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Senin, 08 November 2010

Lecture Notes on Mishkin Chapter 4 ("Understanding Interest Rates") Part A (pp. 67-80) Econ 300: Money, Banking, and Financial Institutions

Department of Economics, Faculty of Economics The University of Lampung Odd Semester (September – December 2010)

Last Updated: September- December , Anno 2010
Latest Course Offering: Odd Semester 2010
Course Instructor: Dr. Yoke Muelgini, M.Sc
ESP 330 email: ekonomimoneter@ymail.com

·         Introduction
·         Basic Types of Debt Instruments
·         The Concept of Present Value
·         Measuring Interest Rates by Yield to Maturity
·         Other Measures of Interest Rates
·         How to Read Financial Bond Pages
·         Interest Rates vs. Return Rates
·         Real vs. Nominal Interest Rates
·         Interest Rate Risk
·         Basic Concepts and Key Issues from Mishkin Chapter 4



Introduction
Assets are stores of value that are primarily held for the services they generate. For example, housing provides shelter, equipment provides capital services in production, and textbooks help to transmit information.
In Chapter 3, Mishkin primarily focuses on money as a medium of exchange and a unit of account. However, money is also an asset that can be held in portfolios just like any other asset. The primary service provided by money is purchasing power.
Temporary control over purchasing power can be obtained by borrowing money via the issue and sale of various types of debt instruments. Roughly speaking, the borrowing fee for money is called "interest," frequently expressed as a percentage (the "interest rate"). Interest is a reward to the lender for parting with purchasing power for a specified period of time.
Following Mishkin's Chapter 4 (Part A = pp.67-78), the sections below focus on the meaning and measurement of interest rates for key types of debt instruments issued and bought in U.S. financial markets. Remaining topics covered in Mishkin's Chapter 4 (Part B = pp.78-92) will be taken up in subsequent notes.
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Basic Types of Debt Instruments
In Chapter 2, Mishkin defines debt instruments (equivalently, credit market instruments) to be particular types of contractual agreements that require the borrower to pay the lender certain fixed dollar amounts at regular intervals until a specified time is reached. In Chapter 4, Mishkin provides a more detailed discussion of four basic types of debt instruments that are distinguished from one another by their payment provisions.
Important Remark: It is assumed below that all bond sales and purchases are for newly issued bonds, so that the sellers and buyers are enabling new borrowing. Bonds can also be re-sold in secondary markets. In secondary market exchanges, the sellers are NOT borrowers; i.e., they are NOT acquiring command over additional purchasing power. Rather, they are simply readjusting the composition of their asset portfolios (more money, less bonds). Moreover, buyers of bonds in secondary markets are not enabling any new borrowing (i.e., they are not original lenders); they are simply acquiring entitlements to payment streams that had previously been owned by others.
·         Simple Loan Contracts:
Under the terms of a simple loan contract, the borrower (contract issuer) receives from the lender (contract buyer) a specified amount of funds (the loan value LV or principal) for a specified period of time (the maturity). The borrower agrees that, at the end of this period of time -- referred to as the maturity date -- the borrower will repay the loan value to the lender together with an additional payment referred to as the interest payment.
    Borrower  
  Receives:  Loan Value LV
                   |                           
             START |___________________________ MATURITY DATE
                                               |
                                               |
  Lender                                  Loan Value LV
  Receives:                           + Interest Payment I
The annual borrowing fee for a simple loan with a loan value LV, an interest payment I, and a maturity of N years is measured by the simple interest rate given by I divided by [LV times N].
Important Remark: Mishkin always implicitly assumes that the maturity N on simple loans is one year (N=1). As will be clarified further below, his assertion "for simple loans, the simple interest rate equals the yield to maturity" is only true if N=1 is assumed for the simple loans and the yield to maturity is calculated as an annual rate.
Example of a Simple Loan Contract:
A borrower receives a loan on January 1, 1999, in amount $500.00, and agrees to pay the lender $550.00 on January 1, 2001. Thus, the loan value is $500.00, the maturity is two years, the maturity date is January 1, 2001, and the interest payment is $50.00. The simple (annual) interest rate for this loan is then $50/[$500*2] = .05, or 5 percent.
Real-World Examples of Simple Loan Contracts: Standard bank deposit accounts take this form. Also, various money market instruments (e.g., commercial loans to businesses) can take this form.
·         Fixed-Payment Loan Contracts:
Under the terms of a fixed-payment loan contract, the borrower (contract issuer) receives from the lender (contract buyer) a specified amount of funds -- the loan value -- and, in return, makes periodic fixed payments to the lender until a specified maturity date. These periodic fixed payments include both principal (loan value) and interest, so at maturity there is no lump sum repayment of principal.

    Borrower  
  Receives:  Loan Value LV
                   |                           
             START |___________________________ MATURITY DATE
                                               |
                                               |
  Lender                                  Loan Value LV
  Receives:                           + Interest Payment I

Example of a Fixed-Payment Loan Contract:
Joe arranges a 15-year installment loan with a finance company to help pay for a new car. Under the terms of this loan, Joe receives $20,000 now to finance the purchase of a new car but must make payments of $2000 every year for the next 15 years to the finance company.
Real-World Examples of Fixed Payment Loan Contracts: Installment loans (e.g., auto loans) and home mortgages typically take this form.
·         Coupon Bond:
Under the terms of a coupon bond, the borrower (bond issuer) agrees to pay the lender (bond buyer) a fixed amount of funds (the coupon payment) on a periodic basis until a specified maturity date, at which time the borrower must also pay the lender the face value (or par value) of the bond. The coupon rate of a coupon bond is, by definition, the amount of the coupon payment divided by the face value of the bond.
As will be clarified in the next section, below, the purchase price of a coupon bond depends on the "present value" of the stream of anticipated coupon payments plus the final face value payment promised by the bond. Coupon bonds that sell above their face value are said to sell at a premium, and those that sell below their face value are said to sell at a discount



  Borrower     Purchase
  Receives:    Price Pb
                 |                                     MATURITY
           START |_______________________ /\/\/\ _____ DATE
                        |           |                |
                        |           |                |
  Lender             Coupon       Coupon    ...    Coupon
  Receives:         Payment C    Payment C        Payment C
                                               + Face Value F


