The following yield data for a number of highest-quality corporate bonds existed at each of the three points in time noted. Yield Time to maturity (years) 5 years ago 2 years ago Today 1 9.1% 14.6% 9.3% 3 9.2 12.8 9.8 5 9.3 12.2 10.9 10 9.5 10.9 12.6 15 9.4 10.7 12.7 20 9.3 10.5 12.9 30 9.4 10.5 13.5 On the same set of axes, draw the yield curve at each of the three given times. Label each curve in part a with its general shape (downward sloping, upward sloping, flat). Describe the general interest rate expectation existing at each of the three times, assuming the expectations theory holds. Examine the data from 5 years ago. According to the expectations theory, what approximate return did investors expect a 5-year bond to pay as of today

5. profit maximization and shutting down in the short run suppose that the market for polos is a competitive market. the following graph shows the daily cost curves of a firm operating in this market. 0 2 4 6 8 10 12 14 16 18 20 50 45 40 35 30 25 20 15 10 5 0 price (dollars per polo) quantity (thousands of polos) mc atc avc for each price in the following table, calculate the firm's optimal quantity of units to produce, and determine the profit or loss if it produces at that quantity, using the data from the previous graph to identify its total variable cost. assume that if the firm is indifferent between producing and shutting down, it will produce. (hint: you can select the purple points [diamond symbols] on the previous graph to see precise information on average variable cost.) price quantity total revenue fixed cost variable cost profit (dollars per polo) (polos) (dollars) (dollars) (dollars) (dollars) 12.50 135,000 27.50 135,000 45.00 135,000 if the firm shuts down, it must incur its fixed costs (fc) in the short run. in this case, the firm's fixed cost is $135,000 per day. in other words, if it shuts down, the firm would suffer losses of $135,000 per day until its fixed costs end (such as the expiration of a building lease). this firm's shutdown price—that is, the price below which it is optimal for the firm to shut down—is per polo.

Bonds often pay a coupon twice a year. for the valuation of bonds that make semiannual payments, the number of periods doubles, whereas the amount of cash flow decreases by half. using the values of cash flows and number of periods, the valuation model is adjusted accordingly. assume that a $1,000,000 par value, semiannual coupon us treasury note with three years to maturity has a coupon rate of 3%. the yield to maturity (ytm) of the bond is 7.70%. using this information and ignoring the other costs involved, calculate the value of the treasury note:

Alarge tank is filled to capacity with 500 gallons of pure water. brine containing 2 pounds of salt per gallon is pumped into the tank at a rate of 5 gal/min. the well-mixed solution is pumped out at the same rate. find the number a(t) of pounds of salt in the tank at time t.