Interest Rates
You are strolling down the sidewalk, wearing your favorite pair of sunglasses on a warm afternoon. Suddenly, you bump into a sort-of friend whom you haven't seen for awhile: Luther Flash. Luther needs to borrow four hundred dollars for needed tattoo enhancements and body piercing to remain on the cutting edge of painful coolness. He knows that you can easily part with the money and promises to pay you back all four hundred dollars in about a year. Luther's reputation is stellar so you aren't worried about him skipping out of town. Plus he is dating Sapphire Slick who is a good friend of yours.
How do you respond to his request?
Perhaps you disagree with Luther's choice of nose rings and adamantly refuse.
Or perhaps you simply whip out four hundred dollars from your wallet and bask in Luther's good will for the next year.
More likely, you draw upon your deepening reservoir of economic knowledge and realize that the four hundred dollars Luther pays you back in a year from now is not going to be worth as much as the four hundred dollars that you lend to him today.
Money loses value over time due to inflation or rising prices. To adjust for inflation, the price of money is measured in terms of the interest rate. There are many different interest rates representing different lengths of time and levels of risk.
Acting fast, you collar a disoriented businessman and grab his copy of the Wall Street Journal and flip to the table of market interest rates. You see that the one-year Treasury Bill is at 5.80% before flinging the newspaper at the fleeing businessman who catches it in stride.
Since Luther is good for his word, you offer to make the loan with a 5.80% interest rate, deciding that there is no need to add a risk premium. In another year, Luther will pay you back your $400 and an additional 5.80% of that amount.
Equity and Debt
Equity gives the owner a share or partial ownership in the company. Typically, equity is in form of common stock. Equity acts as a claim to a share in the net income (income after expenses and taxes) and to the assets of a business firm.
Debt is usually in the form of a bond and represents a loan from the saver to the firm or government that issues the debt. By definition, debt is a promissory note held by investors.
There are four important pieces of information in regards to a bond.
- Par Value is the value or price of the bond when it matures. When they are first sold, corporate bonds are typically sold in $1,000 increments (each bond has a par value of $1,000) and U.S. government Treasury debt is usually in $10,000 denominations. The par value of a bond remains fixed throughout the life of the bond.
Note that bond issuers can call in bonds before they mature and pay them off at their par value plus a penalty. This often occurs when interest rates have fallen substantially below the rate present when the bond was issued.
- Maturity represents the date when the par value of a bond is to be repaid to the bond holder. For example, a ten-year bond will mature ten years after the date it is issued. The bond's par value is then returned to the owner by the bond issuer.
- The majority of bonds pay annual interest to the bondholder. The annual payment to the bondholder is known as the coupon. Coupons are expressed in terms of annual interest rates. It is important to note that the coupon on a bond remains fixed throughout the life of the bond.
- The current market price of a bond adjusts for changing market conditions. Since the par value and coupon of a bond remain fixed, the price at which a bond sells in secondary markets constantly adjusts to changes in market interest rates.
Coupon and Discount Bonds
The majority of corporate or government bonds are issued with a coupon, however much of the short-term government debt such as Treasury-bills are sold at a discount. A coupon bond pays annual interest based on the coupon interest rate and when the bond matures, the full value of the debt is paid back to the lender.
A discount bond usually has no coupon payments but sells at a lower price (or discount) to the par value. The return is based on the difference between the price the bond is purchased at and the par value that the bond is worth when it matures.
The Relationship of Bond Prices to Market Interest Rates
Market interest rates are constantly adjusting to new information, yet the par value and coupon of existing bonds remains fixed. It is the market price at which a bond trades that adjusts to changes in the market interest rates. The relationship is simple: a bond's market price varies inversely to market interest rates. If market interest rates are rising, the prices of outstanding bonds will be falling and vise versa.
The reason for this relationship is that given a fixed coupon associated with a bond, in order to equalize the yield of existing bonds with new bonds, the price of existing bonds will change to adjust for changes in current market interest rates. If current market interest rates rise, then the prices of existing bonds will fall. Or, the prices of existing bonds will increase when current market interest rates fall.
T 7 1/2% 06 (NYSE:T7F06) - More Info: N/A Last Trade
1:03PM · 105 5/8Change
+1/8 (+0.12%)Prev Close
105 1/2Volume
41Day's Range
105 1/2 - 105 5/8Bid
N/AAsk
N/AOpen
105 1/2Ex-Div
N/A52-week Range
100 - 107Earn/Shr
N/AP/E
N/ADiv/Shr
N/AYield
7.11To illustrate the relationship between bond prices and interest rates, consider the above table that summarize information about an AT&T (symbol T) bond traded in July 1997. In the table, the top row reads T 7 1/2% 06.
This information translates into an AT&T bond (AT&T's general trading symbol is T), that pays a 7 1/2% coupon and matures in the year 2006. The par value on corporate bonds is $1,000 each. If you own one of these bonds your annual interest payment is reflected by the coupon = 7 1/2%, or $75 a year.
Note the table cell that notes the Last Trade = 105 5/8 . 105 5/8 is the most recent price at which the bond has traded. If you were to purchase this bond it would cost you
105 5/8 * 10, or $1056.25. In other words, although the par value of the bond equals $1,000 (an amount you would receive if you held the bond until maturity in the year 2006), if you purchased the bond at the price represented here, you would have to pay $1056.25The price of this bond commands a premium since interest rates have fallen since the bond was issued. If AT&T were to issue a new bond that matures in the year 2006, the coupon would be about 7 1/8%, rather than the 7 1/2% coupon found with the existing bond listed here. How do we know this? The information is contained in the bottom right-hand corner of the table where the yield equals 7.11%. The yield represents the rate of return that another debt instrument of similar attributes would yield in today's markets.