Example of a Coupon Bond:
Suppose a coupon bond has a face value of $1000, a maturity of five years, and an annual coupon payment of $60. Then, at the end of each year for the next five years, the borrower (bond issuer) must pay the lender (bond buyer) a coupon payment of $60. In addition, at the end of five years (the maturity date), the borrower must pay the lender the face value of the bond, $1000. The coupon rate for this coupon bond is $60/$1000 = .06, or 6 percent.
Real-World Examples of Coupon Bonds: Capital market instruments such as U.S. Treasury bonds and notes take this form. Corporate bonds also typically take this form.
·         Discount Bond (or Zero-Coupon Bond):
Under the terms of a discount bond, the borrower (bond issuer) immediately receives from the lender (bond buyer) the purchase price Pd of the bond, which is typically less than the face value F of the bond. In return, the borrower promises that, at the bond's maturity date, he will pay the lender the face value F of the bond.

  Borrower      Purchase
  Receives:     Price Pd
                    |                         
              START |_________________________ MATURITY DATE
                                              |
                                              |
  Lender                                 Face Value F
  Receives:           

Important Cautionary Remarks:
The above definition of a discount bond follows the definition used in Mishkin. Some other authors refer to zero-coupon bonds as PURE discount bonds, labelling as a "discount bond" any bond that sells at a discount in the sense that its market price is less than its face value.
Also, Mishkin asserts that discount bonds make no interest payments. While this is literally true, in the sense that only a face value payment is made, it is NOT true that the interest RATE on discount bonds is zero. Indeed, as will be seen below, the most basic measure of interest rates in use today is the annual "yield to maturity" i. For a one-year discount bond, the formula for calculating i reduces to i = [F-Pd]/Pd, hence i is only zero in the highly unlikely event that F=Pd.
Discount Bond Example:
On January 1, 1999, a borrower gives a lender a discount bond with a face value of $200 and a maturity of 2 years, and the lender gives $150 to the borrower. The borrower must then pay the lender $200 on January 1, 2001.
Real-World Examples of Discount Bonds: U.S. Treasury bills and U.S. savings bonds take this form.
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The Concept of Present Value
Suppose someone promises to pay you $100 in some future period T. This amount of money actually has two different values: a nominal value of $100, which is simply a measure of the number of dollars that you will receive in period T; and a present value (sometimes referred to as a present discounted value), roughly defined to be the minimum number of dollars that you would have to give up today in return for receiving $100 in period T.
Stated somewhat differently, the present value of the future $100 payment is the value of this future $100 payment measured in terms of current (or present) dollars.
The concept of present value permits debt instruments with different associated payment streams to be compared with each other by calculating their values in terms of a single common unit: namely, current dollars.
Specific formulas for the calculation of present value for future payments will now be developed and applied to the determination of present value for debt instruments with various types of payment streams.
Present Value of Payments One Period Into the Future:
If you save $1 today for a period of one year at an annual interest rate i, the nominal value of your savings after one year will be

(1)           (1+i)*$1   ,
where the asterisk "*" denotes multiplication.
On the other hand, proceeding in the reverse direction from the future to the present, the present value of the future dollar amount (1+i)*$1 is equal to $1. That is, the amount you would have to save today in order to receive back (1+i)*$1 in one year's time is $1.
More generally, let V(1) denote any amount of money to be received at the end of one year from now, and suppose the annual interest rate is i. Then, the present value of V(1) -- that is, the value of V(1) measured in dollars today -- is defined to be
                                     V(1)
(2)      Present Value    =  -----   .
           of V(1)           (1+i)
In effect, then, the payment V(1) to be received one year from now has been discounted back to the present using the "discount factor" (1+i), so that the value of V(1) is now expressed in current dollars.
Present Value of Payments Multiple Periods Into the Future:
If you save $1 today at a fixed annual interest rate i, what will be the value of your savings in one year's time? In two year's time? In n year's time?
For any integer n, let V(n) denote the nominal value obtained after n years when a single dollar is saved for n successive years at the fixed annual interest rate i. If you save $1 for one year, the nominal value of your savings in one year's time will be V(1)=(1+i)*$1. If you then put aside V(1) as savings for an additional year rather than spend it, the nominal value of your savings at the end of the second year will be
(3) V(2) = (1+i)*V(1) = (1+i)*(1+i)*$1 = (1+i)2*$1 .
And so forth for any number of years n.


(4)    START --------------------------------/\/\/\-------->YEAR
            |         1             2                 n
            |

Nominal                              2                 n
Value of   $1      (1+i)*$1     (1+i) *$1         (1+i)  * $1
Savings:

Now consider the present value of V(n) = (1+i)n*$1 for any year n. By construction, V(n) is the nominal value obtained after n years when a single dollar is saved for n successive years at the fixed annual interest rate i. Consequently, the present value of V(n) is simply equal to $1, regardless of the value of n.
Notice, however, that the present value of V(n) -- namely, $1 -- can be obtained from the following formula:

                                    V(n)
(5)      Present Value     =     ------------    .
         of V(n)                   (1+i)n     
                               
