write one essay on finance topic about 600 words
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05InterestRates.pptxTopic 3 Interest rates
Fundamentals of Finance
Fall 2017
Zhun Liu
APR – Annual Percentage Rate
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The rate that is quoted on a credit card or car loan is the annual percentage rate or APR (this is required by the law: Truth in Lending Laws).
That means that if a credit card company is offering a rate of 1 percent per month, it must quote a 12 percent APR (simple multiplication). This is simple interest.
But does this mean that the consumer is actually paying 12% interest. The somewhat disturbing answer is no!
The effective annual rate under monthly compounding is (1+0.12/12)^12 – 1 = 0.1268 = 12.68%
You are paying an effective rate that is .68% more than what is displayed.
The APR differs from the EAR because it does not take into account compounding. It does not take into account interest on interest. It is simple interest. APR assumes that you cannot reinvest the interest.
An Example
Suppose you can invest $100 in an account that compounds every six months and pays you 5% every six months
How much do you have in six months? In a year?
Is this the same as 10% compounded annually?
Is this the same as 10.25% compounded annually?
Which rate is defined as APR? Is this precise?
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Another Example
Anna is charged 1% interest when she borrows $2000 for one week. What is the annual percentage interest rate (APR) on the loan?
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Effective Annual Interest Rate
Effective Annual Interest Rate (EAR)
EAR accounts for the number of compounding periods and adjusts the annualized interest rate for the time value of money
ACTUAL(!) interest earned in one year
More accurate measure of the rates involved in lending and investing
Can have effective rates at other frequencies as well (quarterly, biennially, every 5 years, etc.)
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Revisit the Example
Anna is charged 1% interest when she borrows $2000 for one week. What is the effective annual interest rate (EAR)?
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Credit cards: EAR and APR
The APR of your credit card is 18%, charged monthly (i.e. 18% per year with monthly compounding)
What does the above statement mean?
Annual Percentage Rate (APR): The way a rate is quoted converted to one year
To calculate the rate each month, divide the APR by 12.
For example, if the APR is 18%, the interest rate on carrying a balance is 1.5% per month.
Effective annual rate (EAR): The actual interest rate earned by the end of the year
Considers the effect of compounding during the year
What is the EAR?
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Converting APR to EAR
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Compounding Frequency
Which loan is cheapest?
6%, compounded annually
6%, compounded semi-annually
6%, compounded monthly
6%, compounded daily
What happens to the EAR as the compounding frequency increases?
Resources: Pearson Education
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Suppose the quoted rate is given and equal to 100%, we all dream of that.
Consider increasingly frequent compounding:
Annually: 100%; Semi-annually: 125%; Quarterly: 144.14% ; Monthly: 161.30%; Daily: 171.46%
every second,…
What happens to the EAR as the number of times we compound m goes to infinity? It increases, but does it increase indefinitely?
You add more and more compounding interval
But the interest rate over that compounding interval gets smaller and smaller
That’s a race!
The limit is e^1-1 = 171.83%
When compounding happens“all the time,” it is called continuous compounding so the daily compounded rate is already pretty close to the continuous compounded rate.
A technical aside:
Lim_{m goes to infinity} (1 + 1/m)^m = e =2.7182. Lim_{m goes to infinity} (1 + Q/m)^m = e^Q. Therefore, Lim_{m goes to infinity} (1 + Q/m)^m -1 = e^Q -1. You do not have to know this, but if you like math you can try to prove it.
Q: How much is 1000$ invested today at 7% continuously compounded worth after 1 year? A: 1000*e^.07 = 1072.508. Annually compounded: 1000*(1+.07) = 1070. The difference is interest on interest coming from compounding with a year.
Q: How much is 100$ invested today at 7% continuously compounded worth after 2 years? A: 1000*e^.07*e^.07 = 1000*e^2*.07=1000*e^.14 = 1150.274. Annually compounded: 1000*(1+.07)^2 = 1149. The difference is interest on interest coming from compounding within each year.
Suppose you were told that the annual rate without compounding is 6%.
Q: What is the equivalent continuously compounded rate? A: e^x=1.06 => x = ln(1.06) = 0.05827 = 5.827%.
The general formula is FV=PV e^rt . Inverting the formula, we get PV = e^(-rt) FV. e^(-rt) is the discount factor under continuous compounding. 1/(1+r)^t is the disocunt factor under annual compounding. Handout 4 reviews continuous compounding. If you need more background please read this handout at home.
Continuous Compounding
When compounding happens “all the time,” it is called continuous compounding
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The Determinants of Interest Rate Levels
Interest Rate
The fee for borrowing money expressed as a percentage of a loan
Determinants
Return on investment
Time preference for consumption
Trade-off between saving (for future consumption) and current consumption
Compensation for expected inflation
Types
Nominal interest rate
The rate the money will grow in a certain period
Real interest rate
Reflect purchasing power, adjusted for inflation
Interest rate that would exist in the absence of inflation (deflation)
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Real and Nominal Interest Rate
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Real and Nominal Interest Rate
An Example
Real rate = 4% Inflation rate = 10%
Nominal rate = ?
Fisher Equation
Nominal rate = Real rate + Inflation rate + (Real rate*Inflation rate)
= 0.04 + 0.10 + (0.04 * 0.10)
= 0.1440 or 14.40%
Simplified
Nominal rate = Real rate + Inflation rate
= 0.04 + 0.10 = 0.14 or 14.00%
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Real and Nominal Interest Rate
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The Determinants of Interest Rate Levels
Cyclical & Long-term Interest Rates
Interest rates tend to rise and fall with changes in the rate of inflation
Interest rates tend to rise when the growth rate of the economy increases (expansion)
Interest rates tend to fall when the growth rate of the economy slows (contraction, recession)
Interest rates tend to follow the business cycle
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Relation Between Annual Inflation Rate and Long-Term Interest Rate
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The Term Structure of Interest Rates
The term structure of interest rates
the relationship between yield to maturity and term-to-maturity on a bond
the graph of the term structure of interest rates is yield curve
The shape and position of the yield curve are not constant.
As the overall level of interest rates changes, the yield curve shifts up and down and changes its shape and slope.
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Yield Curve
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The Term Structure of Interest Rates
Basic shapes (slopes) of yield curves
Ascending or normal yield curves slope upward from left to right and imply higher interest rates are likely
Descending or inverted yield curves slope downward from left to right and imply lower interest rates are likely
Flat yield curves imply interest rates unlikely to change
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Shape of the Yield Curve
Three factors that influence the shape of the yield curve
1) Real rate of interest
2) Expected rate of inflation
3) Interest rate risk
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The Real Rate of Interest
The real rate of interest changes with the business cycle.
Highest rates occur at the end of an economic expansion.
Lowest rates occur at the end of an economic contraction.
Changes in the expected future real rate of interest can affect the slope of the yield curve.
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The Expected Rate of Inflation
If higher inflation is forecast, the yield curve will slope upward because longer-term yields will contain a larger inflation premium than shorter-term yields
If investors believe inflation will subside, the yield curve will slope downward
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Interest Rate Risk
The longer the maturity of a security, the greater its interest rate risk – the risk of selling the security at a lower price - and higher yield-to-maturity
The interest rate risk premium adds upward bias to the slope of the yield curve
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Current Yield Curve
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The Structure of Interest Rates
Cumulative Effect of Factors
In an economic expansion, the real rate of interest and the inflation premium increase monotonically. Interest rate risk increases.
In an economic contraction, the real rate of interest and inflation premium decrease monotonically. Interest rate risk decreases.
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