Weekly Nugget: Analyzing financial situations

Arbitrage opportunities with Forward contracts on Gold

Using the Forward price sub menu item

April 11th, 2025

Introduction

Gold is a precious metal with multiple uses ranging from investments to electronics, to jewelry to aerospace. It used to be the underlying medium of exchange until Central Banks stopped using it as a standard. But Central Banks still keep gold as part of their reserves, and places like Fort Knox hold large amounts of it.

It is interesting that during periods of peace and stability, its price doesn't vary by much, but because us human beings are prone to conflict, eventually markets crash and gold price rises.

We may easily observe, from figure 1 below, how the price of gold rose after the financial market crash of 2007-2008 and how current political turmoil has also resulted in a considerable price hike.

Preparing the Analysis

According to https://goldprice.org/ if you had kept an investment in gold over the past 20 years, you would have obtained a 638% benefit.

Pretty impressive as long as you knew when to enter and exit the investment! After the peak in gold price in 2012 there was a considerable drop in its price in the span of a year. But enough of looking backwards. What may happen in the future? and how do we analyze potential investments?

The future is uncertain, but there are Options, Futures and Forward contracts. As of today, April 11th, the price of an ounce of gold ranged from 3,193 to 3,263 USD. Looking at the futures contracts the one expiring in one-year GCJ6 at CME, had an open price of 3,356.0, a daily high of 3,362.7 and a low of 3,339.7 and 118 contracts, each for 100 ounces, were traded. So, the price is expected to increase, but not in keeping with the recent growth trend. Why? Because future's prices should not give opportunity for setting up an arbitrage.

Futures and forwards are similar instruments, but positions in futures' accounts are adjusted daily, margin calls may be made, and positions are often closed prior to the termination date. However, let us look at how we could use forward contracts to arrive at a fair future price of gold.

Forward Contract Arbitrage Opportunities

Assume that the current price you are analyzing is right in the middle of the current price range observed today: 3,228.0. Consider neither storage nor transaction costs and a quick search lets us know that in the US the one-year treasury bill interest rate is currently 4.048%

With the preceding information, go to the Forward prices sub-menu and, after providing the current price, choosing 'asset', 'compute' and providing the interest rate to maturity you get that the arbitrage free price in a year is 3,358.67 per ounce.

What if you could find someone willing to enter into a forward contract with you for a different future price?

The arbitrage free price is very close to the future open price observed today at CME, so let's consider both the higher price and the lowest.

If you enter the 3,362.7 forward price, you get the result shown in figure 2. You could set up an arbitrage by borrowing money at the risk-free rate, use that money to buy gold keep it for one year and then deliver it and obtain the agreed upon forward price per ounce. With that money, you would have enough to pay back the debt and keep an arbitrage. If you use 1 contract (100 ounces) the arbitrage is 403.1 or 0.12% with respect to the amount you borrowed.

On the other hand, if you enter the 3,339.7 forward price, you get the result shown in figure 3, where you would have to play the opposite role and make an arbitrage of 1,897 USD or 0.59% with respect to the amount you shorted.

Conclusion

Yes, the price fluctuations could result, in theory, in potential arbitrage opportunities but the benefits are small given the current daily level of price fluctuations. Future and forward contracts are not used so much to speculate as options may be, and they play a crucial role in guaranteeing input prices or revenue levels. On the other hand, there are currently one year call options being actively traded on gold with strike prices of 3,700 USD.

Let us know what you think. Until the next post.

Previous Posts:

• April 2nd, 2025: Trading strategies: Bear Spread with Tesla Options.
• March 26th, 2025: Valuing Options, an example with a share of Apple.
• March 19th, 2025: Investment Analysis in Stock Markets
• March 12th, 2025: Cash Flow Transformations in Debt Management
• March 4th, 2025: Term structures of interest rates in financial planning
• Feb 24,2025: How Good is an investment in government bonds in the US and Mexico today?

Trading strategies: Bear Spread with Tesla Options

Using the Payoff function sub menu item

April 2nd, 2025

Introduction

Options can be used to hedge against variations in prices of its underlaying asset, or to speculate on what future prices may be. Among market participants speculators play a very important role by contributing to the liquidity of the various available instruments.

In recent weeks there has been considerable speculation regarding Tesla's shares. Its 52-week high was 488.54 USD and its 52-week low was 138.8 USD, while today's closing price was 282.76 USD

Consider a speculator who believes that the price is going to keep on decreasing, but who also wants to limit her potential losses to 100 USD, in case her intuition turns out to be wrong. Then, she could set up a Bear spread with call options as follows: Buy five call options with strike price 285 and sell five call options with strike price 245.

