2. Interest Rates

2.1. Introduction

The pricing of financial assets is always relative to some benchmarks that we believe are well priced. In order to compare cash flows occurring at different points in time, we use risk-free bonds as the relevant benchmarks. Unfortunately, real financial markets are rarely fully integrated, so the landscape of interest rates is varied and complex.

In order to start thinking about interest rates, consider a zero-coupon bond that pays for certain $100 in 2 years. If you were to purchase the bond, you would pay less than $100. Indeed, most people would prefer to consume earlier than later, and therefore would like to be compensated for having to wait to receive their money. With this in mind, say that the price of the bond is $95.

This means that every dollar paid for certain in two years today costs $0.95. Thus, a bond that pays for certain $50,000 in two years today should cost \(0.95 \times 50{,}000 = \$47{,}500.\)

What if this new bond trades for a different price, say $47,000? Then there would be an arbitrage opportunity. Note that 500 bonds paying $100 in two years are equivalent to a single bond paying $50,000 in two years. Therefore, it makes sense to buy the bond with face value $50,000 expiring in two years and short-sell 500 bonds with face value $100 expiring at the same date. This long-short strategy would provide today with a positive cash-flow of $500 and in two years requires us to pay $50,000 for the bonds that we sold-short which is exactly offset by the bond we bought.

Selling an asset short involves borrowing the asset first, which of course requires us to repay it later. It might be harder perhaps for a small investor to short-sell zero-coupon bonds but it will not be a problem for a big institutional investor such as a hedge fund. Also, making $500 in one trade does not amount for much, but the idea here is to lever the transaction as much as possible. If your arbitrage capital is $100 million, then you can short-sell 1,000,000 bonds with face value $100 and purchase 2,000 bonds with face value $50,000 expiring in two years. The arbitrage profit gets amplified by 2,000, so you make $1,000,000 in the trade.

Arbitrage opportunities cannot last for long as they generate free money for those that exploit them. The asset that is sold short should decrease in value whereas the price of the asset being bought should increase. Therefore, in the absence of arbitrage opportunities, the price of the bond with face value $50,000 must be $47,500.

The fact that $100 paid in two years costs today $95 can also be expressed as a percentage rate of return per year. Indeed, it must be the case that

\[\begin{equation*} 95 (1 + r)^{2} = 100 \end{equation*}\]

if the annualized rate of return is \(r.\) Therefore,

\[\begin{equation*} r = \left( \frac{100}{95} \right)^{1/2} - 1 = 2.60\%. \end{equation*}\]

In other words, the implicit interest rate paid by the 2-year zero coupon bond is 2.60% per year compounded annually.

As we can see, interest rates are usually derived from bond prices although some interest rates are set directly in loans. The arbitrage opportunity we analyzed previously basically allowed us to borrow money at 2.60% and invest it at a higher rate.

In our example, the 2-year bond is free of the risk of default. We say that the 2-year risk-free rate is 2.60% per year. Risk-free rates for different maturities are fundamental to price derivatives. Therefore it is important have a good proxy for it. In the next section I review several important interest rates found in financial markets, and discuss which one is the best proxy to use as a risk-free rate in the pricing of futures, forwards and options.

2.2. Different Types of Interest Rates

2.2.1. Treasury Rates

The United States government through the Department of the Treasury issues Bills, Notes and Bonds to finance government activities. The demand for Treasury securities is high, since for many investors Treasury bonds represent an investment that is free of the risk of default.

As of January 2022, the total amount of public Treasury debt outstanding is $29.70 trillion. If we compare this figure with the market capitalization od the U.S. stock market which is at $53.37 trillion as of 12/31/2021, this makes the U.S. Treasury debt market a big contender for worldwide investors.

Fig. 2.1 The figure shows the evolution of the public Treasury debt from 1990 until 2023.

Notwithstanding the sheer size of the U.S. Treasury market, the demand for Treasury securities often surpasses its supply, which is capped in size by the United States congress. For this reason, the yield-to-maturity (YTM) of Treasury bonds is usually thought to be lower than the rate of a fully collateralized loan. Therefore, Treasury rates are commonly not used as benchmark rates to price derivative securities.

