# United States Savings Bond Interest Calculator

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```					FNCE 3020
Financial Markets and Institutions
Fall Semester 2005

Lecture 2: Understanding Interest Rates
and Calculation of Returns
Interest Rate

   What do you think of when you hear this
term?
Interest Rate Defined

   “Two-sided” Definition:
   Borrowing: the cost of borrowing or the price paid
for the rental of funds.
   From the standpoint of “deficit” entities.
   Saving/Investing: the return from lending funds.
   From the standpoint of “surplus” entities.
   Both concepts usually expressed as a
percentage per year (per annum).
Interest Rates: Borrowing and Savings
Rates
   Yields in percent per annum
Sep 2, 2004   Aug 29, 2005 Change

   Direct Market
   Aaa Industrial Bonds:   5.55%        5.01%     - 54

   Indirect (Intermediary) Market
 CDs (6 months)        1.76%           4.03%       +227
 Prime Loans           4.50%           6.50%       +200
 Spread              2.74%           2.47%        - 27
   Source: http://www.federalreserve.gov/releases/h15/update/
Savings (Investing) versus Lending
Rates
Important Terms in Lending
   Loan Principal: the amount of funds the lender provides to
the borrower.
   Maturity Date: the date the loan must be repaid
   Loan Term: the time period from initiation to maturity date.
   Interest Payment: the cash amount that the borrower
must pay the lender for the use of the loan principal.
   Simple Interest Rate: the interest payment divided by the
loan principal.
   the percentage of principal that must be paid as interest to the
lender. Convention is to express on an annual basis, irrespective
of the loan term.
Types of Debt Instruments

   Simple Loan: Principal and all interest both paid at
maturity (when loan comes due).
   Borrow \$1,000 today at 5% and in 6 months pay \$1,025.
   Commercial bank loans to businesses.
   Fixed-payment Loan: Equal monthly payments
representing a portion of the principal borrowed plus
interest. Paid for a set number of years, at which
time the principal amount is fully repaid.
   Home mortgages (conventional), automobile loans.
Amortization Example
   Mortgage Loan
   Amount \$500,000
   Years: 30 (Monthly payments)
   Interest rate: 7% (fixed)
   Monthly Payment
   \$3,326.51 (for 360 months)
   First Month Payment (n = 1):
   Principal: \$409.84; Interest: \$2,916.67 (or, \$3,326.51)
   Last Month Payment (n = 360):
   Principal: \$3,307.22; Interest: \$19.29 (or, \$3,326.51)
Any Ideas as to why this Cartoon Would
get a Bigger Laugh in England.
Types of Debt Instruments
   Discount Bond (Zero-coupon Bond): Purchased at a
price below face (or par) value with face value paid
at maturity. There are no interest payments.
   Treasury bills
   Coupon Bond: periodic interest payments (stated as
the coupon rate) for a specified period of time after
which the total principal (face or par value) is repaid.
   Treasury and corporate bonds.
   These may be callable!
Treasury Bill Market
   Treasury bills (along with notes and bonds) are
marketable securities the U.S. government sells in
order to pay off maturing debt and raise the cash
needed to run the federal government.
    Treasury securities are sold in primary markets
through:
   Competitive bid. In this form of bidding, investors specify
the rate or yield they will accept.
   Treasury will either accept or reject the bid.
   Noncompetitive bid. By bidding noncompetitively
investors agree to accept whatever rate or yield is
determined at the auction .
Returns on Discount Instruments
   91-day Treasury Bills Issue date: September 1, 2005
   Price 99.117 (per \$100; or \$991.17 on a \$1,000 T-bill)
   At maturity (12/1/05) the bill pays \$1,000.00 (face value)
   Gain = \$8.83
   Two Calculated Yields on T-Bills
   Discount rate (yield):         3.49%
   The discount rate (yield) takes into account the return as a percent of
the face value (\$1,000) of the T-bill.
   Investment rate (yield):       3.57%
   The investment rate (yield) relates the investor's return to the
purchase price (\$996.12) of the T-bill.
   Source: http://wwws.publicdebt.treas.gov/AI/OFBills
   Note: 14 day, 28 day and 91 day T-bills are auctioned every
week.
Calculating Treasury T-Bill Yields
   Computations of yields on Treasury bills depend on the
face value, purchase price and maturity of the issue.
   There are two methods for determining yields:
  The discount method relates the investor's return to
the bill's face value;
 The investment method relates the investor's return to
the bill's purchase price.
   Thus, the investment discount method tends to overstate
yields (results in higher calculated yields) relative to
those computed by the discount method.
 Purchase price will always be lower than the face
value.
Discount Yield Formula
   The following formula is used to determine the
discount yield for T-bills:
Discount yield = [(FV - PP)/FV] * [360/M]
   FV = face value
PP = purchase price
M = maturity of bill.
   M = number of days to maturity
   360 = the number of days used by banks to
determine short-term interest rates.
Investment Yield Formula
   The following formula is used to calculate the investment
yield:
Investment yield = [(FV - PP)/PP] * [365 or 366/M]
 FV = face value
PP = purchase price
M = maturity of bill
   Note: use 366 for leap year calculations.
   When comparing the return on investment in T-bills to
other short-term investment options, the investment yield
method is generally used.
 This yield is alternatively called the bond equivalent

yield, the coupon equivalent rate, the effective yield
and the interest yield.
Example: Calculating T-Bill Yields
   Using data for September 1, 2005 (see previous slide)

