the incandescent bulbs instead, that would be $77less than you would have had over the 5 years. Granted, these savings aren't going toallow you to buy a Lamborghini for your next car, but it's something that not only savesyou money, but saves you time as well, since you won't have to change the bulb 12 timesduring those 5 years! Multiply that by the number of bulbs in your house. Or if you havea business! Even a small business like a medical center could have a hundred or morelight bulbs which are on all day during business hours. The savings could be quitesubstantial. Not only that, but you help to conserve energy and protect the world from
global warming!Since this money is earned free of all taxes, we can also calculate a taxable equivalentyield. The formula for this is:
Taxable Equivalent Yield = (Tax-free yield)/(100% - Tax%)
We assumed that you pay 33% of your income in federal, state, and local taxes. Thisyields:
88.33%/(100% - 33%) = 88.33%/67%= .8833/.67 = 1.318358... = 131.84%
For a more detailed discussion of taxable equivalent yields, see Bond Yields: NominalYield, Current Yield, Yield to Maturity, Yield to Call, True Yield, Taxable EquivalentYield.
That's a taxable equivalent yield of 131.84%, compounded every year, for 5 years. Inother words, you would have to earn that rate of return to equal the tax-free yield of 88.33%. Compare that to a savings account!
Obviously, this rate of return will vary depending on the actual value of the variables. For instance, if the light bulb was not on so much, and it lasted 10 years, then obviously this
will lower the equivalent interest rate, but it will still be substantial.
Conclusion
Here we can see that, even though a zero and a fluorescent bulb pay no actual interest, wecan find an equivalent interest rate that's compounded daily, weekly, quarterly, or
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whatever, so we can compare it to investments that do pay interest. This is the value of the formulas for the present value and the future value of money!
The present value and future value of a dollar is a lump sum payment. A series of equallump sum payments over equal periods of time is called an annuity. This is a more
general concept than the insurance product that most people think of when they see theword annuity; it includes loans, interest payments from bonds, and so on—even theannuity insurance product. Like the present value and future value of a dollar, the presentvalue and future value of an annuity allows you compare investments, or the costs of loans. For instance, you can answer questions such as, How much would my payments beon a $200,000 mortgage with a 6% interest rate? or How much of a mortgage could I get,if the interest rate is 5%, but I can only afford to pay $1,000 per month? next: The Present Value and Future Value of an Annuity