Jules van Binsbergen, William Diamond, and Peter Van Tassel
One of the most fundamental concepts in finance is the notion of a risk-free rate. This interest rate tells us how much money investors are guaranteed to receive in the future by saving one dollar today. As a result, risk-free rates reflect investors’ preferences for payoffs in the future relative to the present. Yields on U.S. Treasury securities are generally viewed as a standard benchmark for the risk-free rate, but they may also feature a “convenience yield,” reflecting Treasuries’ special, money-like properties. In this post, we estimate a risk-free rate implicit in the prices of S&P 500 index options—called the box rate—to measure investors’ time preference separate from Treasury convenience yields.
Risk-Free Payoffs from Options
Options with a European-style expiration can be used to replicate a risk-free payoff using the put-call parity relationship. By buying a put option and selling a call option of the same strike price and maturity, an investor receives the strike price in exchange for delivering the underlying asset on which the option is written. By then selling a put option and buying a call option for a different strike price with the same maturity, an investor can construct a risk-free payoff equal to the difference in strike prices, as shown in the chart below. In industry jargon, this trade is sometimes called the box spread. The difference between the price of the box spread portfolio today and its payoff at maturity reveals a risk-free rate that we call the box rate—the rate at which investors can borrow or lend in the option market. Historical estimates of the box rate from a recent paper are available here.
The Box Spread Trade: Replicating Risk-Free Payoffs with Options
Source: Authors’ calculation. Notes: The chart plots the combined payoff at maturity (solid blue line) from purchasing a 95-strike call and selling a 95-strike put (squares) while also selling a 105-strike call and buying a 105-strike put (triangles). Regardless of the value of the underlying asset at maturity, the combined payoff is a constant value of $10, equal to the difference in the strike prices. This trade is known as the “box spread.”
Estimating Box Rates
We estimate box rates using S&P 500 index options (SPX options). SPX options are among the most liquid and heavily traded options in the world. They have a European-style expiration and long time-series of available historical data, serving as the basis for the Cboe Volatility Index (VIX Index).
The chart below presents an example of estimating box rates on March 15, 2022, the day before the Federal Open Market Committee (FOMC) began its most recent rate hiking cycle. The top panel estimates the one-year box rate from an ordinary least squares (OLS) regression that exploits put-call parity. The box rate implied by the slope coefficient is 1.59 percent. Since option markets are close to arbitrage-free because of the competitive forces in financial markets, put-call parity holds almost exactly. In this example, the R-squared is .99999992 out to seven nines and the standard error of the box rate estimate is less than .01 percent, or one basis point.
The bottom panel extends the analysis to multiple maturities. The box rates and Treasury yield curve have a similar upward slope. The convenience yield, which is the spread between these curves, ranges from 10 to 30 basis points across different maturities.
Example of Estimating the Box Rate
Sources: OptionMetrics; Federal Reserve Board. Notes: The top panel plots put minus call mid-quote prices for the same strike price and maturity on March 15, 2022, alongside fitted values from an ordinary least squares (OLS) regression. The box rate implied by the slope coefficient from the regression is 1.59 percent for a maturity in 367 calendar days on March 17, 2023. The bottom panel plots the term structure of box rates from index options of different maturities alongside estimates of Treasury rates from a smoothed yield curve found here. All rates are zero-coupon discount rates with continuous compounding. Years-to-maturity is actual calendar days divided by 365. The option data is from OptionMetrics for S&P 500 index options with maturities between one month and five years whose bid quotes are greater than zero.
Box Rates and Treasury Convenience Yields over Time
Analyzing box rates and Treasury yields over time, we find three main results. First, from January 1996 to April 2023, the box rate is 35 basis points above the Treasury rate on average, implying a 35 basis point convenience yield. Second, convenience yields grow dramatically during the financial crisis of 2007-09, reaching a peak of roughly 130 basis points in October 2008 as a one-month moving average. Third, the average term structure of convenience yields is almost flat across maturities out to three years.
The chart below illustrates the first two results. The top panel plots the time series of the one-year box rate and Treasury rate over time. The bottom panel plots the convenience yield, which equals the difference between the two rates. Box rates and Treasury rates closely comove, with the Treasury rate consistently below the box rate throughout the sample. In addition to being consistently positive, the convenience yield also exhibits some time-series variation, spiking most significantly during the 2007-09 financial crisis, while staying at more stable levels between 20 and 40 basis points in recent years.
Box Rate Closely Tracks the Treasury Rate and Implies a Positive Convenience Yield
Sources: OptionMetrics; Federal Reserve Board. Notes: The chart plots the one-year box rate, Treasury rate, and convenience yield estimate from January 1996 through April 2023 as a twenty-one-day moving average across trading days. The box rate is estimated by ordinary least squares (OLS) from put-call parity. To obtain a constant one-year maturity, we linearly interpolate the nearest S&P 500 index option maturities whose put-call parity regressions have an R-squared of at least .99999 out to five nines. Treasury rates are from a smoothed yield curve to obtain a one-year, constant maturity, zero-coupon rate. Both rates are zero-coupon discount rates with continuous compounding. Results are similar using the Theil-Sen estimate of the box rate from the box spread trade.
The chart below illustrates our third result by plotting the average term structure of convenience yields. As we saw in the example above, box rates can be estimated for different index option maturities. We find that the average level of convenience yields is relatively stable and close to 35 basis points across maturities. This means that when the Treasury issues debt at any maturity out to three years, it tends to save around 35 basis points relative to the risk-free rates implicit in the option market.