As we can see, current market interest rates for similar debt instruments (based on attributes such as risk and date of maturity) are lower than the coupon of this AT&T bond. As a result, the market price of the bond has risen above its par value in order to equalize rates of return. If you purchase this bond based on the information provided here, you will pay a premium over the par value ($1056.25) and your annual rate of return, or yield will equal to 7.11%.
Remember the bond's coupon remains fixed, so you will receive $75 a year from AT&T in annual bond coupon payments. But the return of the coupon (7 1/2%) is based on a purchase price of $1,000.
T 8 1/8% 22 (NYSE:T8A22) - More Info: N/A Last Trade
3:49PM · 105 1/4Change
-1/8 (-0.12%)Prev Close
105 3/8Volume
580Day's Range
104 5/8 - 105 1/2Bid
N/AAsk
N/AOpen
105Ex-Div
N/A52-week Range
98 7/8 - 105 1/2Earn/Shr
N/AP/E
N/ADiv/Shr
N/AYield
7.74We can perform a similar analysis on the second bond presented here that reads T 8 1/8% 22. This bond pays an annual coupon of 8 1/8% and matures in the year 2022. Since market interest rates have fallen below the original coupon's rate of return, the price of this bond commands a premium.
Present and Future Value
As noted earlier, the link between the present and future in monetary terms is the interest rate (the price of money)
For example, if the interest rate equals 10% then a dollar today is worth $1.10 in one year.
Present Value ($1.00) ---» 10% ---» Future Value ($1.10)
As we can see, the present and future values of money are linked by the interest rate. Here are some important formulas:
FV = PV x (1 + i)
where
FV = Future Value
PV = Present Value
i = the interest rate
PV = FV / (1 + i)
and for multiple years we need to modify the formula to:
PV = FV / (1 + i)n
where n refers to a given year.
Calculating the Present Value for a Coupon Bond
Example 1
Assume that we have a bond with a par value of $1,000 that has a 7% coupon and matures in one year. Market interest rates are expected to fall to 6%. How much should we pay for this bond today?
PV = 70/(1+.06) + 1000/(1+.06) = 1070/(1+.06) = $1,009.43
The first expression in the formula is 70/(1+.06), where $70 is the annual coupon payment on a $1,000 bond with a 7% coupon yield. The expected or current market interest rate is represent by the .06 or 6%. The second part of this formula takes the par value of the bond ($1,000) and discounts it by the market interest rate.
Given our assumptions, the price we should pay for this bond today is $1,009.43.
Example 2
Assume that we have a bond with a par value of $1,000 that has a 7% coupon and matures in two years. Market interest rates are expected to equal 6% during the next two years. How much should we pay for this bond today?
PV = 70/(1+.06)1 + 70/(1+.06)2 + 1000/(1+.06)2
= 70/(1+.06)1 + 1070/(1+.06)2
= 66.04 + 955.36 = $1,021.40
Given our assumptions, the price we should pay for this bond today is $1,021.40
Note that in this example, the coupon in the numerator remains fixed throughout the life of the bond. The only factor allowed to change in the present value calculation is the assumed market interest rate.
A Look at Discount or Zero-coupon Bonds
Example 3
How much to pay for a one-year, $10,000, Treasury-bill (T-bill) issued today, assuming a 5% i rate?
PV = $10,000/(1 + .05) = $9,523.80
Given our assumptions, this bond is sold at a discount for $9,523.80 when it is issued and has no (zero) coupon. In one year's time, when the bond matures, it will pay $10,000 to the bond holder.
Example 4
How much to pay for a three-month, $10,000, Treasury-bill issued today assuming a 5% i rate?
PV = $10,000/(1 + (.05/4)) = $9,876.54
Here the calculation is the same, but since this is only a three-month T-bill, we take the annual 5% return and break it into a quarterly return (.05/4).
Example 5
Assume that we purchase a one-year T-bill for $9,500, what is the associated yield?
The general formula is:
Return (R) = (FV - PV) / PV
or
($10,000 - $9,500) / $9,500 = 5.26%
Example 6
Assume that we purchase a three-month T-bill for $9,800, what is the associated annual yield?
The general formula is:
($10,000 - $9,800) / $9,800 = 2.04% * 4 = 8.16%
We take the quarterly return, 2.04% and annualize it by multiplying by 4.
The Real Interest Rate
Real variables are equal to their nominal value minus the inflation rate. Thus the real rate of interest is calculated by taking the nominal interest rate minus the inflation rate. For example, if the market (nominal) interest rate equals 7% and the annual inflation rate equals 3%, then the real rate of interest equals 4%.
Rather than using the actual inflation rate, the Fisher hypothesis uses the expected inflation rate to determine the real rate of interest.
r = i - the expected inflation rate
Often lenders such as banks desire to maintain a minimum real rate of return. Using the above example, assume that the lender desires to maintain the real return at 4%. If expected future inflation rates rise to 4% (although the current inflation rate is 3%), then the nominal lending rate will be raised to 8% to maintain a real return of 4%.
Overall, the key to the determination of nominal interest rates is not actual inflation rates but expected future rates of inflation.
Readings
Hubbard Chapter 4