16. You are faced with the probability distribution of the HPR on the stock market index fund given in Spreadsheet 5.1 of the text. Suppose the price of a put option on a share of the index fund with exercise price of $110 and time to expiration of 1 year is $12. a. What is the probability distribution of the HPR on the put option? b. What is the probability distribution of the HPR on a portfolio consisting of one share of the index fund and a put option? c. In what sense does buying the put option constitute a purchase of insurance in this case? increase in the value of the investment over this holding period is r(T)=P(T)1001 We can equivalently write this rate of return over the holding period as Holdingperiodreturn=r(T)=P(T)Priceincrease+Income=P(T)100P(T)+0 percentage increase in funds per year. and presented in column 3. The final column re-expresses the total return as an effective annual return. investment increases, is found by compounding as follows: 1+EAR=1.02712=1.0549, implying that EAR=5.49%. Table 5.1 Annualized rates of return. Prices and returns on zero-coupon bonds with face value of $100 and different maturities. page 121 25-year bond in Table 5.1 grows by its maturity by a multiple of 4.2918 (i.e., 1+3.2918 ), so its EAR is found by solving (1+EAR)25=4.29181+EAR=4.29181/25=1.0600 Throughout the chapter, we will examine and compare rates of return on various securities over many different investment periods. Before we can do this properly, however, we first need to learn how to compare investment returns over different holding periods. We start with the simplest security, a zero-coupon bond. This is a bond that pays its owner only one cash flow, for example, $100, on the maturity date. The investor buys the bond for less than face value (face value is just the terminology for the ultimate payoff value of the bond, in this case, \$100), so the total return is the difference between the initial purchase price and the ultimate payment of face value. If we call P(T) the price paid today for a zero-coupon bond with maturity date T, then over the life of the bond, the value of the investment grows by the multiple 100/P(T). The percentage increase in the value of the investment over this holding period is r(T)=P(T)1001 We can equivalently write this rate of return over the holding period as Holdingperiodreturn=r(T)=P(T)Priceincrease+Income=P(T)100P(T)+0 It is no surprise that if you are willing to invest your money for longer periods, you should expect to earn higher total returns. The zero-coupon bond with longer maturity will have a lower present value and a lower price, therefore providing a higher total return. But this observation raises the obvious question of how we should compare returns on investments with differing horizons. The answer is that we must re-express each total return as a rate of return over a common period. We typically express an investment return as an effective annual rate (EAR), defined as the percentage increase in funds per year. Table 5.1 illustrates the technique. Column 2 presents prices of zero-coupon bonds with $100 face value and various maturities. The total return on each security is calculated from Equation 5.1 and presented in column 3 . The final column re-expresses the total return as an effective annual return. For the one-year investment, the EAR is simply the total return on the bond, the percentage increase in the value of the investment, which is 4.69%. For investments that last less than one year, let's say six months, we would compound the half-year return. In the top row of the table (labeled half-year), we see that the semiannual return is 2.71%, so the EAR, the annual rate at which the investment increases, is found by compounding as follows: 1+EAR=1.02712=1.0549, implying that EAR=5.49%. Table 5.1 Annualized rates of return. Prices and returns on zero-coupon bonds with face value of $100 and different maturities. 16. You are faced with the probability distribution of the HPR on the stock market index fund given in Spreadsheet 5.1 of the text. Suppose the price of a put option on a share of the index fund with exercise price of $110 and time to expiration of 1 year is $12. a. What is the probability distribution of the HPR on the put option? b. What is the probability distribution of the HPR on a portfolio consisting of one share of the index fund and a put option? c. In what sense does buying the put option constitute a purchase of insurance in this case? increase in the value of the investment over this holding period is r(T)=P(T)1001 We can equivalently write this rate of return over the holding period as Holdingperiodreturn=r(T)=P(T)Priceincrease+Income=P(T)100P(T)+0 percentage increase in funds per year. and presented in column 3. The final column re-expresses the total return as an effective annual return. investment increases, is found by compounding as follows: 1+EAR=1.02712=1.0549, implying that EAR=5.49%. Table 5.1 Annualized rates of return. Prices and returns on zero-coupon bonds with face value of $100 and different maturities. page 121 25-year bond in Table 5.1 grows by its maturity by a multiple of 4.2918 (i.e., 1+3.2918 ), so its EAR is found by solving (1+EAR)25=4.29181+EAR=4.29181/25=1.0600 Throughout the chapter, we will examine and compare rates of return on various securities over many different investment periods. Before we can do this properly, however, we first need to learn how to compare investment returns over different holding periods. We start with the simplest security, a zero-coupon bond. This is a bond that pays its owner only one cash flow, for example, $100, on the maturity date. The investor buys the bond for less than face value (face value is just the terminology for the ultimate payoff value of the bond, in this case, \$100), so the total return is the difference between the initial purchase price and the ultimate payment of face value. If we call P(T) the price paid today for a zero-coupon bond with maturity date T, then over the life of the bond, the value of the investment grows by the multiple 100/P(T). The percentage increase in the value of the investment over this holding period is r(T)=P(T)1001 We can equivalently write this rate of return over the holding period as Holdingperiodreturn=r(T)=P(T)Priceincrease+Income=P(T)100P(T)+0 It is no surprise that if you are willing to invest your money for longer periods, you should expect to earn higher total returns. The zero-coupon bond with longer maturity will have a lower present value and a lower price, therefore providing a higher total return. But this observation raises the obvious question of how we should compare returns on investments with differing horizons. The answer is that we must re-express each total return as a rate of return over a common period. We typically express an investment return as an effective annual rate (EAR), defined as the percentage increase in funds per year. Table 5.1 illustrates the technique. Column 2 presents prices of zero-coupon bonds with $100 face value and various maturities. The total return on each security is calculated from Equation 5.1 and presented in column 3 . The final column re-expresses the total return as an effective annual return. For the one-year investment, the EAR is simply the total return on the bond, the percentage increase in the value of the investment, which is 4.69%. For investments that last less than one year, let's say six months, we would compound the half-year return. In the top row of the table (labeled half-year), we see that the semiannual return is 2.71%, so the EAR, the annual rate at which the investment increases, is found by compounding as follows: 1+EAR=1.02712=1.0549, implying that EAR=5.49%. Table 5.1 Annualized rates of return. Prices and returns on zero-coupon bonds with face value of $100 and different maturities