Indeed, given any fixed annual interest rate i, and any nominal amount V(n) to be received n years from today, the present value of V(n) is defined as in formula (5).
Present Value of Any Arbitrary Payment Stream:
Now suppose you will be receiving a sequence of three payments over the next three years. The nominal value of the first payment is $100, to be received at the end of the first year; the nominal value of the second payment is $150, to be received at the end of the second year; and the nominal value of the third payment is $200, to be received at the end of the third year.
Given a fixed annual interest rate i, what is the present value of the payment stream ($100,$150,$200) consisting of the three separate payments $100, $150, and $200 to be received over the next three years?
To calculate the present value of the payment stream ($100,$150,$200), use the following two steps:
·         Step 1: Use formula (5) to separately calculate the present value of each of the individual payments in the payment stream, taking care to note how many years into the future each payment is going to be received.
·         Step 2: Sum the separate present value calculations obtained in Step 1 to obtain the present value of the payment stream as a whole.
Carrying out Step 1, it follows from formula (5) that the present value of the $100 payment to be received at the end of the first year is $100/(1+i). Similarly, it follows from formula (5) that the present value of the $150 payment to be received at the end of the second year is
                       $150
(6)                 ----------
                      (1+i)2
Finally, it follows from formula (3) that the present value of the $200 payment to be received at the end of the third year is
                       $200
                    ----------
(7)                   (1+i)3
Consequently, adding together these three separate present value calculations in accordance with Step 2, the present value PV of the payment stream ($100,$150,$200) is given by
(8)
PV =
$100   +
$150   +
$200  
(1 + i)1
(1 + i)2
(1 + i)3
More generally, given any fixed annual interest rate i, and given any payment stream (V1,V2,V3,...,VN) consisting of individual payments to be received over the next N years, the present value of this payment stream can be found by following the two steps outlined above.
In particular, then, given any fixed annual interest rate, and given any debt instrument with an associated payment stream paid out on a yearly basis to the lender (debt instrument buyer), the present (current dollar) value of this debt instrument is found by calculating the present value of its associated payment stream in accordance with Steps 1 and 2 outlined above.
KEY POINT: Regardless how different the payment streams associated with two different debt instruments might be, one can calculate the present values of the payment streams for these debt instruments in current dollar terms and hence have a meaningful way to compare them.
Technical Note:
The above procedure for calculating the present value of debt instruments whose payments are paid out on an annual basis to lenders can be generalized to debt instruments whose payments are paid out at arbitrary times to lenders. To do this, one only needs to be sure that the interest rate used to discount each payment covers a period of time that matches the period of this payment.
For example, to convert an annual interest rate to a monthly interest rate, you divide the annual interest rate by 12 (the number of months in a year). Thus, for example, if the annual interest rate is .12 (i.e., 12 percent), this is equivalent to a monthly interest rate of .01 (i.e., 1 percent). Consequently, if a payment V is to be received at the end of the next month, and the annual interest rate is .12, then the present value of this payment V is V/(1+.01).
Similarly, to convert an annual interest rate to a quarterly (three month) interest rate, you divide the annual interest rate by 4. Thus, a 12 percent annual interest rate is equivalent to a 3 percent quarterly interest rate.
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Measuring Interest Rates by Yield to Maturity
By definition, the current yield to maturity for a marketed debt instrument is the particular fixed annual interest rate i which, when used to calculate the present value of the debt instrument's future stream of payments to the instrument's holder, yields a present value equal to the current market value of the instrument.
Mishkin discusses and illustrates the calculation of the yield to maturity for the four basic types of debt instruments introduced in the first section of these notes, above. Below we review this calculation for two of these debt instrument types: fixed-payment loan contracts and coupon bonds.
Yield to Maturity for Fixed-Payment Loan Contracts:
Recall from previous discussion the general form of a fixed-payment loan contract:

  Borrower
  Receives:  Loan Value LV
                  |                                   MATURITY
            START |__________________________________ DATE
                            |           |            |
                            |           |            |
  Lender                  Fixed       Fixed        Fixed
  Receives:            Payment FP   Payment FP   Payment FP


Consider a particular fixed-payment loan contract with a loan value LV = $5000, annual fixed payments FP = $660.72, and a maturity of N = 20 years. What is the yield to maturity for this loan contract?
The first question that must be answered is what is the current value of this loan contract at the date of its issuance?
The borrower (contract issuer) receives from the lender (contract buyer) the $5000 loan value at the date the loan contract is issued. This $5000 loan value, then, constitutes the current value of the loan contract. It is, in effect, the price paid by the lender to purchase the loan contract from the borrower.
By definition, then, the yield to maturity of this fixed-payment loan contract is the particular fixed annual interest rate i which, when used to calculate the present value of the loan contract, results in a present value that is exactly equal to $5000, the current value of the loan contract.
More precisely, for any fixed annual interest rate i, let PV(i) denote the present value of the lender's payment stream under this fixed payment loan contract when calculated using this interest rate i. Then the way you determine the yield to maturity on this fixed payment payment loan contract is you calculate the particular interest rate i that satisfies the formula
(9) $5000 = PV(i) .
This formula will now be developed step by step.
Using the discussion in the previous section, given any fixed annual interest rate i, the present value of the fixed-payment loan contract at hand -- that is, the present value of the payment stream to the lender generated by this loan contract -- is found as follows.
The payment stream to the lender generated by this loan contract consists of twenty successive yearly fixed payments, each having the nominal value FP=$660.72. Using formula (3), given any year n, n = 1,...,20, and any fixed annual interest rate i, the present value of the particular fixed payment FP = $660.72 received at the end of year n is
FP/(1+i)n .
Consequently, given any fixed annual interest rate i, the present value PV(i) for the fixed payment loan contract as a whole is given by the sum of all of these separate present value calculations for the fixed payments FP received by the lender (debt instrument holder) at the end of years 1 through 20, i.e.,
(10) PV(i) = FP/(1+i) + FP/(1+i)2 + ... + FP/(1+i)20.
Since the current value of the loan contract is $5000, the desired yield to maturity is then found by solving equation (9) for i with PV(i) given explicitly by equation (10).
Because the present value PV(i) depends in a rather complicated way on i, the determination of i from formula (9) is not straightforward. To make life easier, tables have been published that can be used to determine yields to maturity for various types of fixed-payment loan contracts once the current value and fixed payments of the loan are known. For example, using such tables, it can be shown that the solution for i in equation (9) above is approximately i = .12. That is, the yield to maturity i for a fixed-payment loan contract with a current value of $5000, with annual fixed payments of $660.72, and with a maturity of twenty years, is approximately 12 percent.
Yield to Maturity for Coupon Bonds:
Recall from previous discussion the basic contractual terms of a coupon bond:


 Borrower     Purchase
  Receives:    Price Pb
                 |                                     MATURITY
           START |_______________________ /\/\/\ _____ DATE
                        |           |                |
                        |           |                |
                     Coupon       Coupon    ...    Coupon
  Lender            Payment C    Payment C        Payment C
  Receives:                                    + Face Value F
Consider a coupon bond whose purchase price is Pb=$94, whose face value is F = $100, whose coupon payment is C = $10, and whose maturity is 10 years. By definition, the coupon rate for this bond is equal to C/F = $10/$100 = .10 (i.e., 10 percent).
The payment stream to the lender generated by this coupon bond is given by
(11) ( $10, $10, $10, $10, $10, $10, $10, $10, $10, [$10 + $100] ).
For any given fixed annual interest rate i, the present value PV(i) of the payment stream (11) is given by the sum of the separate present value calculations for each of the payments in this payment stream as determined by formula (5). That is,
(12) PV(i) = $10/(1+i) + $10/(1+i)2 + ... + $10/(1+i)9 + [$10 + $100]/(1+i)10 .
The current value of the coupon bond is its current purchase price Pb = $94. It then follows by definition that the yield to maturity for this coupon bond is found by solving the following equation for i:
(13) Pb = PV(i) ,
where PV(i) is as given in (12). The calculation of the yield to maturity i from formula (13) can be difficult, but tables have been published that permit one to read off the yield to maturity i for a coupon bond once the purchase price, the face value, the coupon rate, and the maturity are known.
For example, using such tables, it can be shown that the yield to maturity i for the coupon bond currently under consideration, which has a purchase price of $94 per $100 of face value, a coupon rate of 10 percent, and a maturity of 10 years, is approximately equal to 11 percent.
More generally, given any coupon bond with purchase price Pb, face value F, coupon payment C, and maturity N, the yield to maturity i is found by means of the following formula:
(14a) Pb = PV(i) ,
where the present value PV(i) of the coupon bond is given by
(14b) PV(i) = C/(1+i) + C/(1+i)2 + ... + C/(1+i)N-1 + [C+F]/(1+i)N .
Some Final Important Observations on Yield to Maturity:
For any coupon bond with a given coupon payment C, face value F, and maturity N, the purchase price Pb of the bond is equal to the face value F if and only if the yield to maturity i for the bond is equal to the coupon rate C/F.
This observation follows directly from the structure of a coupon bond. When the purchase price equals the face value, the coupon bond essentially functions as a bank deposit account into which a principal amount (the face value) is deposited by a lender, earns a fixed annual interest rate (the coupon rate) for N years, and is then recovered by the lender.
Illustration for a One-Period Coupon Bond:
For any coupon bond with a given coupon payment C, a given face value F, a given maturity N=1, and a given purchase price Pb, the formula Pb = PV(i) for determining the yield to maturity i can be written as

                            F  +  C
(15)             Pb   =    ----------  .
                             (1+i)

Dividing each side of formula (15) by the face value F, one obtains

                                  1  +  C/F
(16)             Pb/F   =       -----  .
                                (1+i)

Given C, F, and N=1, formula (16) implies that Pb equals F (i.e., the left-hand side equals 1) IF AND ONLY IF i equals C/F (i.e., the right-hand side equals 1).
More generally, for any coupon bond with a given coupon payment C, given face value F, and given maturity N, the purchase price Pb of the bond is lower (higher) than F if and only if the yield to maturity i is higher (lower) than the coupon rate C/F. This follows directly from formula (14) for determination of the yield to maturity, using the previously noted fact that the purchase price Pb is equal to F if and only if the yield to maturity i is equal to the coupon rate C/F.
For example, suppose formula (14) holds with Pb = F and i = C/F. Taking C, F, and N=1 as given, consider what happens if the yield to maturity i now increases, so that i exceeds the coupon rate C/F. Since C and F are given, PV(i) decreases, which implies that Pb must also decrease. Since F is given, and Pb was originally equal to F, this implies that Pb must now be lower than F.
Moreover, for any coupon bond with a given coupon payment C, face value F, and maturity N, the yield to maturity i of the bond is inversely related to the purchase price Pb of the bond. That is, the higher the yield to maturity i, the lower the purchase price Pb, and conversely. This inverse relationship also follows directly from formula (14).
To see this, consider what happens when i increases in formula (14), keeping C, F, and N fixed. When i increases, the denominator (1+i) of the discounted coupon payment C/(1+i) appearing in PV(i) in formula (14) increases, implying that the ratio C/(1+i) is smaller than before, and similarly for each of the other discounted coupon payments that are summed to obtain PV(i) in (14). Consequently, PV(i) decreases. It then follows from formula (14) that Pb also decreases.
This inverse relationship between the yield to maturity of a debt instrument and its purchase price actually holds in general. For any debt instrument with any given payment stream, when the yield to maturity for the debt instrument rises, the purchase price of the debt instrument must fall, and vice versa. This follows directly from the general definition for the yield to maturity, applicable to all debt instruments.
Other Measures of Interest Rates
Mishkin observes that the yield to maturity is the most accurate measure of interest rates and notes that he will henceforth use the terms "interest rate" and "yield to maturity" interchangeably throughout the remainder of his text.
Nevertheless, since the yield to maturity can be difficult to calculate, other less accurate measures of interest rates are commonly used in the financial pages of newspapers and elsewhere to report the properties of debt instruments. Mishkin discusses two such measures at some length: "current yield" and "discount yield."
Current Yield:
The current yield is an approximation to the yield to maturity for coupon bonds. More precisely, letting Pb denote the purchase price of a coupon bond, and C denote its coupon payment, the current yield, denoted below by ic (as in Mishkin), is given by:
                           C
(1)        ic      =     -----   .
                           Pb
In general, for most coupon bonds, the current yield will differ in value from the yield to maturity. However, it can be shown that the current yield equals the yield to maturity for a special type of coupon bond, called a "consol." A consol is a coupon bond that has an infinite maturity and hence never repays its face value F. Rather, the holder of a consol receives a coupon payment C in perpetuity -- that is, in each future payment period without end -- implying that the payment stream to the holder takes the special form (C,C,C,...).
As detailed by Mishkin, the formula Pb = PV(i) for determining the yield to maturity i for a consol reduces to
                    C                                     C
(2)        Pb  =   ---    ,   which implies that  i   =  ---  .
                    i                                     Pb

