FNCE 4070FINANCIAL MARKETS AND INSTITUTIONS

Professor Michael PalmerUniversity of Colorado at BoulderSpring Semester 2011Lecture 3: Understanding Interest Rates

Can you explain this?

Treasuries Decline as Weekly Jobless Claims Drop Treasuries declined as first-time claims for

unemployment insurance fell to the lowest since July 2008.

Interest Rate Defined “Dual” Definition:

Borrowing: the cost of borrowing or the price (%) paid for the “rental” of funds. A financial liability for “deficit” entities.

Saving: the return from investing funds or the price (%) paid to delay consumption. A financial asset for “surplus” entities.

Both concepts are expressed as a percentage per year (Percent per annum – p.a.). True regardless of maturity of instrument of the

financial liability or financial asset. Thus, all interest rate data is annualized. See: http://www.federalreserve.gov/releases/h15/update/

Savings and Borrowing Rates: They Move Together, 1990 – 2010

Regression analysis: 1964 – 2010 (monthly data, 564 observations); CD rate as dependent variable. R-squared = 88.55%

Commonly Used Interest Rate Measures There are four important ways of measuring

(and reporting) interest rates on financial instruments. These are: Coupon yield: The “promised” annual percent return

on a coupon instrument. Current Yield: Bond’s annual coupon payment divided

by its current market price. Discount Yield and Investment Yield: The yield on T-

bills (and other discounted securities, such as commercial paper) which are selling at a discount of their maturity values.

Yield to Maturity: The interest rate that equates the future payments to be received from a financial instrument (coupons plus maturity value) with its market price today (i.e., to its present value).

Benchmarking with Interest Rates Interest rates can be used for cross-country

assessments or changes in individual country assessments over time.

The most common benchmark rates are yields to maturity on 10-year Government U.S. Treasuries and German Bunds.

We assume these are “default-free.” Thus we can compare other sovereigns to these

(and to one another) to assess : Credit ratings risk Inflation risk The market’s overall assessment of country risk

See next slide for benchmark data. Source: http://markets.ft.com/markets/bonds.asp

FT Reported 10-Year Government Benchmark Rates, February 8, 2011Country  Latest

Spreadvs Bund

Spread vsT-Bonds

US 3.66% +0.41 0.00

UK 3.84% +0.59 +0.17

Switzerland 1.94% -1.31 -1.73

Sweden 3.46% +0.21 -0.20

Spain 5.31% +2.05 +1.64

Portugal 7.23% +3.97 +3.56

New Zealand 5.57% +2.31 +1.90

Netherlands 3.44% +0.19 -0.22

Japan 1.33% -1.93 -2.34

Country  LatestSpreadvs Bund

Spread vsT-Bonds

Greece 11.21% +7.96 +7.55

Germany 3.25% 0.00 -0.41

France 3.64% +0.38 -0.03

Finland 3.47% +0.22 -0.19

Denmark 3.31% +0.05 -0.36

Canada 3.47% +0.22 -0.19

Belgium 4.24% +0.99 +0.58

Austria 3.71% +0.46 +0.04

Australia 5.72% +2.46 +2.05

Coupon Yield Coupon yield is the annual interest rate which was promised

by the issuer when a bond is first sold. Information is found in the bond’s indenture.

The coupon yield is expressed as a percentage of the bond’s par value. Par value is also called the maturity value (or face value).

In the United States, all bonds have a par value of $1,000 (Government bonds called Treasuries). UK Government bonds (£100 par value; called gilts) Japanese Government bonds (¥10,000 par value; called JGBs) German Government bonds (minimum amount of €100 par value,

called bunds) Canadian Government bonds (CAD$1,000 par value)

If a U.S. bond has a stated coupon yield of 4.5%, this means that it will pay the holder $45 per year (0.045 x $1,000).

The coupon yield on a bond will not change during the lifespan of the bond.