2.2.2. LIBOR

LIBOR rates have been at the heart of the financial system for many decades. The acronym stands for London Interbank Offered Rate. For all this time, LIBOR has provided a reference for pricing derivatives, loans and securities.

From the beginning, LIBOR was meant to proxy for unsecured short-term borrowing between banks, with borrowing periods ranging from one day to one year. Also, LIBOR rates used to be quoted for different currencies.

Large corporate loans used to be indexed to LIBOR. A company that would borrow money from a bank would then reimburse the loan by making quarterly payments indexed to the applicable three-month LIBOR. Many corporate borrowers, however, would like to to pay a fixed interest instead. This was made possible by combining the loan with an interest rate swap. Thus, one of the most important derivatives that used LIBOR as a reference rate were interest rate swaps.

Fig. 2.2 The figure shows the evolution of 3M LIBOR from January 1986 until May 2023.

In response to recommendations and objectives set forth by the Financial Stability Board and the Financial Stability Oversight Council to address risks related to USD LIBOR [Official Sector Steering Group, 2014], the Federal Reserve Board and the New York Fed jointly convened in 2014 the Alternative Reference Rates Committee (ARRC), which is a group of private-market participants whose role is to ensure a successful transition from U.S. dollar (USD) LIBOR to a more robust reference rate. The recommended alternative was the Secured Overnight Financing Rate (SOFR). A good summary and analysis of the reform is presented in Duffie and Stein [2015].

There were two important reasons to abandon USD LIBOR in favor for SOFR. First, there were several cases of attempted market manipulation and false reporting of LIBOR that came to light in 2012. Second, there has been a decline in the liquidity of interbank unsecured funding post 2009. Therefore, the information content coming from the unsecured financing market might not be fully representative of the level of risk-free rates.

Even though USD LIBOR is still reported for some tenors, its use has stopped and has been officially replaced by SOFR.

2.2.3. OIS and Overnight Rates

In the United States, banks are required to maintain reserves in cash with the Federal Reserve. When a bank needs to increase their reserve, they usually borrow overnight from another bank that might have a reserve surplus. The weighted-average rate of these brokered transactions is termed the effective federal funds rate (EFFR).

Fig. 2.3 Evolution of the EFFR from January 1960 until April 2023.

We can observe from the figure that there have been periods of time in which the federal funds rate has been very high, reaching almost 20% per year. Since 2009, the federal funds rate has been very low and in many cases close to zero.

When the Federal Reserve determines the target federal funds rate, they implement their policy by making sure that the EFFR is close every day to their target. Since 2009, this is done by making sure that the EFFR lies between lower and upper limits, as the next figure shows.

Fig. 2.4 The Federal Reserve makes sure that the EFFR stays within the target range.

The COVID-19 pandemic has pushed these limits back to where they were after the financial crisis of 2008. The recent inflation surge, though, has surprised markets and made the Federal Reserve to increase the reference interest rate.

Traditionally, practitioners have used LIBOR and LIBOR-swap rates as proxies for risk-free rates when valuing derivatives. The financial crisis of 2007 put this practice into question. Banks became reluctant to lend to each other and the TED spread, the difference between 3-month LIBOR and the 3-month Treasury rate, surged significantly above its normal level. It was clear that LIBOR was capturing the credit risk of the underlying banks and therefore could not be used to proxy for the risk-free rate.

The alternative chosen by practitioners was the overnight indexed swap rate. An overnight indexed swap (OIS) is an over-the-counter financial contract in which one party pays the compounded EFFR over a certain period, say three months, in exchange for a fixed payment. For many years since the financial crisis of 2008 the OIS rate has been as a proxy for the risk-free rate [Hull and White, 2013]. Recently, the ARRC has put this practice into question citing the relative illiquidity of the OIS market. The recommendation of the committee has been to switch to SOFR.

2.2.4. SOFR and Repo Rates

A repurchase agreement or repo, a financial institution or trader sells some securities to a counterparty with the agreement to repurchase them back later for a slightly higher price. The implicit interest rate in this transaction is the repo rate. Unlike LIBOR and the EFFR, repo rates are secured borrowing rates.

The weighted average of these repo transactions is called the secured overnight financing rate (SOFR). Effective 2022, this rate has replaced LIBOR USD.