A Bear spread may be easily computed using finnugget's Payoff function submenu.

Preparing the Analysis

Prior to using the Payoff function submenu, a few values need to be computed. First, use the three-rate menu to estimate the effective interest rate from the current yearly nominal rate of 4.34%. It is readily found to be 0.948% for the period ending on June 20th. Next, use the Value Put or Call submenu and use it to estimate the Standard Deviation of the log of the rate of growth. This is accomplished by loading the historical data for Tesla. The problem here is to determine how far back in time should we go. Using the most recent 3500 days you will obtain one estimate, but if you use 3000 days you'll get another and by using 2500 days yet another. Then, you must decide whether to use the opening price or the daily closing price. The two daily prices will likely generate estimates with small variations. How to solve this problem? In our experience using 3000 days, which is roughly equivalent to 8.2 years, seems about right and easier to remember than the 2922 days needed for 8 years.

Use Stock History in Excel to get the opening prices since January 14, 2017, until April 1st, 2025. Upload that file, to get the standard deviation. The result is 0.64003 Then compute the price of an American Call for TSLA with initial price 282.7 and strike price 245, which is 55.49 USD and if the strike price is 285, the result is 34.07 USD

We can compare the above results with the prices reported by Yahoo Finance for the two corresponding contracts: TSLA250620C00245000 and TSLA250620C00285000 See figures 1 and 2. The standard deviations reported are higher than the one we used, but the price variation throughout the day, in both contracts, was quite significant: from 38.1 to 58.39 for the contract with strike price 245 and from 21.91 to 36.7 for the one with strike price 285. Both of the prices we had computed are contained in those ranges.

The only thing missing is the time in years until expiration. There are 56 trading days (until June 20th because there are two holidays), then divide that number by 252 corresponding to the number of trading days in a year. The result is 0.22222

The Bear Spread

We are now ready to enter all of the data. To setup the Bear spread consider buying 5 Calls with strike price 285 and selling 5 Calls with strike price 245. Enter 282.7 as the current price. The results of the Bear spread are shown in figure 3. Notice that the maximum benefit would be 106.71 USD if the price of Tesla dropped to 245 or lower. The maximum loss occurs at a price of 285 signifying a net loss of 93.28 USD.

Figure 4 shows the price where the net benefit is closest to zero, at a price of 266.3 USD. Any price higher than that value generates a net loss, any lower price, generates a net benefit. No transaction costs were considered in this exercise.

As a final consideration, the Payoff function requires all options to be of type European, but you may use American Calls as well because it may be proved that it is never a good strategy to exercise an American call prior to its expiration date.

Conclusion

Options may help us in setting bounds to potential future prices. It is easy to construct Bull, Bear or Butterfly spreads, or any other spread that may satisfy your particular needs.

Can you think of situations where some spread may provide an interesting solution? Let us know by entering the main menu and sending us an email through the contact form. Until the next post.

Valuing Options, an example with a share of Apple.

Using the Options menu item

March 26th, 2025

Introduction

Options are among the most interesting financial instruments. They are contracts with a buyer and a seller.

The buyer of the option obtains the possibility of guaranteeing either a buying price (Call contracts) or a selling price (Put contracts) of a certain asset, called the underlying asset. The buyer acquires the right to exercise the option, and so a date must be set called the expiration date. If the buyer may exercise the option only on that expiration date the option is called European. If the buyer may exercise the option at any time up to that expiration date, then it is called American.

The seller of the option gets an immediate benefit: the option's price but has to deliver the option and get an agreed upon price for it, in the case of the Call contract, or buy the option at the agreed upon price in the case of the Put contract. The agreed upon price must be set on the contract and it is called the strike price.

Exploring Options

It sounds pretty convoluted, doesn't it? But it gets easier if you just think of options as insurance. The seller of the option is like the insurance company and the buyer is just someone wishing to guarantee a certain selling (Put) or buying price (Call). Puts and calls are the two types of insurances, and they come in two varieties American or European. The thing that you want to be able to buy or sell, known as the underlying asset has to be something that is being traded in a financial market, it could be a commodity, an exchange rate for currencies or a share of a public company.

In the case of a public company, the company itself may be completely unaware that its shares are being used as the underlying asset; however, it is also quite common to find public companies incentivizing key employees by offering them options to buy shares of the European type.

