Lecture Notes on Mishkin Chapter 4
("Understanding Interest Rates")
Part A (pp. 68-80)
Econ 353: Money, Banking, and Financial Institutions

Last Updated: 28 March 2015
Latest Course Offering: Spring 2011

Course Instructor:
Professor Leigh Tesfatsion
tesfatsi AT iastate.edu
Econ 353 Homepage:
http://www2.econ.iastate.edu/classes/econ353/tesfatsion/

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. 68-80), the sections below focus on the meaning and measurement of interest rates for key types of debt instruments (or "credit market instruments") that are issued and bought in U.S. financial markets. Remaining topics covered in Mishkin's Chapter 4:Part B (pp. 80-89) will be taken up in subsequent notes.

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 five basic types of debt instruments that are distinguished from one another by their payment provisions: simple loan contracts; fixed-payment loans; coupon bonds; discount bonds (or "zero-coupon bonds"); and consols (or "perpetuities") that pay a fixed amount in each payment period forever ("in perpetuity").

The notes below concentrate on the first four types of debt instruments.

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 engaging in portfolio rebalancing, meaning they are adjusting the composition of their asset portfolios (e.g., more money and 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.

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)                         n
                                    (1+i)

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:

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)                 ----------
                            2
                       (1+i)

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)                         3
                       (1+i)

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.

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.

Basic Concepts and Key Issues from Mishkin Chapter 4:Part A

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)
Consol or perpetuity (Mishkin pp. 77-78)
Nominal value
Present value (or present discounted value)
Yield to maturity

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
Calculating the yield to maturity for a consol (Mishkin pp. 77-78)
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

Copyright © 2011 Leigh Tesfatsion. All Rights Reserved.