2.3. Compounding Frequencies

The notion of future value (FV) is fundamental to finance. It allows us to compute how an investment grows over time when you apply a certain rate of interest to it. In these notes and unless otherwise stated, we will express all interest rates as a percentage per year. It makes a difference, though, how often you compound the interest rate.

A very common way to express the interest rate is with annual compounding.

Example 2.1

If the interest rate is 10% per year compounded annually, after a year, $100 invested at this rate will grow to

\[\begin{equation*} \text{FV} = 100 \times 1.10 = \$110.00. \end{equation*}\]

In words, the future value of $100 after a year is $110 when the interest rate is 10% per year with annual compounding.

As a consequence, a rate of 10% per year with annual compounding corresponds to the effective annual rate (EAR) of the investment.

However, there are other ways to express the interest rate that are also used in practice. For example, consider the fixed-income market which in the United States is even larger than the stock market in terms of market capitalization. Bonds in the United States pay coupons every six months. As a consequence, it is very common to express the yield-to-maturity (YTM) of Treasury or corporate bonds with semi-annual compounding. In this case we divide the interest rate by two and compound it twice per year.

Example 2.2

If the interest rate is 10% per year compounded semi-annually, $100 will grow to

\[\begin{equation*} \text{FV} = 100 \times 1.05^{2} = \$110.25. \end{equation*}\]

If that interest rate is 10% per year compounded monthly, $100 will grow to

\[\begin{equation*} \text{FV} = 100 \left(1 + \frac{0.10}{12} \right)^{12} = \$110.47. \end{equation*}\]

If the same interest rate is compounded daily, $100 will grow to

\[\begin{equation*} \text{FV} = 100 \left( 1 + \frac{0.10}{365} \right)^{365} = \$110.52. \end{equation*}\]

Example 2.2 suggests the following rule for the EAR of an investment that uses a specific compounding rule.

Property 2.1 (Effective Annual Rate)

An interest rate \(r\) that is compounded \(n\) times per year will generate an effective annual rate equal to:

\[\begin{equation*} \text{EAR} = \left(1 + \frac{r}{n}\right)^{n} - 1 \end{equation*}\]

As we can see form Property 2.1, the EAR increases as we compound more often.

2.4. Continuous Compounding

When pricing futures, forwards, options, and other derivatives, it is useful to use continuous-compounding to discount cash flows in order to keep the same convention when pricing derivatives in continuous-time. In this section we will learn how to use continuously-compounded rates to compute present and future values, and how to convert between continuously-compounded rates and other conventions used in practice such as annual or semi-annual compounding.

Indeed, there is a limit to the compounding operation in Example 2.2 (see also Example 20.7):

\[\begin{equation*} \lim_{n\rightarrow\infty} 100\left(1+\frac{0.10}{n}\right)^{n} = 100e^{0.10} = \$110.52. \end{equation*}\]

We call this operation continuous compounding, and you can see that compounding daily is already a pretty good approximation of it. ^{1 1}In a spreadsheet, to compute \(100 e^{0.10}\) you need to type =100*exp(0.10).

More generally, an amount \(PV\) invested today in a risk-free deposit account earning an annual interest rate \(r\) compounded continuously will grow to \(FV = PV e^{r t}\) at time \(t.\)

We say that \(FV\) is the future value of \(PV\) at time \(t.\) Equivalently, we say that \(PV = FV e^{-r t}\) is the present value of \(FV.\)

Property 2.2 (Continuously-Compounded Interest Rate)

If we denote by \(r\) the continuously-compounded interest rate, the relationship between present value \((PV)\) and future value \((FV)\) is given by:

\[\begin{equation*} FV = PV e^{r T} \Longleftrightarrow PV = FV e^{-r T}. \end{equation*}\]

Example 2.3

If you invest $500 at 6% per year with continuous compounding, in 16 months the balance of your account will be:

\[\begin{equation*} FV = 500 e^{0.06 \times 16/12} = \$541.64. \end{equation*}\]

To perform the previous computation using a spreadsheet software you need to type =500*exp(0.06*16/12).

Example 2.4

You want to know how much you need to have in a bank account today so that in 18 months from now your balance is $2,000. The interest rate is 7% per year with continuous compounding.

