Unraveling the Cost of Carry Model: A Generalization for Forward and Futures Contracts

In the world of finance, pricing forward and futures contracts can be a complex task. The cost of carry model is widely used, but its various forms have led to confusion among market participants. In this article, we'll delve into the TN05-03 document, which aims to provide a generalized formulation of the cost of carry model.

The cost of carry model is essential for pricing forward and futures contracts because it takes into account the underlying asset's cash flows and discounting methods. The model can be applied to various types of assets, including stocks, currencies, commodities, and fixed-income instruments.

However, the widespread use of different variations has led to a proliferation of models. This article aims to clarify the core concepts and provide a deeper understanding of how the cost of carry model works.

Understanding Time Value and Discounting

To grasp the cost of carry model, it's crucial to comprehend time value and discounting. The concept of time preference is central to this discussion. Time preference reflects the cost of waiting until a later date to receive money. In other words, it captures the discount that people apply to future values to obtain their present value.

Let Bt(T) be the value at time t of $1 or any currency considered to be the home currency of the person trading the contract at time T. This value is determined by supply and demand in the short-term risk-free interest rate market. In other words, it represents the price of a claim on $1 at a future date.

An interest rate, on the other hand, is simply a mathematical relationship that maps a present value to a future value or vice versa. There are an infinite number of interest rate specifications that can convert the value Bt(T) to $1 at T or vice versa.

Discrete Interest with Annual Compounding

One common interest rate specification is discrete interest with annual compounding. Let rt(T) be the interest rate observed at time t for discounting $1 at time T in the following manner:

( )/365 1 ( ) . (1 ( t T t t B T r T − = +

Here, T – t is the number of days over the period, and (T – t)/365 is the number of years over the period. It's worth noting that compounding can occur more frequently than once a year.

LIBOR Interest

The Eurodollar market assumes 360 days in a year and computes interest using add-on interest. Let ιt(T) be the LIBOR rate on day t for discounting $1 on day T. $1 invested for T days at the rate ιt(T) would grow to a value of:

1 + ιt(T)[(T-t)/360]

This represents the present value of $1 at T.

Discount Interest

The U.S. Treasury Bill market and a few other markets use discount interest, which deducts interest from the principal in advance of the loan, thereby raising the effective rate. Let rd tT) be the discount rate:

Letting rd tT) be the discount rate:

Bt(T) = (1 + rd[T-t])[-(T-t)/360]

The discount rate is simply a mathematical relationship that maps a present value to a future value or vice versa.

Portfolio Implications

Now that we have a deeper understanding of the cost of carry model, let's discuss its implications for portfolio management. The model can be applied to various asset classes, including stocks (e.g., C), currencies (e.g., MS), commodities (e.g., QUAL), and fixed-income instruments (e.g., TIP, EEM).

Investors should consider the following:

Stocks: The cost of carry model is essential for pricing forward contracts on stocks. Investors should take into account the underlying asset's cash flows and discounting methods. Currencies: The LIBOR interest rate specification is commonly used in currency markets. Investors should be aware of the add-on interest method used by the Eurodollar market. * Commodities: The cost of carry model can be applied to various commodities, including oil, gold, and agricultural products. Investors should consider the underlying asset's cash flows and discounting methods.

Practical Implementation

So, how can investors apply this knowledge in practice? Here are some key takeaways:

Timing considerations: Investors should carefully consider the timing of their trades and the expiration dates of contracts. Entry/exit strategies: Investors should develop specific entry and exit strategies based on their risk tolerance and investment goals.

Actionable Conclusion

In conclusion, the cost of carry model is a fundamental concept in finance that requires a deep understanding of time value and discounting. By applying this knowledge to various asset classes, investors can make more informed decisions about forward and futures contracts.

Investors should take into account the underlying asset's cash flows and discounting methods when pricing these contracts. The cost of carry model is essential for portfolio management, and its practical implementation requires careful consideration of timing and entry/exit strategies.