Rate, rate who's got the rate?

If this all seems confusing and that you've entered into a gag in Mr. Wonka's factory, you need not worry trained professionals find this baffling as well. In the last post we used a discount rate to discover what the true benefit of a security was in present value terms. Now, we need to add a bit of complexity to obtain a further degree of accuracy. So today will be about zero-rates on those risk-free securities.

First we will just assume that we know what the zero-rate is and then we can later show how to figure out the implied zero rate. The zero rate is basically what we calculated yesterday. However, coupons and dividends do not all come at the end of the security's life. So we need various zero-rates to find out what the correct price of the security is.

Given:
Zero Rate - Maturity
5.0% - 0.5 (Years)
5.8% - 1.0
6.4% - 1.5
6.8% - 2.0

Now we want to price a 2 year Treasury bond with a principal of $1,000 and a 6% coupon paid semiannually.
0.0 - ?
0.5 - 30
1.0 - 30
1.5 - 30
2.0 - 1,030. That is 1,000 in principal and 30 for the coupon payment.
We utilize the formula from yesterday and use the spot/zero-rate for each maturity.
So we find that:
30e^(-.05*.5)+30e^(-.058*1.0)+30e^(-.064*1.5)+1030e^(-.068*2.0)= 983.85
29.2593+28.3095+27.2539+899.0279=983.85

This result in and of itself is rather interesting. See the security pays 6% annual via two coupons of $30. However, it sells at a discount. This is because the majority of the bond is paid in the final payment where the interest rate is greatest and is discounted by the most periods as well. We can figure out the bond's yield maturity by utilizing a financial calculator. Entering 4 for the period, Present value of -983.85, Future Value of 1000 and Payments of 30. Then solving for the Yield to Maturity it shows 3.439% and multiplying this by two gives the annual rate of 6.89%

What happens when there is not a corresponding Treasury? That is the million dollar question.

Say you have these 5 Treasury securities:We now employ the bootstrap method.
So to find the quarterly zero-rate we know that you will be repaid your principal in 1/4 a year. Thus you gain 25 on your 975 worth of investment. If we annualize that figure you would have:
(4 x 25)/975= 10.26% Which we should change into continuous compounding.
So 4 x ln(1 +(.1026/4)= 10.13% We do the same with the other non-coupon bearing instruments and come out with continuous compounding rates of 10.47% and 10.54%. Now is where the bootstrapping begins in earnest. So the 1 1/2 year bond will pay out coupons semiannually( all Treasury securities do) so it will pay 40, then 40 and finally 1,040. So we can use the zero-rates we have already applied to find the zero-rate for 1.5 years.

40e^(-.1047x0.5)+40e^(-.1054*1.0)+1040e(-R*1.5)=960
37.9599+35.9986+1040e^(-R*1.5)=960
e^(-R*1.5)=886.0415/1040
e^(-R*1.5)=.85196
-1.5R=ln.85196
R=10.68%

Then the last one:
60e^(-.1047*0.5)+60e^(-.1054*1.0)+60e^(-.1068*1.5)+1060e^(-R*2.0)=1016
56.9398+53.9979+51.1184+1060e^(-R*2.0)=1016
e^(-R*2.0)=853.9439/1060
-2R=ln(.8056)
R=10.81%

Now we have a zero yield curve

Homework:
Calculate the zeroes for 1/2 year, 1 year, 1 and 1/2 year and 2 years.

Stupid thought of the day

Ever hear about the mathematician who drowned in the river that was on average 2-inches deep? The joke is slightly bemusing, however, it did start me pondering how the river could be set up so it could drown a man. I mean we most likely have heard that a baby can drown in two-inches of water, but a man is fairly improbable.

Here was the set up I came up with: You have a river that has the same characteristics up and down its length. So a cross-section will mimic the entire river system. It is 1,000 feet wide. The 1st section is 445 feet wide and is 1-inch deep. The middle section is 10 feet wide and is 100 inches deep, (8 1/3 feet.) The last section from the middle to the far bank is 445 feet wide and is 1-inch deep. To find the average depth of the river we add the 3 sections together after we have multiplied them by their depth. Then we divide it by the total width of the river. So

(445*1)+(10*100)+(445*1)=445+1000+445=1990
1990/1000=1.99 inches

So on average the river is 1.99 inches deep, but you can see how this can be tricky for the man fording it.

Answer to Nominal Profits

Your savings account pays 12% annually continuously compounded but pays out the interest to you in quarterly installments. How much interest will be paid on your 1,000-dollar deposit each quarter?

e^R=(1+r/m)^m, where e^.12=(1+r/4)^4. We take the natural log of each side
.12=4 ln(1+r/4)then we move the 4 to the left side so that .03=ln(1+r/4) Then we raise both sides by e so that e^.03 = (1 +r/4) so that 4(e^.03 -1)= r and therefore r=12.18%. For future reference we can just note that r=m(e^(R/m)-1)

To fully answer the question, we must take the 1000*(.1218/4)= 30.46 which is the dollar amount paid out per quarter on the account.