Comparing (1) and (2), it follows that -- for a consol -- the current yield ic equals the yield to maturity i because both are equal to C/Pb.
For coupon bonds with less than infinite maturities, the current yield ic no longer coincides with the yield to maturity i. However, the current yield becomes an increasingly better approximation for the yield to maturity as the maturity of a coupon bond becomes longer and longer (hence closer and closer to the infinite maturity of a consol). That is, all else remaining the same, ic provides an increasingly accurate approximation to i as one considers coupon bonds with successively longer maturities N.
For fixed C, F, and Pb:

                           implies
(3)  Maturity N increases  -------> ic approaches i

Another aspect of a coupon bond that determines how accurate an approximation ic provides to the yield to maturity i is the difference between the bond's purchase price Pb and its face (or par) value F.
From formula (1), it is seen that the current yield ic is equal to the coupon rate C/F for a coupon bond when the purchase price Pb equals the face value F. As previously seen, however, the coupon rate C/F equals the yield to maturity i when the purchase price Pb of a coupon bond equals its face value F.
Consequently, all else remaining the same, the current yield ic provides an increasingly more accurate approximation to the yield to maturity i as one considers coupon bonds whose purchase prices Pb are successively closer to their face values F.
Given C, F, and N:

                         implies
(4)    Pb approaches F   ------->   ic approaches i  .

Finally, it follows directly from definition (1) for the current yield ic that, given any fixed value for the coupon payment C, ic is inversely related to the bond purchase price Pb. That is,
Given C:


                     if and only if
(5)   ic increases  <-------------->  Pb decreases  .

Recall from previous discussion that, for given C, F, and N, the yield to maturity i is also inversely related to a bond's purchase price Pb. Consequently, one obtains the following important conclusion:
For a coupon bond with a given coupon payment C, face value F, and maturity N, the current yield ic and the yield to maturity i always move together in response to changes in the purchase price Pb.
Note that this positive co-movement between ic and i holds even if ic is a bad approximation to i in level terms in the sense that the difference between ic and i is large.
Discount Yield:
U.S. Treasury bills are an example of a discount bond. For ease of calculation, interest rates on many discount bonds such as Treasury bills and commercial paper are quoted on a 360-day "discount yield" basis (or "bank discount basis") rather than on a yield-to-maturity basis, as follows.
Let F denote the face value of a discount bond, and let Pd denote the purchase price of the discount bond. Then the discount yield, denoted below by idb (as in Mishkin), is given by:
                   F - Pd          360
(6)      idb   =  -------- * -----------------   .
                     F       Days to Maturity
Let us see how idb compares, for example, to the yield to maturity i for a one-year discount bond. As discussed by Mishkin, in the case of a one-year discount bond the usual formula Pd = PV(i) for determining the yield to maturity takes the form
                    F  -  Pd
(7)         i  =   ----------  .
                       Pd
Comparing (7) with (6) for the special case of a discount bond with a one year maturity (i.e., days to maturity = 365), it follows that
                          F         365
(8)     i   =    idb *  -----  *  -------  .
                          Pd        360

Consequently, recalling that discount bonds are typically priced at a discount (Pd < F), it follows that the yield to maturity i for a discount bond with a one-year maturity is typically greater than the discount yield idb.
Another implication of formula (6) is that, for any given discount bond with a fixed face value F and a fixed maturity N, the discount yield idb is inversely related to the price Pd of the discount bond -- that is, when idb increases, Pd decreases, and vice versa.
Recall from previous notes that the yield to maturity i on a discount bond is also inversely related to the purchase price Pd. This follows directly from the general formula Pd=PV(i) used to determine i for discount bonds -- see, for example, relation (7), which is what the general formula Pd=PV(i) reduces to when the discount bond has a one-year maturity.
Consequently, as for the current yield, one obtains the following important observation:
For any given discount bond with a fixed face value F and a fixed maturity, the discount yield idb and the yield to maturity i always move together in response to changes in the purchase price Pd.
As for the current yield, this positive co-movement between idb and i holds even if idb is a bad approximation to i in level terms in the sense that the difference between idb and i is large.
How to Read Financial Bond Pages
Mishkin provides a detailed discussion concerning how an understanding of the previously discussed interest rate measures will permit you to make sense out of the tables found in the financial sections of newspapers and magazines that report on U.S. Treasury debt instruments and corporate bonds traded on stock exchanges. He illustrates his discussion using financial pages from the the Wall Street Journal.
This section reviews the main points of Mishkin's discussion, using the financial pages from the February 17, 1999 issue of the New York Times to illustrate and elaborate key points. As will be seen, the form in which financial information is reported in the New York Times is essentially the same as in the Wall Street Journal, with only minor notational differences.
Treasury Bonds and Notes:
Treasury bonds (T-bonds) are coupon bonds with a maturity greater than ten years, and Treasury notes (T-notes) are coupon bonds with a maturity of between one and ten years. As in the Wall Street Journal, the New York Times provides a single table reporting on T-bonds and T-notes because both have the same structure.
Below is a sample listing from the T-bonds and T-notes table appearing in the New York Times (February 17, 1999, p. C17) which reports information for the previous trading day, February 16, 1999:

