Financial Markets -- Problems
Problem 1 -- Simon's Bond
Simon wants to sell a bond which will pay $ 1060 at the end of one year.
- The bond has
- a principal amount or face value of $1000
- a coupon payment of $ 60 a year.
- a coupon rate of 6 percent.
- a term of three years, of which two have already gone by.
- The price for the bond now (when it has only one year to maturity) is given by the present value formula
PV = 1060 / 1 + r
- if r = .03, PV = $ 1029.13
- if r = .06, PV = $ 981.48
- if r = .10, PV = $ 963.64
Note that as the interest rate rises, the price of the bond falls.
- Any risk of non-payment (perhaps the company issuing the bond has had a bad year) will lower the price of the bond.
Problem 2 -- Brothers Grimm
Shares in Brothers Grimm will are expected to pay a dividend of $ 5 and to be worth $ 100 a year from now. If you intend to sell a year from now, you will price the share of stock in accordance with the formula:
PV = 105 / (1 + r + rp)
- if r = .05 and rp = 0, PV = $ 100.00
- if r = .10 and rp = 0, PV = $ 95.45
- if r = .05 and rp = .03, PV = $ 97.22
Without a dividend, the calculations are:
PV = 100 / (1 + r + rp)
- if r = .05 and rp = 0, PV = $ 95.23
- if r = .10 and rp = 0, PV = $ 90.91
- if r = .05 and rp = .03, PV = $ 92.59
Problem 3 -- Changing valuations of stocks and bonds
- if interest rates on newly issued government bonds rise, the government bonds will definitely be cheaper. You will be unable to sell an equivalent maturity value corporate bond for more than a new government bond, so its price will also fall.
Since bonds and stocks react the same way to changes in interest rates, the price of a share of stock will also fall, other things equal.
- if inflation is expected to drop, the Fisher effect will lead to nominal interest rates falling. The prices of bonds and stocks will both rise.
- if perception of stock market risk increases, the risk premium for stocks will increase, and hence their price will fall.
The price of U.S. Government bonds will not be affected.
- if a new product promises to increase profits 5 years from now, the stock price will increase now, although they will be discounted by (1 + r + rp) to the fifth power.
- if dividends are not paid next year, the price of a share of stock will go down right now.
- if price controls are imposed on prescription drugs, the profits of pharmaceutical companies will decline, and the stock price of pharmaceutical companies will fall.
Problem 4 -- Diversification
The risk inherent in the stock market can be lessened for any individual by portfolio diversification .
Note that the formula to compute expected value in an uncertain situation is found by multiplying the value of each outcome by the probablity of that outcome, and then adding up all the results.
Consider the problem of buying a pharmaceutical stock (BigPharma) or a stock in a pollution control company (CleanGreen).
You have $ 1000 to invest in the shares of one or the other stock.
If the Democrats win the next election, they will impose price controls on BigPharma and give government contracts to CleanGreen.
Hence BigPharma stock will not appreciate in price and will not be able to pay a dividend, although we assume it will not lose value.
CleanGreen stock will appreciate by 10 percent.
If the Republicans win the next election, they will not impose price controls, and BigPharma stock will appreciate by 8 percent.
We assume CleanGreen stock will stay at the same price of $ 1000.
If we assume that there is a 40 percent chance the Republicans win the next election and a 60 percent chance that the Democrats win, we can make the following calculations:
- to maximize expected return, we can consider the expected value of investing in:
So that if we invest in only one stock, BigPharma has the highest expected value.
- BigPharma only = .40 (0) + .60 (80) = $ 48
- CleanGreen only = .40 (100) + .60 (0) = $ 40
- if we split our portfolio, so we invest $ 500 in each stock, we will gain .10 * $500 = $ 50 if the Democrats win and .08 * $ 500 = $ 40 if the Republicans win.
The expected value of the portfolio is
.40 * $50 + .60 * $40 = $ 20 + $ 24 = $ 44
- diversifying the portfolio has a lower expected return than investing in BigPharma only, but guarantees a return of at least $ 40. The BigPharma only investment can result in a return of zero.
- if you want to guarantee a return of at least 4.4 percent or $ 44 , you need an investment of:
That is, you must put at least $ 440 in CleanGreen and at least $ 550 in BigPharma.
- at least .10 x = $ 44 in CleanGreen
- and at least .08 x = $ 44 in BigPharma
You are able to do this because the two add up to $ 990;
note that you would not have been able to get a 5 percent guaranteed return, which would have required
or a total of $ 1125 to invest ($ 125 more than you have available).
- $ 50 / .10 = $ 500 in CleanGreen
- and $ 50 / .08 = $ 625 in BigPharma
- What is the maximum guaranteed return you can get, and how can you get it?
If you place $ X in Big Pharma, your return will be
Hence you can guarantee the same return no matter who wins by splitting your portfolio so that
- .08 X if the Republicans win
- .10 (1000 - X) if the Democrats win
.08 X = .10 (1000 - X)
Hence X = $ 555.56 and 1000 - X = $ 444.44, guaranteeing a return of
Problem 5 -- Data on saving, investment and trade
Use Table 5.1 (Saving and Investment)
and Table 4.1 (Foreign Transactions)
In case the above links change, go directly to the Bureau of Economic Analysis and search for the tables.
The annual data for 2005 (the data are preliminary, and may change) indicate:
- Private net saving is +$ 426.2 billion
- Government saving is -$ 320.0 billion
- National saving is +$ 106.1 billion
The total national saving of $ 106 billion is not enough to finance net domestic investment.
Net domestic investment is calculated by subtracting deprciation or Consumption of Fixed Capital from Gross Domestic Investment :
2505.3 - 1574.1 = 931.2
The saving-investment balance is restored by bringing borrowing from the rest of the world into the picture.
Our borrowing from the rest of the world is $ 787.0 billion.
If you do the math, your answer will not quite balance until you take the statistical discrepancy of $ 42.8 billion into account.
Looking at Table 4.1 indicates that most of that goes to finance a current account deficit of $ 782.3 billion.
Problem 6 -- International capital flows - examples
Problem 7 -- International capital flows - graphs
Problem 8 -- Algebra of capital flows