Nominal Profits

Before we move further with options, we need to provide a sturdy foundation for understanding them. That is, in the last post we talked about how much money various option strategies made, but what we didn't consider what the opportunity cost of the strategy was. By that I mean there is a cost of the funds. Consider that you could have left the 9,400 dollars in your bank account earning 5% interest. It took one month for the strategy to play out. Thus, we need to consider how much money we could have earned. The formula is A(1 + R/m)^mn where

* A equals the deposit
* R equals the interest rate
* m equals the compounding frequency
* n equals the period of time the investment is held

Therefore, 9,400*(1+(.05/12))^(12*1/12)= $9,439.17 that is we could have earned $39.17 on the investment risk-free. Conversely, we should take the profit we earned and discount it backwards by this same discount rate, 1.004167 to find out what we made in real terms.

In the real world though we would need to find a "better" or more suitable risk-free rate. While it is true that your deposit at the bank is FDIC insured, it is more than likely that the bank is using that subsidy to achieve funding versus seeking funds in the capital markets. Most people would point to the US Treasury rates or LIBOR. The latter consisting of the rate at which one bank will lend funds to another bank. Thus, it is considered the opportunity cost of capital in the specified time period, 1, 3 6 and 12 month durations. There is also a super-short duration that is sometimes used which is the Repo rate. Basically, one bank sells a security to another bank with an agreement to buy the security back at a slightly higher price. A quick example, I sell a MBS to JP Morgan for 99.98 dollars of its face value and buy it back for 100 or par two days later. Thus, JPM lent me 99.98 dollars for two days. I paid two cents for the privilege. This is basically what the Federal Reserve did during the credit crunch in 2008, they would extend these loans at very low rates, so that banks could remain liquid and meet their liabilities. Now the Fed is actually buying securities, which is an entirely different matter.

However, in derivative securities when using a rate like OIS, LIBOR or Treasury rates they are continuously compounded. So in the bank account example, the bank only credits my account at month end. So I end up with slightly more than 5% more on my account at the end of the year. That is I receive 1.004167 more each month, at the end of two months I would have 1.008351 until I ended up at 1.051162 at the end of the year. If instead it compounded only twice or four times a year I would have less money the less amount of compounding terms. Conversely the more times the money is compounded the more money I would have at year-end. So when a financial asset is continuously compounded it reaches the maximum it could possibly grow at a stated rate of interest. We can then use the exponential factor e^x so that the equation can be written Ae^xn or substituting x for the interest rate the equation is Ae^Rn. Thus, continuous compounding brings up the factor to 1.051271.

However, in the financial markets quotes are given in all variants of time, quarterly, monthly and we need to be able to go back and forth between them.

Ae^(Rn)=A(1+ r/m)^(mn) which you can subtract n from both sides and A as well to end up with e^R=(1+r/m)^m.

Now we can go back and forth. So if a broker quotes you 10% interest compounded semiannually that gives you a m of two and a r of .1. Therefore, e^R=(1+r/2)^2 so take the natural log of both sides so that R=2 ln (1+.1/2) finding that R equals 0.09758

A basic rule is that when you go from the various compounding to continuous compounding the continuous compounding will always be at a lower rate because it compounds more often. In the reverse going from continuous to quarterly you should expect that quarterly rate would be higher.

Homework problem:

Your savings account pays 12% annually continuously compounded but pays out the interest to you in quarterly installments. How much interest will be paid on your 1,000-dollar deposit each quarter?

Answer to Options Uncovered

A July European call sells for $4.70 and the strike price is $95. The stock last traded today at 94. An investor is trying to decide whether to buy 20 call contracts or buy 100 shares, both scenarios cost the same amount. Under which scenarios is the option strategy more profitable, the stock, when are they equivalent?

I drew the graph for you charting the Profit/Loss potential for both the option and the stock strategy. You can see that the loss rate is severe for the options, meaning if the stock does not move up and quickly the option will expire and you will be out the $9,400 in premium. However, once the stock moves above 100 the benefit of leverage comes into play and the upside is also more severe as well. The two strategies are equivalent at a share price of $100.

I would not guess that most clients would go after the call options because there are only 14 more trading days until expiration and the stock would have to move 6.38% up to equal the stock strategy. However, if this option were further out like an October call this might be a reasonable strategy.

Options uncovered

I was recently reading an article in Fortune magazine about derivative contracts. It hilariously made reference to the 1st derivative contract being contrived around 1994. Well, let me quote directly. "In a 1994 cover story by this writer (Carol Loomis), Fortune called derivatives, then relatively new on the scene, "The Risk That Won't Go Away." Knowing this was not true but not knowing when the first derivative contract "appeared on the scene" I Googled it. Lo and behold Aristotle wrote about them in his seminal work Politics. This recorded work was said to have been writing around 350 years before the common era.