Current Yield Since bond prices are likely to change, we often

refer to the “current yield” which is measured by dividing a bond’s annual coupon payment by its current market price. This provides us with a measure of the interest yield

obtained at the current market price (i.e., cost) Current yield = coupon payment/market price

So, if our 4.5% coupon bond is currently selling at $900 the calculated current yield is: $45/$900 = 5.00%

And if the bond is selling at $1,100, the current yield is: $45/$1,100 = 4.09%

Discount and Investment Yield Discount yields and investment yields are calculated

for T-bills and other short term money market instruments (e.g., commercial paper and bankers’ acceptances) where there are no stated coupons (and thus the assets are quoted at a discount of their maturity value).

The discount yield relates the return to the instrument’s par (or face or maturity) value. The discount yield is sometimes called the bank discount

rate or the discount rate. The investment yield relates the return to the

instrument’s current market price. The investment yield is sometimes called the coupon

equivalent yield, the bond equivalent rate, the effective yield or the interest yield.

Calculating the Discount Yield Discount yield = [(PV - MP)/PV] * [360/M] PV = par (or face or maturity) value MP = market price M = maturity of bill.

For a “new” three-month T-bill (13 weeks) use 91, and for a six-month T-bill (26 weeks) use 182.

For outstanding issues, use the actual days to maturity.

Note: 360 = is the number of days used by banks to determine short-term interest rates.

Discount Yield Example

What is the discount yield for a 182-day T-bill, with a market price of $965.93 (per $1,000 par, or face, value)?

Discount yield = [(PV - MP)/PV] * [360/M]

Discount yield = [(1,000) - (965.93)] / (1,000) * [360/182]Discount yield = [34.07 / 1,000] * [1.978022]Discount yield = .0673912 = 6.74%

Investment Yield The investment yield is generally calculated so

that we can compare the return on T-bills to “coupon” investment options. The calculated investment yield is comparable to the

yields on coupon bearing securities, such as long term bonds and notes.

As noted: The investment yield relates the return to the instrument’s current market price.

In addition, the investment yield is based on a calendar year: 365 days, or 366 in leap years.

Investment yield = [(PV - MP)/MP] * [365 or 366/M]

Investment Yield Example

What is the investment yield of a 182-day T-bill, with a market price of $965.93 per $1,000 par, or face, value?

Investment yield = [(PV - MP)/MP] * [365/M]Investment yield = [(1,000 – 965.93) / (965.93)] * [365/182]Investment yield = [34.07] / 965.93] * [2.0054945]Investment yield = .0707372 = 7.07%

Comparing Discount and Investment Yields Looking at the last two examples we found:

Discount yield = [(PV - MP)/PV] * [360/M]Discount yield = [(1,000 - 965.93)] / (1,000) * [360/182]Discount yield = [34.07 / 1,000] * [1.978022]Discount yield = .0673912 = 6.74%

Investment yield = [(PV - MP)/MP] * [365/M]Investment yield = [(1,000 – 965.93)] / (965.93) * [365/182]Investment yield = [34.07 / 965.93] * [2.0054945]Investment yield = .0707372 = 7.07%

Note: The discount formula will tend to “understate” yields relative to those computed by the investment method, because the market price is lower than the par value ($1,000). However, if the market price is very close to the par value, the yields will be

similar. See: http://www.ustreas.gov/offices/domestic-finance/debt-

management/interest-rate/daily_treas_bill_rates.shtml And: http://www.treasurydirect.gov/RI/OFBills

Bloomberg and Reported Yields on T-Bills Go to http://www.Bloomberg.com Go to Market Data Go to Rates and Bonds You will see for “U.S. Treasuries” the following data (note:

this is an example from the Feb 4, 2011 site):Coupon Maturity Current

Date Price/Yield3-month 0.000 05/05/2011 0.14/.156-month 0.000 08/04/2011 0.16/.1712-month 0.000 01/12/2011 0.27/.28

Key: These are T-bills, thus the coupon is 0% (recall they are sold at a discount). At maturity date they will pay the holder $1,000. The current price is the discount yield (bank discount yield) and the current yield is the investment yield (bond or coupon equivalent yield).

Yield to Maturity The yield to maturity uses the concept of present value

in its determination. Yield to maturity is the interest rate at which if we

discount the incomes (i.e., cash-flows) of a bond, we get the par value exactly (or the net present value = 0).

