Finance Answer Note #9

Question 1 Assuming the expectations theory is the correct theory of the term structure, calculate the interest rates in the term structure for maturities of one to five years, and plot the resulting yield curves for the following paths of one-year interest rates over the next five years: a) 15%, 17%, 17%, 17%, 17% b) 15%, 14%, 14%, 14%, 14% Answer According to the expectations theory of the term structure, the long-term interest rates are determined by the market's expectations of future short-term interest rates. This theory assumes that investors are indifferent between holding short-term bonds or rolling over short-term bonds in the future. Therefore, the yield curve reflects the expected path of future short-term interest rates. In this scenario, we are given two different paths for oneyear interest rates over the next five years and we will use the expectations theory to calculate the interest rates for maturities of one to five years and plot the corresponding yield curves. First, for path a) where the one-year interest rates are 15%, 17%, 17%, 17%, and 17% for the next five years, we can calculate the interest rates for maturities of one to five years using the following formula: R(n) = ([(1 + r(n))/ (1 + r(1))]-1) x 100, Where R(n) is the interest rate for maturity of n years and r(n) is the one-year interest rate for year n. For maturity of one year, R(1) = ([(1+0.15)/(1+0.15)]-1) x100 = 0% For maturity of two years, R(2) = ([(1+0.17)/(1+0.15)]-1) x100 = 1.33% For maturity of three years, R(3) = ([(1+0.17)/(1+0.15)]-1) x100 = 1.33% For maturity of four years, R(4) = ([(1+0.17)/(1+0.15)]-1) x100 = 1.33% For maturity of five years, R(5) = ([(1+0.17)/(1+0.15)]-1) x100 = 1.33% Using these calculations, we can plot the yield curve for path a) as shown below: [picture of yield curve for path a with points (1,0), (2,1.33), (3,1.33), (4,1.33), (5,1.33)] Now, let's move on to path b) where the one-year interest rates are 15%, 14%, 14%, 14%, and 14% for the next five years. Using the same formula, we can calculate the interest rates for maturities of one to five years as follows: For maturity of one year, R(1) = ([(1+0.15)/(1+0.15)]-1) x100 = 0% For maturity of two years, R(2) = ([(1+0.14)/(1+0.15)]-1) x100 = -0.67% For maturity of three years, R(3) = ([(1+0.14)/(1+0.15)]-1) x100 = -0.67% For maturity of four years, R(4) = ([(1+0.14)/(1+0.15)]-1) x100 = -0.67% For maturity of five years, R(5) = ([(1+0.14)/(1+0.15)]-1) x100 = -0.67% The yield curve for path b) can be plotted as shown below: [picture of yield curve for path b with points (1,0), (2,-0.67), (3,-0.67), (4,-0.67), (5,0.67)] We can see from these yield curves that the shape of the curve for path a) is upward sloping, while the curve for path b) is inverted (downward sloping). This indicates that the market is expecting short-term interest rates to increase in the future for path a), while expecting a decrease in short-term interest rates for path b). In terms of useful sources, the calculations were based on the expectations theory of the term structure, which is a well-established theory in finance literature. Some useful academic sources for this theory include "The Theory of Interest Rates" by Irving Fisher and "The Expectations Theory of the Term Structure of Interest Rates" by Antii Ilmanen. Moreover, for the step-by-step breakdown of the calculations, "Investments" by Bodie, Kane, and Marcus provides a comprehensive guide on bond valuation and yield curve calculations. This textbook also includes practical examples and problems to reinforce the understanding of this concept. In conclusion, as a University student with high marks, I have applied the expectations theory of the term structure to calculate the interest rates for maturities of one to five years and plot the corresponding yield curves for two different paths of one-year interest rates. This answer considers both the theoretical and Question 2 The price of bonds and the interest rate are inversely related, that is, when one increases, the other decreases. True False Answer True. According to the principles of bond valuation, there exists an inverse relationship between bond prices and interest rates. This means that as interest rates rise, bond prices fall and vice versa. Bonds are essentially a form of debt instrument issued by companies or governments to raise capital. They provide investors with a fixed rate of return, usually through semiannual interest payments, and the face value or principal amount is repaid upon maturity. The price of a bond is the present value of its future cash flows, and the interest rate (also known as the yield) is used to discount these cash flows to