      Month         Rate      Bid      Ask     Chg     Yld

    Feb  00    p    7 1/8   102.08    102.10  +0.01    4.80

    Aug 03-08       8 3/8   112.25    112.27  -0.05    5.14

Remark on Notation: For expositional simplicity, T-bonds and T-notes will hereafter be lumped together and simply referred to as "bonds."
The first column (Month) of the sample listing identifies the month and year that a bond matures. If two maturity dates are shown, the second is the actual maturity date. The first indicates the first date at which the bond might be called. A bond is called when it is retired early at the discretion of the issuer (borrower), an event that generally happens only when prevailing interest rates lie below the coupon rate.
A footnote may next be provided to indicate that a bond has some special feature. For example: the letter "p" denotes "T-note, non-resident aliens exempt from withholding tax"; the letter "k" denotes "T-bond, non-resident aliens exempt from withholding"; and the letters "zr" denote "zero coupon issue".
Remark on Treasury Zero Coupon Issues: Coupon stripping is the act of removing the individual coupon payments from a coupon bond and treating each payment as a separate zero-coupon bond. The remaining face-only bond is then also in effect a zero-coupon bond. For example, a 20-year bond with a face value of $100,000 and an annual coupon payment of $10,000 could be stripped into 21 separate zero-coupon instruments: namely, 20 "interest strips" consisting of the 20 annual coupon payments of $10,000, each due on the specified annual coupon payment date; and one "principal strip" consisting of an instrument having a face value of $100,000 due in 20 years.
Merrill Lynch began the market for stripped securities in 1982. The U.S. Treasury introduced stripping of its coupon bond issues in February 1985, referring to the resulting zero-coupon securities as STRIPs (Separate Trading of Registered Interest and Principal of Securities). U.S. Treasury strips are mentioned in the explanatory notes for the Wall Street Journal bonds and notes table presented by Mishkin (page 81).
The second column (Rate) identifies a bond's annual coupon rate, i.e., the annual coupon payment as a percentage of face value. Usually this annual coupon payment is paid in two equal semiannual installments.
The third and fourth columns (Bid, Ask) provide information about a bond's bid and asked prices, which by convention are quoted as a percentage per $100 of face value (so that 100 = face value) with fractions in 32s. Unlike the Wall Street Journal, which uses a colon to indicate fractional values in 32s (e.g., 102:08 = 102 8/32), the New York Times uses a decimal point (e.g., 102.08 = 102 8/32).
The New York Times lists Street Software/Bear Stearns as the source of its price quotations. These quotations are not necessarily prices at which an individual could actually have traded on the trading day in question but rather represent the approximate market prices that prevailed.
More precisely, the bid price quoted for a bond is the approximate market price offered by prospective buyers of the bond on the trading day in question, so it indicates approximately how much you would have received if you had sold the bond on that day. In contrast, the asked price quoted for a bond is the approximate market price demanded by prospective sellers of the bond on the trading day in question, so it indicates approximately how much you would have had to pay to purchase the bond on that day.
The asked price minus the bid price -- referred to as the bid-ask spread -- reflects the gross profit margin of the bond dealers who handle trades in this bond. Hence, for obvious reasons, the asked price always exceeds the bid price.
The fifth column (Chg) indicates the change in the BID price from the previous trading day's quotation.
The sixth and final column (Yld) provides the yield to maturity on the bond using the currently quoted ASKED price as the purchase price.
Treasury bills:
Treasury bills (T-bills) are discount bonds with a maturity of one year or less. Consequently, they have no coupon rate and are identified solely by their maturity date.
Below is a sample listing from the T-bills table appearing in the New York Times (February 17, 1999, page C17) which reports information for the previous trading day, February 16, 1999:
      Date         Bid      Ask      Chg      Yield

    Feb 25 99      4.07     4.05   - 0.04      4.11

    Aug 19 99      4.46     4.44   + 0.04      4.61
The first column (Date) gives the month, day, and year of the maturity date.
The second column (Bid) gives the discount yield (6) in percentage terms using as the purchase price Pd the BID price, i.e., the price offered by prospective buyers. The third column (Ask) gives the discount yield (6) in percentage terms using as the purchase price Pd the ASKED price, i.e., the price demanded by prospective sellers.
Recall that the discount yield varies inversely with the purchase price Pd. It follows that, for T-bill issues, the bid discount yield reported in the Bid column is always greater than the asked discount yield reported in the Ask column, indicating that the bid price is less than the asked price.
The fourth column (Chg) reports the change in the asked discount yield from the previous trading day measured in terms of basis points, which are hundredths of a percentage point (e.g., -0.04 means the asked discount yield has fallen in percentage terms by 4 basis points).
The fifth and final column (Yield) provides the yield to maturity using the current ASKED price as the current value.
Corporate Bonds Traded on Stock Exchanges:
As noted by Mishkin, corporate bonds typically take the form of coupon bonds.
A majority of bonds, and all municipal or tax-exempt bonds, are not listed on exchanges; rather, they are traded over-the-counter. However, the New York Stock Exchange (NYSE), and to a much less extent the American Stock Exchange (AMEX), do list various coupon bonds issued by corporations with strong credit ratings.
Below is a sample listing from the NYSE corporate bond table appearing in the New York Times (February 17, 1999, page C17) which reports information for the previous trading day, February 16, 1999:
  Company   Coupon    Mat.   Cur.Yld.   Vol.    Price      Chg.
             Rate

    ATT      5 1/8     01      5.1       60    99 7/8    -   1/8

  ARetire    5 3/4     02      cv        25    89 1/2    +  1 1/2

The first column (Company) shows the issuing company, the second column gives the original coupon rate, and the third column gives the last two digits of the maturity year. The fourth column reports the annual current yield (Cur. Yld.). In some cases, a footnote may instead be inserted to call attention to a special feature of the bond; for example, the letters "cv" in the above table denote "convertible into stock under special conditions".
The remaining three columns report the number of bonds traded for the day measured in $1000 face value (Vol.), the bond's closing price for the day expressed as a percentage of face value with 100 equaling face value (Price), and the difference between the current trading day's closing price and the previous trading day's closing price (Chg).
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Interest Rates vs. Return Rates
Given any asset A held over any given time period T, the return to A over the holding period T is, by definition:
·         the sum of all payments (rents, coupon payments, dividends, etc.) generated by A during period T, assumed paid out at the end of the period,
·         PLUS the capital gain (+) or loss (-) in the market value of A over period T, measured as the market value of A at the end of period T minus the market value of A at the beginning of period T.
The return rate on asset A over the holding period T is then defined to be the return on A over period T divided by the market value of A at the beginning of period T.
More precisely, suppose that an asset A is held over a time period that starts at some time t and ends at time t+1. Let the market value of A at time t be denoted by P(t) and the market value of A at time t+1 be denoted by P(t+1). Finally, let V(t,t+1) denote the sum of all payments accruing to the holder of asset A from t to t+1, assumed to be paid out at time t+1.
Then, by definition, the return rate on asset A from t to t+1 is given by the following formula:
          
(9)  Return Rate on      V(t,t+1)  +  P(t+1) - P(t)
     Asset A From     =  ---------------------------
     time t to t+1                  P(t)

                         V(t,t+1)         P(t+1) - P(t)
                      =  ---------   +    -------------
                           P(t)                P(t)

