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What is compound interest? How does it relate to the formula: X dollars today = (1 + i ) t X dollars in t years? What is present value? How does it relate to the formula: X /(1 + i ) t dollars today = X dollars in t years?
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Compound interest describes how quickly an investment increases in value when interest is paid, or compounded, not only on the original amount invested but also on all interest payments that have been previously made.
This concept relates to the formula (1+i) t X through the variables i , the interest rate, and t , the amount of years (time) X dollars is invested. The first year X dollars are invested the payoff is (1+i)X. If we allow this investment to 'roll-over' another year and invest (1+i)X we will have (1+i)(1+i)X = (1+i) 2 X at the end of year two. That is we earn interest on the principal and interest from the previous year. After t years we have (1+i) t X.
The present value model simply rearranges the equation above to make it easier to transform future amounts of money into present amounts of money. Instead of using the formula (1+i) t X to calculate the 'future value' of X dollars today we can write the formula as X /(1 + i ) t to calculate how much X dollars in the future is worth to us today.
For example, assume I offer you $1100 a year from now or $1000 today that you can't spend for a year (you must save the $1000). Also, assume the current interest rate iS10%. Which would you choose Your answer should be it doesn't matter which I give you. If you take the $1000 today it is worth $1100 a year from now. Thus, the offer of $1100 in the future is equivalent to $1000 (= X /(1 + i ) = $1100/1.1).
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Corporations often distribute profits to their shareholders in the form of dividends, which are simply checks mailed out to shareholders. Suppose that you have the chance to buy a share in a fashion company called Rogue Designs for $35 and that the company will pay dividends of $2 per year on that share every year. What is the annual percentage rate of return? Next, suppose that you and other investors could get a 12 percent per year rate of return by owning the stocks of other very similar fashion companies. If investors care only about rates of return, what should happen to the share price of Rogue Designs? (Hint: This is an arbitrage situation.)
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A yearly dividend of $2 on a $35 share of stock equals a 5.71% annual rate of return ($2/$35 =.0571 = 5.71%).
If stocks returning 12 percent annual rates of return became available, investors would sell shares of Rogue Designs and buy shares in companies earning thE12% return. This would cause the price of Rogue Designs stock to fall. Note that as the stock price falls, the annual percentage return from owning Rogue Designs stock will rise, assuming the $2 annual dividend continues. Eventually the rates of return of the two (or more) similar companies should equalize.
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Suppose that you invest $100 today in a risk-free investment and let the 4 percent annual interest rate compound. Rounded to full dollars, what will be the value of your investment 4 years from now?
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The future value ( FV ) of the investment is calculated using the formula of future value model. The formula for this is given as:
Where
"X " is the amount of money today
" i " is the interest rate.
" t " is the duration or the number of years.
On substitution:
Thus, future value of the investment is $116. 985.
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If we compare the betas of various investment opportunities, why do the assets that have higher betas also have higher average expected rates of return?
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Suppose that you desire to get a lump sum payment of $100,000 two years from now. Rounded to full dollars, how many current dollars will you have to invest today at a 10 percent interest to accomplish your goal?
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In this chapter we discussed short-term U.S. government bonds. But the U.S. government also issues longer-term bonds with horizons of up to 30 years. Why do 20-year bonds issued by the U.S. government have lower rates of return than 20-year bonds issued by corporations? And which would you consider more likely, that longer-term U.S. government bonds have a higher interest rate than short-term U.S. government bonds, or vice versa? Explain.
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Suppose that the city of New York issues bonds to raise money to pay for a new tunnel linking New Jersey and Manhattan. An investor named Susan buys one of the bonds on the same day that the city of New York pays a contractor for completing the first stage of construction. Is Susan making an economic or a financial investment? What about the city of New York?
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Consider an asset that costs $120 today. You are going to hold it for 1 year and then sell it. Suppose that there is a 25 percent chance that it will be worth $100 in a year, a 25 percent chance that it will be worth $115 in a year, and a 50 percent chance that it will be worth $140 in a year. What is its average expected rate of return? Next, figure out what the investment's average expected rate of return would be if its current price were $130 today. Does the increase in the current price increase or decrease the asset's average expected rate of return? At what price would the asset have a zero average expected rate of return?
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What are mutual funds? What different types of mutual funds are there? And why do you think they are so popular with investors?
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How do stocks and bonds differ in terms of the future payments that they are expected to make? Which type of investment (stocks or bonds) is considered to be more risky? Given what you know, which investment (stocks or bonds) do you think commonly goes by the nickname "fixed income"?
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What determines the vertical intercept of the Security Market Line (SML)? What determines its slope? And what will happen to an asset's price if it initially plots onto a point above the SML?
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Suppose that a risk-free investment will make three future payments of $100 in one year, $100 in two years, and $100 in three years. If the Federal Reserve has set the risk-free interest rate at 8 percent, what is the proper current price of this investment? What is the price of this investment if the Federal Reserve raises the risk-free interest rate to 10 percent?
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Suppose initially that two assets, A and B, will each make a single guaranteed payment of $100 in 1 year. But asset A has a current price of $80 while asset B has a current price of $90.a. What are the rates of return of assets A and B at their current prices? Given these rates of return, which asset should investors buy and which asset should they sell?
b. Assume that arbitrage continues until A and B have the same expected rate of return. When arbitrage ends, will A and B have the same price?
Next, consider another pair of assets, C and D. Asset C will make a single payment of $150 in one year while D will make a single payment of $200 in one year. Assume that the current price of C is $120 and that the current price of D is $180.
c. What are the rates of return of assets C and D at their current prices? Given these rates of return, which asset should investors buy and which asset should they sell?
d. Assume that arbitrage continues until C and D have the same expected rate of return. When arbitrage ends, will C and D have the same price?
Compare your answers to questions a through d before answering question e.
e. We know that arbitrage will equalize rates of return. Does it also guarantee to equalize prices? In what situations will it equalize prices?
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Advanced Analysis Suppose that the equation for the SLM is Y = 0.05 + 0.04 X , where Y is the average expected rate of return, 0.05 is the vertical intercept, 0.04 is the slope, and X is the risk level as measured by beta. What is the risk-free interest rate for this SML? What is the average expected rate of return at a beta of 1.5? What is the value of beta at an average expected rate of return is 7 percent?
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Suppose that the Federal Reserve thinks that a stock market bubble is occurring and wants to reduce stock prices. What should it do to interest rates?
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Consider another situation involving the SML. Suppose that the risk-free interest rate stays the same, but that investors' dislike of risk grows more intense. Given this change, will average expected rates of return rise or fall? Next, compare what will happen to the rates of return on low-risk and high-risk investments. Which will have a larger increase in average expected rates of return, investments with high betas or investments with low betas? And will high-beta or low-beta investments show larger percentage changes in their prices?
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Why is it reasonable to ignore diversifiable risk and care only about nondiversifiable risk? What about investors who put all their money into only a single risky stock? Can they properly ignore diversifiable risk?
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LAST WORD Why is it so hard for actively managed funds to generate higher rates of return than passively managed index funds having similar levels of risk? Is there a simple way for an actively managed fund to increase its average expected rate of return?
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