How do you set a price to such a complex instrument? Options have been traded for a long time, but what we could term a fair price has existed only since the last couple of decades of the previous century. The way the price is computed is by ensuring that you could not get a financial arbitrage from a portfolio set up with the option, the underlying asset and a risk free bond. Fortunately, finnugget computes that price for you.

The price you pay for an option will depend on several factors, among the most important ones are: time until the expiration date, the strike price, the current price, the risk-free interest rate and also how much the price of the underlying asset has been changing.

An Example: An Apple Call Option

As an example, let's consider the price of an American Option to buy a share of Apple with expiration date on April 4th. That date is 7 days away from today, the current risk free interest rate in 4.5% per year with daily capitalization, for all other required parameters, we may use the STOCKHISTORY function in Excel and find the daily closing price of Aapl for the last 3000 days (how far back you go is pretty much up to you!). We upload the data into finnugget's Option Value menu and as soon as we input 222 for the current price and 200 as the strike price, we get the results shown in figure 1. The price of an American Call with strike price 220 and current price 222 is today, 5.62 USD.

If you looked for the price of such an option in Yahoo finance, you would get the data shown in figure 2.

In particular you may see that the price of the option has been fluctuating between 5.1 and 7.1 during the day, 1,524 contracts (each representing 100 shares) were bought or sold, and there are 6.66 thousand contracts that have not been settled or closed out. So, that gives you an idea of the importance of the derivatives market. In just one type of contract (AAPL250404C00220000) in one day of trading 77,620 shares of Apple were used as the underlying asset.

Conclusion

Options may help us in setting bounds to potential future prices, they may be very useful in hedging strategies, to create incentives, or as part of a merger and acquisition strategy.

With finnugget you may compute the price of any option for any underlying asset, just prepare the data and input it. We'll take care of the number crunching for you. If you have comments or suggestions, we're looking forward to reading them. Please go to the Contact menu and drop us a line. Until our next posting...

Investment Analysis in Stock Markets

Using the Investment models menu item

March 19th, 2025

Introduction

Public companies have two prices attached to them at all times. The first one is readily available as it is its current market price. The second one is harder to obtain and is known as its intrinsic value. A company's intrinsic value represents its current potential to generate net benefits expressed as net present value and divided by the number of shares outstanding. Public companies have two prices attached to them at all times. The first one is readily available as it is its current market price. The second one is harder to obtain and is known as its intrinsic value. A company's intrinsic value represents its current potential to generate net benefits expressed as net present value and divided by the number of shares outstanding. Clearly computing the second price involves number crunching and it's time consuming.

Anyone with a mild interest in public companies, knows that their prices fluctuate quite a bit. People refer to that fluctuation as volatility and that means there is no certain outcome. Furthermore, you may have heard that every now and then markets crash (1929, 1987, 2007), so there is certainly risk involved in these kinds of investments, but the flip side is that there is also opportunity. A share of Apple in January 2000 was worth 0.94 USD. That share today is worth 212. So, if you had invested 1000 USD in AAPL back then, you would have 212,000 USD on Monday March 18, 2025. That's almost a 24% yearly compound growth! But GE was also considered a solid investment in 2000, and it pretty much ceased to exist as we knew it.

Investing

So, there are at least three lessons here: First you should study the companies that you are planning on investing. Two, diversify. Three: monitor and adjust your investments periodically. Having said that, we can go deeper. Assume you are considering investing in the following five companies: Amazon, Apple, Google, Microsoft and Meta. Because of the recent decrease in their prices, you think it may be a good time to buy some shares, but you don't know how much to invest in each.

Fortunately, you may use Finnugget , but first you have to prepare the input data, using Microsoft Excel's Stockhistory function or a similar one in another spreadsheet software. You could basically obtain, in 15 minutes or less the monthly closing prices and in the bottom row the current stock outstanding (in billions). As shown below in figures 1 and 2.

Middle part of table not shown.

Once you have the data in the format above, you may upload it directly into the Mean Variance menu and you will get the chart shown as figure 3.

Strategy

Clicking on anyone asset you may see its ticker, its monthly estimated expected rate of return and its estimated monthly standard deviation. Figure three shows META. By clicking on the efficient frontier radio button and leaving the is shorting permitted box blank, you get the chart appearing as figure 4.