Yield to maturity (i) is calculated as:

MP = Market price of a bond (i.e., present value) C = Coupon payments (a cash flow) PV = Par, or face value, at maturity (a cash flow) n = Years to maturity

Note: i is also the internal rate of return

nn i

PV

i

C

i

C

i

C

i

CMP

11...

111 32

Yield to Maturity Example Assume the following given variables:

C =$40 (thus a 4.0% coupon issue; paid annually)N =10 PV =$1,000MP =$1,050 (note: bond is selling at a premium of par)

1050 = 40/(1 + i)1 + 40/(1 + i)2 + . . . + 40/(1 + i)10 + 1000/(1 + i)10

Solve for i, the yield to maturity Note: The “i" calculated using this formula will be

the return that you will be getting when the bond is held until it matures and assuming that the periodic coupon payments are reinvested at the same yield. In this example, the “i" is 3.4%.

Yield to Maturity Second Example Now assume the following:

C =$40 N =10PV =$1,000MP =$900.00 (note: bond is selling at a discount of par)

900 = 40/(1 + i)1 + 40/(1 + i)2 + . . . + 40/(1 + i)10 + 1,000/(1 + i)10

Solve for i, the yield to maturity Note: The “i" calculated in this example is 5.315%. What one factor accounts for the yield to maturity

difference when compared to the previous slide, with its i of 3.4%?

Useful Web Site for Calculating a Bond’s Yield to Maturity While yields to maturity can be determined

through a book of bond tables or through business calculators, the following is a useful web site for doing so:

http://www.money-zine.com/Calculators/Investment-Calculators/Bond-Yield-Calculator/

The Yield to Maturity Think of the yield to maturity as the “required return

on an investment.” Since the required return changes over time, we can

expect these changes to produce inverse changes in the prices on outstanding (seasoned) bonds.

Why will the required return change over time? Changes in inflation (inflationary expectations). Changes in the economy’s credit conditions resulting from

change in business activity. Changes in central bank policies.

Impact on shorter term maturities. Changes in the credit risk (i.e., risk of default) associated with

the issuer of the bond. On Governments, also changes in credit ratings risk

Portugal this week.

Illustrating the Relationship Between Interest Rates and Bond Prices

Assume the following: A 10 year corporate Aaa bond which was issued 8

years ago (thus it has 2 years to maturity) has a coupon rate of 7%, with interest paid annually.

Thus, 7% was the required return when this bond was issued. This bond is referred to as an outstanding (or seasoned)

bond. Question: How much will a holder of this bond receive

in interest payments each year? This bond has a par value of $1,000.

Question: How much will a holder of this bond receive in principal payment at the end of 2 years?

What Happens when Interest Rates Rise? Assume, market interest rates rise (i.e., the required return rises)

and now 2 year Aaa corporate bonds are now offering coupon returns of 10%. This is the “current required return” (or “i” in the present value bond

formula) Question: What will the market pay (i.e., market price) for the

outstanding 2 year, 7% coupon bond noted on the previous slide? PV = $70/(1+.10) + $1,070/(1+.10)2

PV = $947.94 (this is today’s market price) Note: The 2 year bond’s price has fallen below par (selling at a

discount of its par value). Conclusion: When market interest rates rise, the prices on

outstanding bonds will fall.

What Happens when Interest Rates Fall? Assume, market interest rates fall (i.e., the required return falls)

and now 2 year Aaa corporate bonds are now offering coupon returns of 5%. This is the “current required return” (or “i” in the present value bond

formula) Question: What will the market pay (i.e., market price) for the

outstanding 2 year, 7% coupon bond? PV = $70/(1+.05) + $1,070/(1+.05)2

PV = $1,037.19 (this is today’s market price) Note: The 2 year bond’s price has risen above par (selling at a

premium of its par value). Conclusion: When market interest rates fall, the prices on

outstanding bonds will rise.

Change in Market’s Required Return Versus Change in Market Demand The examples on the previous slides demonstrated

the impact of a change in the market’s required return on bond prices. Observation: Cause – effect relationship runs from

changes in required return to changes in market prices (which produce the market’s new required return).