                      = payments     +   Capital Gain (if +)
                        received as      or Loss (if -) as a
                        percentage       percentage of P(t)
                        of P(t)

Formula (9) holds for any asset A, whether physical or financial. In particular, it holds for bonds. The question then arises: For bonds, what is the connection between the return rate defined by formula (9) and the interest rate on the bond defined by yield to maturity, current yield, or discount yield?
As discussed by Mishkin, the return rate on a bond is not necessarily equal to the interest rate on that bond, whether defined by yield to maturity, the current yield, or the discount yield.
The reason for this is that the return rate calculated for a particular holding period takes into account any capital gains or losses that occur during this holding period, in addition to payments received during the holding period. In contrast, the current yield ignores capital gains and losses altogether, and the yield to maturity and the discount yield only take into account the overall anticipated capital gain or loss that is incurred when the bond is held to maturity (as measured by the difference between the final face value payment and the initial purchase price).


Example: Coupon Bonds
Suppose you purchase a coupon bond at time t at a purchase price P(t) which has a coupon payment C, a face value F, and a maturity date at time t+k with k GREATER than 1. Suppose you receive a coupon payment C at time t+1, and you also sell the coupon bond in a secondary market at time t+1 at a price P(t+1). By definition, the current yield that you receive on this coupon bond during the holding period from t to t+1 is given by
                       C
(10)       ic(t)  =  ------   .
                      P(t)
Also, the percentage capital gain or loss you incur on the coupon bond during the holding period from t to t+1, denoted by g(t,t+1), is given by
                                 P(t+1)  -  P(t)
(11)          g(t,t+1)     =    -----------------    .
                                       P(t)
It then follows from definition (9) that the return rate on the coupon bond from t to t+1 can be expressed as

           C  +  P(t+1) - P(t)
(12)      ---------------------   =    ic(t)   +   g(t,t+1)  .
                  P(t)

Clearly the return rate (12) coincides with the current yield ic(t) if

(13)           P(t)   =   P(t+1)  .

Condition (13) implies that there are no capital gains or losses on the coupon bond during the holding period from t to t+1, i.e., g(t,t+1) = 0. Conversely, if condition (13) fails to hold, then the return rate (12) does not coincide with the current yield ic(t).
Thus, condition (13) is both necessary and sufficient for the return rate (12) from t to t+1 to equal the current yield ic(t). That is:
                             if and only if
   Return Rate   =  ic(t)   <-------------->   P(t) = P(t+1)
  From t To t+1
Now suppose, instead, that

(14a)   Time t+1 = Maturity Date of the Coupon Bond;

(14b)   Face Value F = Purchase Price P(t+1) of the Coupon Bond at t+1.


Condition (14b) makes sense if the bond is sold at time t+1 AFTER the receipt of the coupon payment C but BEFORE the receipt of the face value payment F. Given conditions (14a,b), the return rate (12) for the coupon bond reduces to the definition of the yield to maturity i(t). To see this, simply apply the usual formula Pb = PV(i) for obtaining the yield to maturity i for a coupon bond held to maturity. Conversely, for a coupon bond for which conditions (14a,b) do NOT hold, the return rate (12) from t to t+1 does NOT coincide with the yield to maturity i(t).
Thus, for a coupon bond purchased at time t, conditions (14a,b) are both necessary and sufficient for the bond's return rate (12) from t to t+1 to equal the bond's yield to maturity i(t). That is:


                         if and only if
   Return Rate  = i(t)  <-------------->  Conditions (14a,b) Hold
  From t To t+1                           



Example: Discount Bond
For a discount bond with a purchase price Pd and a face value F, the return rate (9) over any holding period t to t+1 reduces to

              P(t+1) - Pd
(15)           -----------   .
                   Pd


Recalling definition (6) for the discount yield idb, it is seen that the return rate (15) will generally differ from idb except in the degenerate case Pd = P(t+1) = F when both are zero.