determine the bond's value. To understand the inverse relationship between bond prices and interest rates, we need to look at two key factors: coupon rate and market interest rate. The coupon rate is the fixed interest rate that a bond pays, while the market interest rate refers to the rate at which new bonds are being issued in the market. When a bond is issued, it is set at a specific coupon rate, which remains fixed until maturity. This means that even if the market interest rates change, the bond's coupon rate remains the same. Now, let's consider an example of how changes in market interest rates affect bond prices. Suppose we have a 10-year bond with a coupon rate of 5%, and the market interest rate for similar bonds rises to 6%. This means that investors can now get a higher rate of return by investing in new bonds instead of holding onto the existing 5% bond. As a result, the 5% bond's price will fall to make it more attractive to investors. Conversely, if the market interest rate falls to 4%, the 5% bond's price will increase due to its higher coupon rate compared to new bonds that offer only a 4% return. Question 3 If a bank has $1,000,000 of checkable deposits, what is the maximum amount that they can lend out if the required reserve ratio is 10%? a. $1,000,000 b. Not enough information c. $100,000 d. $900,000 Answer Based on the given information, the maximum amount that the bank can lend out would be $900,000. This is because the required reserve ratio, also known as the cash reserve ratio, is set at 10%. This means that 10% of the $1,000,000 checkable deposits must be held in reserve by the bank, while the remaining 90% can be loaned out as per the bank's discretion. In order to understand this concept better, let us first define what checkable deposits are. According to the Federal Reserve Bank of St. Louis, checkable deposits refer to "amounts in bank accounts that depositors can access by writing a check or using a debit card". These deposits are part of the money supply in an economy and are considered as liabilities for the bank. Now, let us look into the required reserve ratio. As mentioned earlier, this is the percentage of checkable deposits that banks are required to hold in reserve. This is set by the Federal Reserve in order to control the money supply and influence the economy. In the United States, the required reserve ratio is currently set at 10%. This means that for every $100 of checkable deposits, banks are required to hold $10 in reserve. To calculate the maximum amount that a bank can lend out, we can use the formula: Maximum amount available to lend = total checkable deposits x required reserve ratio Substituting the values from the given information, we get: $900,000 = $1,000,000 x 0.10 Therefore, the correct answer to the question is d. $900,000. Question 4 A bond with a face value of $1000 selling at a price equal to $925, has an interest rate equal to _____. a. 0.08% b. 0.925% c. 9.25% d. 8.1% Answer Based on the given information, we can use the formula for calculating bond yield to determine the interest rate of the bond. Bond Yield = (Annual Interest Payment / Current Bond Price) * 100 Firstly, we need to find the annual interest payment, which is equal to the coupon rate (expressed as a percentage) multiplied by the face value of the bond. Since the face value is $1000 and the bond is selling at $925, it can be assumed that the coupon rate is lower than the bond yield. Let's assume that the coupon rate is 8%, which is equal to 0.08 as an interest rate. This means that the annual interest payment will be $80 ($1000 x 0.08). Now, we can plug in the values into the formula: Bond Yield = ($80 / $925) * 100 Bond Yield = 8.6486... Therefore, the interest rate of the bond is approximately 8.65%, which rounds up to the closest given option of 8.1%. Question 5 A 9 percent preferrred stock for Micky's Corp. is currently selling for $75 a share. What is the cost of preferred stock? Answer The cost of preferred stock can be determined by calculating the dividend yield or the dividend rate of the preferred stock. In this case, as stated in the question, the preferred stock for Micky's Corp. has a dividend rate of 9 percent. To calculate the cost of preferred stock, we can use the following formula: Cost of preferred stock = Annual dividend / Current market price. In this question, the annual dividend would be 9% of the current market price, which is $75. Therefore, the annual dividend would be $6.75 ($75 x 0.09). Using this information, the cost of preferred stock can be calculated as follows: Cost of preferred stock = $6.75 / $75 =0.09 or 9% Therefore, the cost of preferred stock for Micky's Corp. is 9%.