   Discount yield = [(FV - PP)/FV] * [360/M]
= [\$1,000 -991.17/1,000]*[360/91]
= 3.49%

   Investment yield = [(FV - PP)/PP] * [365/M]
= [\$1,000 – 991.17/991.17]*[365/91]
= 3.57%

   For a discussion of T-bills rate calculations see:
ml
Returns on Coupon Bonds

   Assume a 10 year, \$1,000 face value,
coupon bond with a coupon rate of 10% (paid
annually).
   You would receive \$100 per year for ten years as
the annual interest payment.
   At the end of ten years (on maturity date) you
would receive \$1,000 (face value)
Bonds and Present Value Concept
   Defined: Present value is today’s value of a
payment (or series of payments) to be received in
the future.
   \$100 to be received in 1 year is worth \$95.24 today if we
assume the interest to be earned is 5%
   If you invested \$95.24 for 1 year at a rate of 5%, at the end
of the year you would have \$100.00
   Importance: provides a mechanism for determining:
   Today’s price of a credit market instrument.
   Thus, allows us to compare the prices of different instruments
with different payment schedules.
   Equivalent (comparable) interest rates on different
instruments.
Determination of Market Price
   Assume a financial instrument offers the following 2
year payment stream:
   Annual (end of the year) interest payments of \$70 (two
payments each of \$70), or a 7% coupon rate.
   A principal (face value) repayment of \$1,000 at the end of
the second year.
   Assume interest rates on financial instruments of
similar risk to the one above are offering returns of
10%
   Question: How much should you pay (i.e., what is
the market price) for the financial instrument in
question?
Present Value (“Market Price”) of a
Security
   The price the market should pay for the security
noted on the previous slide is equal to the present
value of the expected future income stream
discounted at a rate of interest of 10% (this is the
current opportunity cost), or:
   PV = \$70/(1+.10) + \$1,070/(1+.10)2
   PV = \$63.64 + \$1,070/1.21
   PV = \$63.64 + \$884.30
   PV = \$947.94 (this is the market price!)
   Note: The bond is selling at a price below its par
value.
What If?

   What if the opportunity cost (market interest
rate) is less than the coupon rate?
   What will the market price of the bond be if the
discount rate is 5%?
   PV = \$70/(1+.05) + \$1,070/(1+.05)2
   PV = \$66.67 + \$1,070/1.1025
   PV = \$66.67 + \$970.52
   PV = \$1037.19 (this is the market price!)
   Note: Now the bond is selling at a price
above par
Rules #1 and #2

   #1: When the market interest rate rises above
the coupon rate on a bond, the price of the
bond falls (sells at a discount of par).
   #2: When the market interest rate falls below
the coupon rate on a bond, the price of the
bond rises (sells at a premium of par)
   Thus: there is an inverse relationship
between market interest rates and bond
prices.
What If?

   What if the opportunity cost (market interest
rate) is equal to the coupon rate?
   What will the market price of the bond be?
   PV = \$70/(1+.07) + \$1,070/(1+.07)2
   PV = \$65.42 + \$1,070/1.1449
   PV = \$65.42 + \$934.58
   PV = \$1000.00 (this is the market price!)
   Note: Rule #3 -- The price will always equal
par if the market rate equals the coupon rate.
What if We Vary the Time to Maturity
and the Market Rate Rises
   Assume a one year bond.
   PV = \$1,070/(1.10)
   PV = \$972.72 (this is the market price)
   Compare to the two year bond.
   PV = \$70/(1+.10) + \$1,070/(1+.10)2
   PV = \$63.64 + \$1,070/1.21
   PV = \$63.64 + \$884.30
   PV = \$947.94 (this is the market price!)
   Note: The longer the term to maturity, the greater
the price change (i.e., decline).
What if We Vary the Time to Maturity
and the Market Rate Falls
   Assume a one year bond.
   PV = \$1,070/(1.05)
   PV = \$1,019.05 (this is the market price)
   Compare to the two year bond.
   PV = \$70/(1+.05) + \$1,070/(1+.05)2
   PV = \$66.67 + \$1,070/1.1025
   PV = \$66.67 + \$970.52
   PV = \$1037.19 (this is the market price!)
   Note: The longer the term to maturity, the greater
the price change (i.e., increase).
Rule #4