In summary, then, only under special conditions will the return rate for a bond over a given holding period coincide with the yield to maturity, the current yield, or the discount yield.
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Real vs. Nominal Interest Rates
The interest rate measures examined to date have all been "nominal" in the sense that they have not been adjusted for expected changes in prices. What actually concerns a "rational" saver considering the purchase of a debt instrument is not the nominal payment stream he or she expects to earn in future periods but rather the command over purchasing power that this nominal payment stream is expected to entail. This purchasing power depends on the behavior of prices.
Let infe(t) denote the expected inflation rate at time t, and let i(t) denote the (nominal) interest rate for some debt instrument at time t. Then the real interest rate associated with i(t) is defined by the following Fisher equation:
(16)  ir(t) = i(t) - infe(t) .
That is, the real interest rate is the nominal interest rate minus the expected inflation rate.
Note: As explained by Mishkin, the real interest rate defined by (16) is more precisely called the ex ante real interest rate because it adjusts for expected changes in the price level. If the expected inflation rate in (16) is replaced by the actual inflation rate, one obtains the ex post real interest rate.
Real interest rates provide a more accurate measure of the true costs of borrowing and the true gains from lending than nominal interest rates, and hence provide a better indicator of the incentives to borrow and lend. In particular, for any given nominal interest rate i on a debt instrument D, the incentive to borrow (i.e. to issue D) will be higher if the real interest rate associated with i is lower (i.e. if the expected inflation rate is higher). This is so since a higher expected inflation rate means the borrower (i.e. the issuer of D) can expect to pay off his future nominal debt obligations using cheaper dollars than he borrowed. For this same reason, the incentive to lend (i.e. to purchase D) will be lower if the real interest rate associated with i is lower.
A similar distinction is made between the (nominal) return rate defined by (9), which has not been adjusted for expected changes in prices, and the "real return rate" which is subject to such adjustment. More precisely, the real return rate on any asset A over any holding period from t to t+1 is defined to be the (nominal) return rate (9) minus the expected inflation rate infe(t).
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Interest Rate Risk
Consider a bond B held at time t whose maturity date exceeds t. Let the yield to maturity on B at time t be denoted by i(t) and let the price of B at time t be denoted by P(t).
NOTE: It makes no sense to talk about the yield to maturity at time t for a bond that matures AT time t, because there are no future payments to be discounted.
As previously seen, the definition of i(t) implies that i(t) must move inversely to P(t). That is, if one goes up, the other goes down.
Suppose holding periods are measured in years, with t denoting the beginning of year t, and i(t) denotes the yield to maturity on B at time t. An increase in i(t) at time t --- equivalently a fall in P(t) at time t --- results in a decrease in the return rate to B over the holding period from t-1 to t. This is because any holder of B at time t who chooses to sell B at time t would receive a smaller payment for B at time t than what he would have received without the price fall at time t, implying a smaller capital gain (or a larger capital loss) over the holding period from t-1 to t.
The uncertainty regarding return rates that bond holders face due to possible changes in yields to maturity is called interest rate risk.
Mishkin (Table 2) illustrates interest rate risk for bonds of different maturities, each with a coupon payment of $10 and a face value of $1000. This illustration is worth reviewing with some care.
First note that, for each bond in Table 2, the initial yield to maturity i(1) for year 1 ("this year") is equal to the initial current yield ic(1) = 10 percent for year 1 because the initial price of the bond is set at its face value of $1000. However, by assumption, the yield to maturity i(2) for year 2 ("next year") increases to 20 percent.
The coupon bond listed in Table 2 with a 1-year maturity has a price P(2) in year 2 that is fixed (by contract) at the bond's face value, $10000. For all other listed coupon bonds, however, their maturities exceed one year. Consequently, when i(2) increases, their price P(2) in year 2 decreases to some value smaller than their original price P(1) = $1000 in year 1 and hence smaller than the bond's face value of $1000. For example, for the coupon bond with a 30-year maturity, P(2) = $503.
Using representation (12), the return rate from year 1 to year 2 for each of the coupon bonds in Table 2 is given by the sum of the current yield ic(1) = C/P(1) for year 1 and the capital loss g(1,2) = [P(2)-P(1)]/P(1) from year 1 to year 2.
Consequently, except for the bond with a one-year maturity, these return rates are smaller than they would have been without the increase in the yield to maturity in year 2. Indeed, for the coupon bond with a 30-year maturity, the capital loss g(1,2) is so large (-49.7 percent) that it overwhelms the 10 percent initial current yield ic(1) = 10 percent, resulting in a negative return rate of -39.6 percent from year t=1 to t=2.
More precisely, examining the return rates in column (6) of Table 2 as the maturity is decreased from 30 years to 1 year, it is seen that the coupon bonds with longer maturities experience a greater decline in their return rates when the yield to maturity i(2) increases. This is due to the fact that this increase in i(2) results in a smaller decline in the price P(2) for coupon bonds with smaller maturities and hence a smaller capital loss. [To see why, consider the formula Pb = PV(i) from which the yield to maturity i is determined.] Indeed, for coupon bonds with a one-year maturity, P(2) remains fixed at the face value $1000.
An important implication of this illustration, then, is that the return rates of bonds with longer-term maturities respond more dramatically to changes in the yield to maturity than bonds with shorter-term maturities. That is, longer-term bonds are more subject to interest rate risk. This is one reason why investment in longer-term bonds is considered more risky than investment in shorter-term bonds.

                  Increase in yield to maturity
                      from year 1 to year 2

                              /|\
                               |
            i(1)=ic(1)        i(2)
(17)     |--------------|-----------/\/\/\--|   Year
         1              2                   N

        P(1)           P(2)            Maturity Date
                        |                (N > 2)
                       \|/

                   Capital loss
                  from t=1 to t=2
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Basic Concepts and Key Issues from Mishkin Chapter 4
Basic Concepts:
Simple loan contract
Principal
Maturity and maturity date
Interest payment
Simple interest rate
Fixed-payment loan contract
Coupon bond
Face value
Coupon payment
Coupon rate
Discount bond (or zero-coupon bond)
Nominal value
Present value (or present discounted value)
Yield to maturity
Current yield
Consol bond
Discount yield
Capital gain or loss
(Nominal) return rate
Real interest rate
Expected inflation rate
Real return rate
Interest rate risk

Key Issues:
§  Diagrammatic representation of a simple loan contract
§  Diagrammatic representation of a fixed-payment loan contract
§  Diagrammatic representation of a coupon bond
§  Diagrammatic representation of a discount bond
§  Present value of a future payment
§  Present value of a stream of future payments
§  General formula for determining the yield to maturity for any bond
§  Calculating the yield to maturity for a simple loan
§  Calculating the yield to maturity for a fixed-payment loan
§  Calculating the yield to maturity for a coupon bond
§  Calculating the yield to maturity for a discount bond
§  Inverse relationship between the price of a bond and its yield to maturity
·         Relationships connecting a coupon bond's purchase price, face value, yield to maturity, and coupon rate  
·         Calculating the current yield for a consol bond
·         For a coupon bond, how does its maturity affect the relationship between its current yield and its yield to maturity?
·         For a coupon bond, what is the relationship between its current yield, its coupon rate, its purchase price, and its face value?
·         For a coupon bond, what is the relationship between its current yield, its yield to maturity, its purchase price, and its face value?
·         For a coupon bond, what is the relationship between its current yield and its purchase price, given any fixed level for its coupon payment?
·         For a coupon bond, all else remaining the same, why do its current yield and its yield to maturity always move together?
·         For a discount bond, all else remaining the same, why do its discount yield and its yield to maturity always move together?
·         How to read financial bond pages for information on Treasury bonds and notes, Treasury bills, and corporate bonds traded on stock exchanges.
·         Why is the return rate on a bond not necessarily equal to its interest rate?
·         For a coupon bond, when is its return rate equal to its current yield?
·         For a coupon bond, when is its return rate equal to its yield to maturity?
·         What is the relationship between real and nominal interest rates?
·         Why do real interest rates provide a more accurate measure of the true costs of borrowing and the true gains from lending than nominal interest rates?
·         Why do real return rates provide a more accurate measure of the true gains or losses from holding an asset than nominal return rates?
·         How does the maturity of a bond affect its interest rate risk?
·         Why do bonds with long maturities expose bond holders to greater interest rate risk than bonds with shorter maturities?
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