You may click on any point in the frontier and get the corresponding investments that would generate the estimates for both expected rate of return and standard deviation. In the chart we chose a portfolio with a monthly expected rate of return of 0.023 and a standard deviation of 0.074 Such a portfolio, shown in green, would have the following composition: 8% of the amount you intend to invest should be invested in MSFT, 60% in AAPL, and 32% in META. Of course you could incorporate other companies up to 10 of them. You could just as well analyze companies from other exchanges around the world.

As an example from Indian companies, we analyze a potential portfolio using data for HCL Technologies, Bharti Airtel Ltd, Tata Consultancy Services Ltd, Sun Pharmaceutical Ind. Ltd., Reliance Industries and Infosys Ltd. We obtain the following chart in which we chose the point on the efficient frontier with smallest standard deviation yielding the following portfolio composition: 36% in BHARTIARTL, 40% in SUNPHARMA, 24% in INFY. See figure 5.

Conclusion

Always keep in mind that investments in the stock market carry risk, and that they are meant to be long term investments. They may be sound investments as long as the company is well managed, with outstanding services or products and also that you are able to explain to someone else what is it that the company does and why it appeals to you. If at any point in time you are not comfortable with the way the company is being run, its products or services, or you are no longer able to explain either what it does or why it appeals to you, it may be time to explore other investment alternatives. Hopefully, Finnugget will help you out again.

Cash Flow Transformations in Debt Management

Using the Equivalent Cash Flows menu item

March 12th, 2025

When analyzing a project's attractiveness, it is easier to decide based on a single equivalent cash flow, rather than trying to interpret a set of future cash flows. Such transformations are made possible using the set of appropriate interest rates. The interest rates must span the project's horizon.

The equivalent current cash could very well represent the funding necessary to meet certain future obligations. That would be the present value of the future obligations. In that case, the set of interest rates would be the ones that could be obtained for all future payment dates from the best available instrument, such as a set of riskless or high-quality zero-coupon bonds.

For example, Microsoft reported in June 2024 total debt of almost 45 billion USD. That debt included 4 bonds with the data shown in the table below:

Let's assume yearly coupons are payable in March 15 and that Microsoft considers high quality bonds to offer and acceptable level of risk and of course better interest rates than government securities. The interest rates for high grade bonds and the obligations derived from these bonds until their maturities are shown in the table below:

We may easily input that data, using the required format, into the Equivalent Cash Flows menu item and find that the present value of those obligations currently is 12.48 billion USD. See figure 1.

It is just as easy to transform that cash flow into a currently equivalent one in March 2042, that future value is 32.87 billion USD. See figure 2.

Finally, we could transform those obligations into a set of eleven equal payments of 1.44 billion USD beginning now and ending in ten years, March 2035. See figure 3. Why would it be important to consider different sets of transformations? What do you think are the factors involved? How about getting rid of old bonds paying more sizable coupons? When would that be a relevant issue? What interest rates should a company use in the transformation process if it is not considered a high-quality credit company? Should it use one set of rates for investment and paying purposes and another set of rates for other transformations?

Conclusion

Any set of future cash flows may be easily transformed into an equivalent one through the use of the appropriate interest rates, but What do you think happens as interest rates change? We'll pick up this conversation in a later posting.

Term structures of interest rates in financial planning

Using the Term Structure menu item

March 4th, 2025

Interest rates constitute the most important price, typically expressed as a rate, when considering loans or investments. This price represents the cost of financing or the potential benefit to be derived from investing between any initial and final times, and in financial planning it is used to estimate what those financial costs or benefits could be for future periods.

For example, if you knew today, that you will receive an important amount of money, say 1 million USD in one year, and you do not expect that you will need it for another two years hence, could you today make arrangements that will insure how much you will have at your disposal three years from now? The answer is most definitely Yes!

Background

The Federal Reserve Bank of Saint Louis reports every month, the 'spot' rates for high quality rating bonds in the US. High quality means bonds issued by companies that are currently highly unlikely to default on their obligations. These are companies whose obligations are not entirely risk free, such as are the federal government issued obligations in any given currency. But they are companies for whom paying off their debts should not cause them any difficulty whatsoever.

The high-quality 'spot' rates, for January 2025 are shown in figure 1, along with the corresponding rates for January 2024. A given set of 'spot' rates gives rise to a complete set of future implied interest rates. Here, we will show how to make those future interest rates available to you at the same time as those 'spot' rates. Both the 'spot' and the future interest rates constitute what is known as the term structure of interest rates at any given time for a given level of risk.