The required amount is the present value of $2,000 for 18 months at 7%:

\[\begin{equation*} PV = 2000 e^{-0.07 \times 18/12} = \$1{,}800.65. \end{equation*}\]

To perform the previous computation using a spreadsheet software you need to type =2000*exp(-0.07*18/12).

Example 2.5

Say that you invest $100 in a deposit account and after a year your balance is $110. What equivalent continuously compounded interest rate would give you the same amount?

If we denote by \(r\) the continuously compounded interest rate, it must be the case that:

\[\begin{equation*} 110 = 100e^{r} \Rightarrow r = \ln\left(\frac{110}{100}\right) = 9.53\%. \end{equation*}\]

To perform the previous computation using a spreadsheet software you need to type =ln(110/100).

Therefore 10% per year compounded annually is the same as 9.53% per year compounded continuously.

2.5. Zero Rates

A zero-coupon bond pays its principal or face-value \(F\) at maturity but makes no intermediate payments.

Even though zero-coupon bonds do not trade actively in financial markets, it is possible to synthesize a zero-coupon bond from a portfolio of two coupon bonds.

The price of a zero-coupon bond is determined by discounting its face-value at the relevant interest rate. For a given maturity \(T,\) the \(T\)-year zero rate, denoted by \(r_{T},\) is the interest rate that gives the right \(T\)-year zero-coupon bond price \(Z_{T}\). That is, if \(r_{T}\) is a continuously-compounded rate we must have:

\[\begin{equation*} Z_{T} = F e^{-r_{T} T}. \end{equation*}\]

Example 2.6

You have the following information for zero rates expressed per year with continuous compounding:

Maturity (months)	1	3	6	9	12
Zero Rate (%)	6.0	6.4	6.6	6.8	7.0

Consider a zero-coupon risk-free bond with face value $1,000 and expiring in 9 months. To compute the price of the bond, we use the 9-month interest rate which is 6.8%:

\[\begin{equation*} Z = 1000e^{-0.068 \times 9/12} = \$950.28. \end{equation*}\]

Example 2.7

A 7-year zero-coupon risk-free bond with face value $1,000 trades for $650. The 7-year zero rate \(r\) expressed with continuous compounding satisfies:

\[\begin{align*} 1000 e^{-r \times 7} & = 650 \\ e^{-r \times 7} & = \frac{650}{1000} \\ -r \times 7 & = \ln\left(\frac{650}{1000}\right) \\ r & = -\frac{1}{7} \ln\left(\frac{650}{1000}\right) \end{align*}\]

The 7-year zero rate is then 6.15% per year with continuous compounding.

2.6. References

[DS15]

Darrell Duffie and Jeremy C. Stein. Reforming libor and other financial market benchmarks. Journal of Economic Perspectives, 29(2):191–212, 2015.

[HW13]

John C Hull and Alan White. Libor vs. ois: the derivatives discounting dilemma. Journal of Investment Management, 2013.

[OfficialSSGroup14]

Official Sector Steering Group. Reforming major interest rate benchmarks. Financial Stability Board, 2014.

2.7. Practice Problems

Exercise 2.1

You invest $5,000 at 15% per year with continuous compounding. How much will you have in the account after 14 years?

Solution to Exercise 2.1

The balance in the account will be \(5{,}000 e^{0.15 \times 14} = \$40{,}830.85\) after 14 years.

Exercise 2.2

Consider the following cash flows occurring at the times indicated below:

Time (years)	7	14	21
Cash flow	149	240	184

Compute the present value if the discount rate is 9% per year with continuous compounding.

Solution to Exercise 2.2

The present value of these cash flows is:

\[\begin{equation*} \textit{PV} = 149 e^{-0.09 \times 7} + 240 e^{-0.09 \times 14} + 184 e^{-0.09 \times 21} = 175.23. \end{equation*}\]

Exercise 2.3

The interest rate is 8% per year continuously-compounded and assumed to remain constant. Compute the present value of the following cash flows.