However, it is possible for a change in market demand to produce changes in bond prices and thus in market interest rates. For example: Safe haven effects result in changes in

demand for particular assets. Observation: Cause – effect relationship runs from

changes in demand to changes in prices (which have an automatic impact on yields).

QE 2 Impacts on Interest Rates How can we view QE2’s potential impact on long

term interest rates. Should the required return on longer-term, seasoned

issues change? Probably not.

Two possible channels of influence: (1) Demand and Supply of Bonds: Fed purchases

will drive up market prices (increase in demand), and thus drive down yield, or Fed purchases will reduce supply of long term securities, thus drive up prices (and yields will fall).

(2) Demand and Supply of Loanable Funds: Increase supply of long term funds will shift out the supply (of loanable funds) schedule, thus push down yields.

What if the Time to Maturity Varies? Assume a one year bond (7% coupon) and the market

interest rate rises to 10%, or falls to 5%. PV@10% = $1,070/(1.10) PV = $972.72 PV @5%= $1,070/(1.05) PV = $1,019.05

Now assume a two year bond (7% coupon) and the market interest rate rises to 10%, or falls to 5% PV@10% = $70/(1+.10) + $1,070/(1+.10)2

PV = $947.94 PV@5% = $70/(1.05) + $1,070/(1+.05) 2

PV = $1037.19 Conclusion: For a given interest rate change, the longer

the term to maturity, the greater the bond’s price change.

Summary: The Interest Rate Bond Price Relationship #1: When the market interest rate (i.e., the required

rate) rises above the coupon rate on a bond, the price of the bond falls (i.e., it sells at a discount of par).

#2: When the market interest rate (i.e., the required rate) falls below the coupon rate on a bond, the price of the bond rises (i.e., it sells at a premium of par)

IMPORTANT: There is an inverse relationship between market interest rates and bond prices (on outstanding or seasoned bonds).

#3: The price of a bond will always equal par if the market interest rate equals the coupon rate.

Summary: The Interest Rate Bond Price Relationship Continued #4: The greater the term to maturity, the greater the

change in price (on outstanding bonds) for a given change in market interest rates. This becomes very important when developing a bond

portfolio-maturity strategy which incorporates expected changes in interest rates.

This is the strategy used by bond traders: 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? See Appendix 1 for Excel Calculation of bond prices.

Interest Rate (or Price) Risk on a Bond Defined: The risk associated with a reduction in

the market price of a bond, resulting from a rise in market interest rates.

This risk is present because of the “inverse” relationship between market interest rates and bond prices.

Greatest risk (i.e., potential price change) the longer the maturity of the fixed income security you are holding and the greater the interest rate change. For a historical example, see the next slide.

Illustration of Price Risk: 1950 - 1970

Reinvestment Risk on a Bond Reinvestment risk occurs because of the need to “roll

over” securities at maturity, i.e., reinvesting the par value into a new security.

Problem for bond holder: The interest rate you can obtain at roll over is unknown while you are holding these outstanding securities.

Issue: What if market interest rates fall? You will then re-invest at a lower interest rate then the

rate you had on the maturing bond. Potential reinvestment risk is greater when holding

shorter term fixed income securities. With longer term bonds, you have locked in a known

return over the long term. For a historical example, see the next slide

Illustration of Reinvestment Issue: 1990 - 2008

Concept of Bond Duration Issue: The fact that two bonds have the same term to

maturity does not necessarily mean that they carry the same interest rate risk (i.e., potential for a given change in price).

Assume the following two bonds: (1) A 20 year, 10% coupon bond and (2) A 20 year, 6% coupon bond.

Which one do you think has the greatest interest rate (i.e., price change) risk for a given change in interest rates? Hint: Think of the present value formula (market price of a bond)

and which bond will pay off more quickly to the holder (in terms of coupon cash flows).

Solution to Previous Question Assume interest rates change (increase) by 100

basis points, then for each bond we can determine the following market price.

20-year, 10% coupon bond’s market price (at a market interest rate of 11%) = $919.77

20-year, 6% coupon bond’s market price (at a market interest rate of 7%) = $893.22

Observation: The bond with the higher coupon, (10%) will pay back quicker (i.e., produces more income early on), thus the impact of the new discount rate on its cash flow is less.