   Rule #4: The greater the term to maturity, the
greater the change in price for a given
change in market interest rates.
   This becomes very important when developing a
bond maturity strategy to incorporate your
expected changes in interest rates.
   What if you think interest rates will fall? Where should
you concentrate the maturity of your bonds?
   What if you think interest rates will rise? Where should
you concentrate the maturity of your bonds?
Interest Rate Measures
   There are three important ways of calculating
the interest rate on a financial instrument.
These include:
   Discount Yield and Investment Yield: Used to
measure the yield on T-bills (and other discounted
securities) and T-notes and Bonds.
 See Specific formulas noted earlier.
   Yield to Maturity: The interest rate that equates the
future payments to be received from a financial
instrument with its market price today (present value).
   Current Yield: Coupon payment divided by the current
market price of a financial instrument.
 Can be a rough approximation of yield to maturity.
Yield to Maturity
   Uses the concept of present value in the
determination of the yield to maturity.
   Yield to maturity (i) is calculated in the formula below
where:
   P = market price (present value)
   C = coupon payments
   F = face value (at maturity)
   n = years to maturity

C        C       C              C        F
P               2      3  ...        n 
1 i  1  i 1 i          1 i  1  i n
Current Yield:
   Coupon payment divided by the current market
price of a financial instrument.
C
ic 
P
   Two Characteristics of Current Yield
   The nearer the market price is to par and longer is the
maturity of bond, the better approximation to yield to
maturity.
   Change in the current yield always signals change in
same direction as the yield to maturity.
Web Site for Calculating Yields

   Visit the following web site. It allows you to
calculate the current yield and yield to
maturity for data you input.

   http://www.moneychimp.com/calculator/bond
_yield_calculator.htm
Issues of Risk

   Price Risk: Interest rates may move against you and
produce losses on holdings of fixed income
securities.
   Specifically when interest rates rise (fixed income security
prices will fall).
   Greatest risk (potential price change) the longer the
maturity of the fixed income security.
   Reinvestment Risk: Potential if holding short term
fixed income securities. Need to roll them over at
maturity. Interest rate at which you will reinvest is
uncertain.
Duration
   Issue: The fact that two bonds have the same term
to maturity, does not mean that they carry the same
interest rate risk.
   Assume
   A 10 year, 10% coupon bond and
   A 10 year zero coupon bond.
   Which has the greatest interest rate (price) risk for a
given change in interest rates?
   The zero coupon bond because none of its payment occur
until maturity.
   Why: The present value impact will be greater than with the
coupon issue.
Using Duration

   Duration is an estimate of the average
lifetime of a security’s stream of payments.
   Everything else equal:
   The longer the term to maturity, the longer
duration.
   The lower the coupon rate, the longer the
duration.
   And, the greater the duration of a security,
the greater the interest rate (price) risk.
Distinction Between Real and
Nominal Interest Rates
   Real interest rate:
   The interest rate that is adjusted for expected
changes in the price level, and is calculated:
Ir = I - Pe
Where:
Ir = real rate of interest
I = market (nominal) rate of interest
Pe = expected rate of inflation, i.e., price level
changes (over the maturity of the financial asset)
Real Interest Rate Impacts

   Real interest rate more accurately reflect the
true cost of borrowing and returns (e.g., on
lending, investing, savings).
   When the real rate is low (or negative), there
is a greater incentive to borrow and less
incentive to lend.
   Why?
   When real rate is high, there is less incentive
to borrow and more incentive to lend.
   Why?
Calculation of Real Interest Rates

    If I = 10% and Pe = 1% then
Ir = 10% - 1% = 9%

    If I = 10% and Pe = 8% then
Ir = 10% - 8% = 2%
Note: When real rate is high, less incentive to
borrow and more to lend.

Note: When real rate is low (or negative),
greater incentive to borrow and less to lend.
Are Real Rates Subject to Change?

Real and Nominal Interest Rates (Three-Month Treasury Bill), 1953–2004
Are There Differences Globally?
   United States
   2 year government bond rate         4.26%
   Expected rate of inflation (2006)   2.60%
   Real interest rate                  2.06%

   Japan
   2 year government bond rate        1.62%
   Expected rate of inflation (2006)* 0.00%
   Real interest rate                 1.62%

   Australia
   2 year government bond rate        5.04%
   Expected rate of inflation (2006)* 2.60%
   Real interest rate                 2.44%

   Source of date: The Economist, August 30, 2005
Internet Source of Interest Rate Date

   Historical and Current Data for U.S.
   http://www.federalreserve.gov/releases/h15/update/

   Real Time Data (U.S. and other major countries)
 http://www.bloomberg.com
   Go to Market Data and then to Rates and Bonds

   Other Countries:
   Economist.com (both web source or hard copy)

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