Time (years)	2	5	14	19
Cash Flow	100	200	300	150

Solution to Exercise 2.3

\[\begin{equation*} \textit{PV} = 100 e^{-0.08 \times 2} + 200 e^{-0.08 \times 5} + 300 e^{-0.08 \times 14} + 150 e^{-0.08 \times 19} = \$349.97 \end{equation*}\]

Exercise 2.4

An investor receives $1,100 in one year in return for an investment of $1,000 now. Calculate the percentage return per year with:

1. Annual compounding

2. Semiannual compounding

3. Monthly compounding

4. Continuous compounding

Solution to Exercise 2.4

1. Annual compounding:

   \[\begin{equation*} 1,100 = 1,000 (1 + r) \Rightarrow r = \frac{1,100}{1,000} - 1 = 10\% \end{equation*}\]

2. Semiannual compounding:

   \[\begin{equation*} 1,100 = 1,000 \left(1 + \frac{r}{2}\right)^{2} \Rightarrow r = 2 \left(\left(\frac{1,100}{1,000}\right)^{\frac{1}{2}} - 1\right) = 9.76\% \end{equation*}\]

3. Monthly compounding:

   \[\begin{equation*} 1,100 = 1,000 \left(1 + \frac{r}{12}\right)^{12} \Rightarrow r = 12 \left(\left(\frac{1,100}{1,000}\right)^{\frac{1}{12}} - 1\right) = 9.57\% \end{equation*}\]

4. Continuous compounding:

   \[\begin{equation*} 1,100 = 1,000 e^{r} \Rightarrow r = \ln\left(\frac{1,100}{1,000}\right) = 9.53\% \end{equation*}\]

Exercise 2.5

An effective annual rate (EAR) of 11% per year is equivalent to which rate expressed per year with continuous compounding?

Solution to Exercise 2.5

\[\begin{equation*} 1.11 = e^{r} \Rightarrow r = \ln(1.11) = 10.44\%. \end{equation*}\]

Exercise 2.6

You are considering an investment of $100 today that should grow to $167 in 8 years. What rate of return, expressed per year with continuous compounding, is consistent with this information?

Solution to Exercise 2.6

\[\begin{equation*} 100 e^{r \times 8} = 167 \Rightarrow r = \frac{1}{8} \ln\left(\frac{167}{100}\right) = 6.41\% \end{equation*}\]

Exercise 2.7

A bank quotes an interest rate of 14% per year with quarterly compounding. Compute an equivalent rate with:

1. Continuous compounding

2. Annual compounding

Solution to Exercise 2.7

1. Continuous compounding:

   \[\begin{equation*} \left(1 + \frac{0.14}{4}\right)^{4} = e^{r} \Rightarrow r = 4 \ln\left(1 + \frac{0.14}{4}\right) = 13.76\% \end{equation*}\]

2. Annual compounding:

   \[\begin{equation*} \left(1 + \frac{0.14}{4}\right)^{4} = 1 + r \Rightarrow r = \left(1 + \frac{0.14}{4}\right)^{4} - 1 = 14.75\% \end{equation*}\]

Exercise 2.8

A deposit account pays 12% per year with continuous compounding, but interest is actually paid quarterly. How much interest will be paid each quarter on a $10,000 deposit?

Solution to Exercise 2.8

The interest paid quarterly should be \(10,000 e^{0.12 \times 0.25} - 10,000 = \$304.55.\)

Exercise 2.9

Suppose that 6-month, 12-month, 18-month, 24-month, and 30-month zero rates are, respectively, 4.0%, 4.2%, 4.4%, 4.6%, and 4.8% per year, with continuous compounding. Estimate the cash price of a bond with a face value of $100 that will mature in 30 months and pays a coupon of 4% per year semiannually.

Solution to Exercise 2.9

\[\begin{equation*} B = 2 e^{-0.040 \times 0.5} + 2e^{-0.042 \times 1.0} + 2 e^{-0.044 \times 1.5} + 2 e^{-0.046 \times 2.0} + 102 e^{-0.048 \times 2.5} = \$98.04. \end{equation*}\]

Exercise 2.10

You have information of cash flows and zero-coupon rates (per year with continuous compounding) for different maturities as shown below:

Time (years)	4	10	17
Zero-coupon rate (%)	5.8	6.9	7.3
Cash flow	387	473	276

Compute the present value of those cash flows.

Solution to Exercise 2.10

The present value of these cash flows is:

\[\begin{equation*} \textit{PV} = 387 e^{-0.058 \times 4} + 473 e^{-0.069 \times 10} + 276 e^{-0.073 \times 17} = 623.91. \end{equation*}\]

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