There is the anecdote of Thales the Milesian and his financial device, which involves a principle of universal application, but is attributed to him on account of his reputation for wisdom. He was reproached for his poverty, which was supposed to show that philosophy was of no use. According to the story, he knew by his skill in the stars while it was yet winter that there would be a great harvest of olives in the coming year; so, having a little money, he gave deposits for the use of all the olive-presses in Chios and Miletus, which he hired at a low price because no one bid against him. When the harvest-time came, and many were wanted all at once and of a sudden, he let them out at any rate which he pleased, and made a quantity of money. Thus he showed the world that philosophers can easily be rich if they like, but that their ambition is of another sort.

Apparently arriving on the scene in Loomis' meaning entails a period of twenty-three hundred and forty-four years. It must be quite the scene for the party to keep going on that long. I digress.

So I wanted to go over derivatives including options, then futures and finally dreaded credit derivative products. Because we must crawl before walk and walk before run, this may take a post or two dozen.

Options come in two basic varieties that can be combined in countless ways to hedge or speculate in the capital markets. The two basic types are call and puts. Now sense these securities are contracts they are very carefully worded. So here it is:

1. Call option - the right to buy an asset by a certain date for a certain price
2. Put option - the right to sell an asset by a certain date for a certain price
   

Easy, right? Well there are a couple more moving parts, which is where everyone gets wound up. There is the strike/exercise price. This is the certain price. Then there is the expiration date or maturity which is the certain date.

Not so bad. Well, here comes the kicker. The option is a right but just like the American voter who has the right to vote does not mean he will show up to vote. Thus, an option purchaser may not exercise the right. All American options, at least the ones on exchanges are for 100 shares. One last thing, there is still the current price or the spot price, which is what the asset is currently worth. Now for an example.

Today is June 29, 2009 The SPY ETF which mimics the S&P500 currently sells for 92.70. The July 2009 contract with a 93.00 strike price sells for $1.62. So if today you wanted to buy the option you would contact a broker, he would find someone to sell a call and then you would pay 162 dollars. This payment gives you the right to purchase the 100 shares of SPY by July 17th.

So two things can happen, either the option is worth money or it isn't come July 17th. If SPY has been bid up to 95 dollars. Then you could use your option right to buy 100 shares of SPY at 93.00 and immediately sell them in the market for 95.00 Gaining 200 dollars. Then you would figure your net takeaway is 38 dollars because it cost you 162 dollars to set up the contract. On the flip side if SPY is at 94.62 or less it would not make sense for you to exercise your option. That is at 94.62 your gain on the sale once you exercised the right would by 162, less the fee to purchase the contract you are at zero. So you do not exercise the option contract.


Now why would someone do this instead of just purchasing the shares outright? There are countless reasons but the most simple explanation is leverage. You only used 162 dollars to purchase an investment that was worth 9,270 dollars in today's market.

The same is true for puts. This time you instead have a hunch that the market is going to fall in the next couple of months. So instead you buy a July put on the SPY ETF with an expiration price of 90.00 for $0.98. So on July 17th the SPY is trading at 85 dollars. You had sent your 98 dollars to the broker on June 29th and now you can sell SPY shares for 90 dollars when the market is trading them at 85. So two things can happen, either you already own a lot (100 shares of SPY) as part of your market portfolio and you just complete the transaction. Or you can buy the shares and add it to your portfolio now at 85 dollars and sell them. Either way you make 500 dollars on the trade and net out the cost of the contract which was 98, so final total is 402 dollars in profit.



Finally, there are two more positions a person could take with these options, which is selling them. An easy example is insurance. I own 1000 shares of Google (GOOG,) but I think the US government may enact anti-trust litigation against them. Thus, I want to insure my portfolio. You look at the case and decide that there is no way the US government will act on such flimsy evidence. You in fact own GOOG stock as well, to enhance your holdings you sell a put to me. That is, I pay you an upfront premium in case the stock falls. If the stock does not fall, or doesn't fall enough, then you picked up free premium plus you still own the stock and its dividend rights. However, the downside risk is steep. If GOOG did start dropping you would end up doubling down at a time when the majority of stockholders are selling.

The other side is the selling of calls. Just like puts, most calls are never exercised. Thus, if you think that MSFT is not going to appreciate in the next month, then you can sell a call, take the premium and sleep peacefully. Of course ...

Here is a quick homework example and I will post the answer up tomorrow.
A July European call sells for $4.70 and the strike price is $95. The stock last traded today at 94. An investor is trying to decide whether to buy 20 call contracts or buy 100 shares, both scenarios cost the same amount. Under which scenarios is the option strategy more profitable, the stock, when are they equivalent?