#  The Effective Annual Rate (EAR) ◦ Indicates the total amount of interest that will be earned at the end of one year ◦ The EAR considers the effect of.

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 The Effective Annual Rate (EAR) ◦ Indicates the total amount of interest that will be earned at the end of one year ◦ The EAR considers the effect of compounding  Also referred to as the effective annual yield (EAY) or annual percentage yield (APY)

 Adjusting the Discount Rate to Different Time Periods ◦ If 5% is the effective annual rate what is the 6- month rate? ◦ Earning a 5% return annually is not the same as earning 2.5% every six months.  (1.05) 0.5 – 1= 1.0247 – 1 =.0247 = 2.47%  Note: n = 0.5 since we are solving for the six month (or 1/2 year) rate

 Problem ◦ Suppose your bank account pays interest monthly with an effective annual rate of 5%. What is the interest you will earn in one month? What is the interest you will earn in two years? ◦ We know from above that a 5% EAR is equivalent to earning (1.05) 1/12 -1 =.4074% per month. ◦ We can use the same formula to find the two year rate (1.05) 2 – 1 = 10.25% (in particular, not 10%).

 The annual percentage rate (APR), indicates the amount of simple interest earned in one year. ◦ Simple interest is the amount of interest earned without the effect of compounding. ◦ The APR is typically less than the effective annual rate (EAR). ◦ APR is simply a communication device.

 The APR itself cannot be used as a discount rate (except by accident). ◦ The APR with k compounding periods each year is simply a standardized way of quoting the actual interest earned each compounding period:

 Converting an APR to an EAR ◦ The EAR increases with the frequency of compounding.  Continuous compounding is compounding every instant.

◦ If the APR is 6% what are the EARs for different compounding intervals?  Annual compounding: (1 + 0.06/1) 1 – 1 = 6%  Semiannual compounding: (1 + 0.06/2) 2 – 1 = 6.09%  Monthly compounding: (1 + 0.06/12) 12 – 1 = 6.1678%  Daily compounding: (1 + 0.06/365) 365 – 1 = 6.1831%  A 6% APR with continuous compounding results in an EAR of approximately 6.1837%. This is an abstraction we will not make much use of in this course.

 You are offered a chance to buy a perpetuity paying a \$100 per year (beginning in one year) for \$1,650 today. You are able to borrow and lend money at a 6% APR with monthly compounding. Should you buy the perpetuity?  First we must find the EAR since the perpetuity makes annual payments. The effective annual rate is (1+0.06/12) 12 – 1 = 6.16778%.  The value of the perpetuity is \$100/0.0616778 = \$1,621.33.

 Inflation and Real Versus Nominal Rates ◦ Nominal Interest Rate: The rates quoted by financial institutions and used for discounting or compounding cash flows ◦ Real Interest Rate: The rate of growth of your purchasing power, after adjusting for inflation

 The Real Interest Rate

 Problem ◦ In the year 2006, the average 1-year Treasury rate was about 4.93% and the rate of inflation was about 2.58%. ◦ What was the real interest rate in 2006?  Solution ◦ Using the equation given above, the real interest rate in 2006 was:  (4.93% − 2.58%) ÷ (1.0258) = 2.29%  Which is approximately equal to the difference between the nominal rate and inflation: 4.93% – 2.58% = 2.35%

 An increase in interest rates will typically reduce the NPV of an investment. ◦ Consider an investment that requires an initial investment of \$3 million and generates a cash flow of \$1 million per year for four years. If the interest rate is 4%, the investment has an NPV of: ◦ Recall the number \$0.629 million means something very specific. What is it?

 If the interest rate jumps to 15%, the NPV of this investment becomes negative and we would not undertake such a project.

 Term Structure: The relationship between the investment term and the interest rate  Yield Curve: A graph of the term structure

 When the yield curve is not flat we must discount future cash flows at the appropriate rates. ◦ Compute the present value of a risk-free three-year annuity of \$500 per year, given the following yield curve:

 Solution ◦ Each cash flow must be discounted by the corresponding interest rate: ◦ It is very important to note that even though this is a 3-year annuity, we cannot use the annuity formula to find its present value.

 The shape of the yield curve is influenced by interest rate expectations. ◦ An inverted yield curve indicates that interest rates are expected to decline in the future.  Because interest rates tend to fall in response to an economic slowdown, an inverted yield curve is often interpreted as a negative forecast for economic growth.  Each of the last six recessions in the United States was preceded by a period in which the yield curve was inverted.

 The shape of the yield curve is influenced by interest rate expectations. ◦ A steep yield curve generally indicates that interest rates are expected to rise in the future.  The yield curve tends to be sharply increasing as the economy comes out of a recession and interest rates are expected to rise.

 Risk and Interest Rates ◦ U.S. Treasury securities are considered “risk-free.” All other borrowers have some risk of default, so investors require a higher rate of return. ◦ The yield curve is commonly presented in terms of risk free yields but one can also examine a AAA or AA yield curve at any point in time.  When computing the present value of future cash flows that are risky we adjust the discount rate to account for the risk.

 Taxes reduce the amount of interest an investor can keep, and we refer to this reduced amount as the after-tax interest rate.

 Opportunity Cost of Capital: The best available expected return offered in the capital market on an investment of comparable risk and term (investment horizon) to the cash flow being discounted ◦ Also referred to as Cost of Capital

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