Solution

Those future interest rates play a fundamental role when negotiating terms for various transactions to take place in the future. Figure 2 shows the implied future interest rates that the Term Structure menu item computes for you.

To verify that you could set up a contract today that guarantees that you will be receive anyone of the implied future interest rates, take a look at figure 3, which shows how you could guarantee the 4.85% interest rate between years 1 and 3 (i.e. 'future 1,3' from january 2026 till january 2028). You could repeat the exercise with any initial and final times. Of course, we assume that shorting any fractional amount of any bond is possible. Bond A in this example is a zero coupon bond maturing in one year, and bond B is a zero coupon bond maturing in three years. Both bonds with a 100 USD face value.

Investing one million in one year would require you to short 10,000 bonds A sell them today at their price today which is $95.62058 USD each and invest the total of $956,205.8USD buying 10,993.548 bonds B (at 86.9788 USD each). Notice you don't really put any money initially, but you owe 10 thousand bonds A and that means one year from now, in January 2026 you will have to pay $1million. But that is the amount of money you will receive then. Finally, in January 2028 you will get paid $100 for each of the 10,993.548 bonds B you bought. That is $1,099,354.8 USD Meaning the interest you obtained on the $1 million you paid in 2026 is 99,354.8 or 4.85% per year for each of the two years. But that is precisely the future rate in figure 2 between year 1 and 3!

Conclusion

Term structure of interest rates offers the possibility of managing risk by planning what future income or obligations will entail in terms of monetary flows. You can do that as long as there are 'spot' rates available with a horizon encompassing your planning needs.

How Good is an investment in government bonds in the US and Mexico today?

Using the Three rate menu item

February 24, 2025

A key aspect when analyzing investments, is to understand the difference between effective and real interest rates for a given time period. Those two rates are related by the inflation rate. Let's look at the potential benefit of investing in a 'treasury bill' in the US and a 'cete' in Mexico, for one year, in order to look at how those interest rates are related. Both the 'treasury bill' and the 'cete' are zero coupon bonds with a given maturity.

According to the results of the auction reported by the Department of the Treasury of the US, a 182-day Treasury Bill was priced today at 97.8868 USD and will pay 100 on August 28, 2025.
That means that each Bill will pay an effective rate of 2.1588% for one semester (26 weeks out of 52 in a year).

In order to analyze a complete year we need to assume that the bi-annual rates will remain pretty much the same for the next semester. With that assumption, we may use the three rates menu and enter 0.021588 in the effective rate input, and then select the bi-annual radio button in the Choose capitalization frequency table. Enter 2 in the number of periods box, and then using the expected inflation rate of 3% for the next 12 months, from the New York Federal Reserve Bank, we arrive at a Real interest rate of 1.3244%
That is the growth in purchasing power that investing in bi-annual treasury bills over the next twelve months would give you. So, you would have 4.36% more money, but things are expected to be more expensive resulting in just 1.3% of purchasing power improvement. See figure 1.

Of course, you may as well look for a one-year treasury bill, but the most recent auction for 52 weeks maturity bills was held on January 21,2025 and at that time the price from the auction was 95.93028 That meant a 4.24237% yearly effective rate, and with a 3% inflation resulted in a 1.206% real interest rate. See figure 2.

In the case of the Mexican 'cete', the auction of february 20 for a 350 day bond resulted in a price of 9.16999 for 10 MXN to be received in february 5 2026. That means an effective rate of 9.051% for a period two weeks shy of a year. Mexico's Central Bank has estimated the inflation for 2025 at 3.83% If we use that inflation for the next twelve months and take the 50 week Cete as a one year investment, the real interest rate in this case results in 5.47%

However, you must also keep in mind that interest received from investing in either treasury bills or in cetes is taxable. That means a further deduction in your change in purchasing power is applicable.

Conclusion

Investments for a one-year period in US government 'treasury bills' will currently result in an expected improvement of your purchasing power of around 1.2 to 1.3%, probably closer to 1% once IRS deductions have been considered. In the case of Mexico the 5.47% improvement in purchasing power would probably end up being closer to 5% once income tax deductions are taken into account. The higher rate paid by Mexican Cetes also reflects the fact that historically Mexican governments and its financial agent, the Mexican Central Bank have had to incurr in MXN devaluations to be able to meet its obligations. In the US we don't currently see a big incentive for considering investments in treasury bills, and in Mexico it is worth considering as long as the current government manages its public finances responsibly. What do you think?