Duration and Interest Rate Risk Duration is an estimate of the average lifetime of

a security’s stream of payments. Duration rules: (1) The lower the coupon rate (maturity equal), the

longer the duration. (2) The longer the term to maturity (coupon equal), the

longer duration. (3) Zero-coupon bonds, which have only one cash

flow, have durations equal to their maturity. Duration is a measure of risk because it has a

direct relationship with price volatility. The longer the duration of a bond, the greater

the interest rate (price) risk and the shorter the duration of a bond, the less the interest rate risk.

Calculated Durations

Duration for a 10 year bond assuming different coupons yields: Coupon 10% Duration 6.54 yrs Coupon 5% Duration 7.99 yrs Zero Coupon Duration 10 years

Duration for a 10% coupon bond assuming different maturities: 5 years Duration 4.05yrs 10 years Duration 6.54 yrs 20 years Duration 9.00 yrs

Note: See Appendix 2 for Excel calculations

Using Duration in Portfolio Management Given that the greater the duration of a bond, the

greater its price volatility (i.e., interest rate risk), we can apply the following:

(1) For those who wish to minimize interest rate risk, they should consider bonds with high coupon payments and shorter maturities (also stay away from zero coupon bonds). Objective: Reduce the duration of their bond portfolio.

(2) For those who wish to maximize the potential for price changes, they should consider bonds with low coupon payments and longer maturities (including zero coupon bonds). Objective: Increase the duration of their bond portfolio

The Real Interest Rate

Real interest rate: This is the market (or nominal) interest rate that

is adjusted for expected changes in the price level (i.e., inflation) and is calculated as follows:irr = imr - pe

Where:

irr = real rate of interest (% p.a.)

imr = market (nominal) rate of interest (% p.a.)

pe = expected annual rate of inflation, i.e., the

average annual price level change over the maturity of the financial asset (% p.a.)

Real Interest Rate Impacts on Borrowing and Investing We assume that real interest rates more accurately

reflect the true cost of borrowing and true returns to lenders and/or investors. Assume: imr = 10% and pe = 12% then irr = 10% - 12% = -2%

When the real rate is low (or negative), there should be a greater incentive to borrow and less incentive to lend (or invest). Assume: Imr = 10% and pe = 1% then Irr = 10% - 1% = 9%

When the real rate is high, there should be less incentive to borrow and more incentive to lend (or invest).

3-41

U.S. Real and Nominal Interest Rates: 1953-2007

Real Interest Rate as an Indicator of Monetary Policy The real interest rate (on the fed funds rate) is also

assumed to be a better measure of the stance of monetary policy than just the market interest rate. Why: Real rate affects borrowing decisions. If the real rate is negative, or very low, monetary

policy is very accommodative and borrowing will be encouraged.

If the real rate high, monetary policy is very tight and borrowing will be discouraged.

A neutral monetary policy occurs when the real rate is zero.

Example of Nominal Versus Real RateEconomic Background U.S. experiences the 2000

“dot-com” stock market crash and “terrorist- attack” induced recession of 2001: March 11, 2000 to October 9,

2002, Nasdaq lost 78% of its value.

In response the Fed pushed the fed funds rates to 1.0% (levels not seen since the 1950s)

Nominal Fed Funds Rate

Real Fed Funds RateReal Rate Goes Negative 2003/04

Where is it today? Effective Rate: ______

Go to: http://www.bloomberg.com/apps/quote?ticker=FEDL01%3AIND

Latest Inflation: ______Go to:

http://www.bls.gov/bls/inflation.htm

Your analysis of monetary policy and credit conditions in the economy?

Another Web Site for Calculating Yields Visit the web site below. It allows you to calculate the current

yield and yield to maturity for specific data you input on: Current Market Price Coupon Rate Years to Maturity

It also allows you to calculate present values.

