Interest Rates Interest Rate Quotes and Adjustments The
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Interest Rates
Interest Rate Quotes and Adjustments � 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)
Interest Rate Quotes and Adjustments � 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
Example � 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%).
Annual Percentage Rates � 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.
Annual Percentage Rates � 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:
Annual Percentage Rates � Converting an APR to an EAR ◦ The EAR increases with the frequency of compounding. �Continuous compounding is compounding every instant.
Annual Percentage Rates ◦ 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.
Example � 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.
Determinants of Interest Rates � 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
Determinants of Interest Rates � The Real Interest Rate
Example � 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%
Investment Policy and Interest Rates � 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 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?
Investment Policy and Interest Rates � If the interest rate jumps to 15%, the NPV of this investment becomes negative and we would not undertake such a project.
The Yield Curve and Discount Rates � Term Structure: The relationship between the investment term and the interest rate � Yield Curve: A graph of the term structure
Term Structure of Risk-Free U. S. Interest Rates, January 2004, 2005, and 2006
Example � 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:
Example � 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 Yield Curve and the Economy � 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 Yield Curve and the Economy � 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 Discount Rates � 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.
After-Tax Interest Rates � Taxes reduce the amount of interest an investor can keep, and we refer to this reduced amount as the after-tax interest rate.
The Opportunity Cost of Capital � 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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