Use this web site to test your understanding of the relationship between bond prices and interest rates. See what happens to the calculated interest rates when you

change the bond price above and below the par value. Note the inverse relationship.

http://www.moneychimp.com/calculator/bond_yield_calculator.htm

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)

Appendix 1

Using Excel to Calculate the Market Price (Present Value) of a Bond

Using Excel to Calculate Bond Price Go to Formulas in Microsoft Excel Go to Financial Go to Price Insert Your Data:

Example for 20 year, 10% coupon bond with market rate of 11%: Settlement: DATE(2009,2,1) Assume, Feb 1, 2009 Maturity: DATE(2029,2,1) Note: 20 years to maturity Rate: 10% (this is the coupon yield) Yld: 11% (this is the yield to maturity) Redemption: 100 (this is the price per $100) Frequency: 2 (assume interest is paid semi-annually) Basis: 3 (this basis uses a 365 day calendar year)

Formula result (i.e., price per $100 face value) = 91.97694 (or $919.77)

Appendix 2

Using Excel to Calculate the Duration of a Bond

Using Excel to Calculate Duration Go to Formulas in Microsoft Excel Go to Financial Go to Duration Insert Your Data:

Example for 10 year, 10% coupon bond with market rate of 10%: Settlement: DATE(2009,2,1) Assume, Feb 1, 2009 Maturity: DATE(2019,2,1) Note: 10 years to maturity Rate: 10% (this is the coupon yield) Yld: 10% (this is the yield to maturity) Frequency: 2 (assume interest is paid semi-annually) Basis: 3 (this basis uses a 365 day calendar year)

Formula result = 6.54266

Appendix 3

The Real Interest Rate during a period of deflation

What if the Rate of Inflation is Negative (i.e., Deflation) Assume the following: imr = 3% and pe = -2% Then the calculated real rate would be: irr = 3% - (-2%) = 5% Issues:

1. What will be the economy’s incentive to borrow? High or low.

2, What are the issues facing the central bank when the economy is experiencing deflation? How can borrowing be encouraged?

Appendix 4

Types of Debt Instruments and Lending Terms

2 Basic Types of Debt Instruments Discount Bond (Zero-coupon Bond):

A bond whose purchase price is below the face (or par) value of the bond (i.e., at a discount)

The entire face (par) value is paid at maturity. There are no interest payments. U.S. Treasury bills are an example of a discount security (as is

commercial paper and bankers’ acceptances). Coupon Bond:

A bond that pays 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. In the United States and Japan, interest payments are

typically made every six months and in Europe typically once a year.

coupon bonds can sell at either a discount or premium (of par value).

These bonds are generally callable. Issuer can “retire” them before their stated maturity date. Why do you think they might do this?

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 or refinanced.

(Loan) Term: the time period from initiation of the loan to the 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 annual interest payment divided by the loan principal. In bond terminology, the coupon interest rate is the annual interest

payment divided by the par value.

Types of Loans Simple Loan: Principal and all interest both paid at

maturity (i.e., date when loan comes due). Borrow $1,000 today at 5% and in 1 year pay $1,050 Commercial bank loans to businesses are usually simply

loans. 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 (maturity date) the principal amount is fully repaid. Referred to as an amortized loan. Home mortgages (conventional), automobile loans.

Amortization Loan Example: Real Estate

Mortgage Loan Principal Amount: $500,000 Years To Maturity: 30 years (with monthly payments) Interest rate: 7% (fixed rate mortgage)

Monthly Payment $3,326.51 (for 360 months, i.e., 30 years)

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)

Appendix 5

Quoting Treasury Notes and Bonds

Treasury Prices in 32nds Treasury note and bond prices are quoted in dollars and

fractions of a dollar. By market convention, the normal fraction used for Treasury

security prices is 1/32 (of $1). In a quoted price, the decimal point separates the full dollar

portion of the price from the 32nds of a dollar, which are to the right of the decimal. Thus a quote of 100.08 means $105 plus 8/32 of a dollar, or $100.25, for

each $100 face value of the note. Note: the symbol + refers to ½ of 1/32nd.

Change data is the difference between the current trading day's price and the price of the preceding trading day. It, too, is a shorthand reference to 32nds of a point. For example, a +16 refers to a change of 16/32 or 50 cents from the

previous day.