Convenience Yield Term Structure Relatively Flat out to Three Years, 35 Basis Points on Average
Sources: OptionMetrics; Federal Reserve Board. Notes: The chart plots the nonparametric binned regression of convenience yield onto time-to-maturity using the binsreg package. It reports a 95 percent uniform confidence band and pointwise confidence bands after partitioning time-to-maturity into ten bins. We use a binned regression approach to account for the fact that option maturities are fixed in calendar time and are not constant maturity. The regression includes maturities from six months to three years. The sample period is January 1996 to April 2023.
Interpretation and Applications
Treasuries are generally considered to be one of the most liquid securities in the world, as they can be traded quickly, in large scale, and at low cost. Treasuries are also a common form of collateral that play a unique role in regulatory capital and liquidity constraints. Because options are risky financial derivatives, the box rate is an alternative risk-free rate benchmark that is based on option prices that do not embed a safe asset premium. Note that the recent effort to replace LIBOR with a new benchmark interest rate shows the importance of finding risk-free rate alternatives. The box rate, which is based on market prices, is one candidate that may be considered alongside other robust reference rates such as SOFR (Secured Overnight Financing Rate) to support financial stability.
The convenience yield estimated with the box rate is also a potentially useful barometer for stress in the financial system. It measures how much investors are willing to pay to hold Treasury securities instead of less money-like assets with identical cashflows. In historical data, the convenience yield was largest during the financial crisis of 2007-09. Investors and policymakers who want a real-time measure of the scarcity of safe assets may therefore find convenience yields based on box rates useful.
Economic Magnitude of Treasury Convenience Yield
As a back-of-the-envelope approximation, if we apply the one-year convenience yield estimated with the box rate to the amount of Treasury marketable debt outstanding each year, the Treasury convenience yield has saved taxpayers around $35 billion per year over the past twenty years. Given the increase in debt following the COVID-19 crisis, this amount has increased to $70 billion per year since 2020. These significant taxpayer savings bring into focus the importance of suggested reforms and recent research on the Treasury market, where improving the resiliency and functioning of the Treasury market may help to maintain Treasuries’ convenience yield.
Jules H. van Binsbergen is the Nippon Life Professor in Finance at the Wharton School of the University of Pennsylvania.
William Diamond is an assistant professor of finance at the Wharton School of the University of Pennsylvania.
Peter Van Tassel is a financial research economist in Capital Markets Studies in the Federal Reserve Bank of New York’s Research and Statistics Group.
How to cite this post:
Jules van Binsbergen, William Diamond, and Peter Van Tassel, “Options for Calculating Risk-Free Rates,” Federal Reserve Bank of New York Liberty Street Economics, October 2, 2023, https://libertystreeteconomics.newyorkfed.org/2023/10/options-for-calculating-risk-free-rates/.
Disclaimer The views expressed in this post are those of the author(s) and do not necessarily reflect the position of the Federal Reserve Bank of New York or the Federal Reserve System. Any errors or omissions are the responsibility of the author(s).
In this specific predicament, U.S. officials have to choose a strategy to deliver the aid without the perception of benefiting Hamas, a group the U.S. and Israel both classify as a terrorist organization.
When aiding people in war zones, you can’t just send money, a development strategy called “cash transfers” that has become increasingly popular due to its efficiency. Sending money can boost the supply of locally produced goods and services and help people on the ground pay for what they need most. But injecting cash into an economy so completely cut off from the world would only stoke inflation.
So the aid must consist of goods that have to be brought into Gaza, and services provided by people working as part of an aid mission. Humanitarian aid can include food and water; health, sanitation and hygiene supplies and services; and tents and other materials for shelter and settlement.
Due to the closure of the border with Israel, aid can arrive in Gaza only via the Rafah crossing on the Egyptian border.
The U.S. Agency for International Development, or USAID, will likely turn to its longtime partner on the ground, the United Nations Relief and Works Agency, or UNRWA, to serve as supply depots and distribute goods. That agency, originally founded in 1949 as a temporary measure until a two-state solution could be found, serves in effect as a parallel yet unelected government for Palestinian refugees.
USAID will likely want to tap into UNRWA’s network of 284 schools – many of which are now transformed into humanitarian shelters housing two-thirds of the estimated 1 million people displaced by Israeli airstrikes – and 22 hospitals to expedite distribution.
Since Biden took office, total yearly U.S. assistance for the Palestinian territories has totaled around $150 million, restored from just $8 million in 2020 under the Trump administration. During the Obama administration, however, the U.S. was providing more aid to the territories than it is now, with $1 billion disbursed in the 2013 fiscal year.
The United Nations Relief and Works Agency is a U.N. organization. It’s not run by Hamas, unlike, for instance, the Gaza Ministry of Health. However, Hamas has frequently undermined UNRWA’s efforts and diverted international aid for military purposes.
Humanitarian aid professionals regularly have to contend with these trade-offs when deciding to what extent they can work with governments and local authorities that commit violent acts. They need to do so in exchange for the access required to help civilians under their control.
Similarly, Biden has had to make concessions to Israel while brokering for the freedom to send humanitarian aid to Gaza. For example, he has assured Israel that if any of the aid is diverted by Hamas, the operation will cease.
This promise may have been politically necessary. But if Biden already believes Hamas to be uncaring about civilian welfare, he may not expect the group to refrain from taking what they can.
Security best practices
What can be done to protect the security of humanitarian aid operations that take place in the midst of dangerous conflicts?
Under International Humanitarian Law, local authorities have the primary responsibility for ensuring the delivery of aid – even when they aren’t carrying out that task. To increase the chances that the local authorities will not attack them, aid groups can give “humanitarian notification” and voluntarily alert the local government as to where they will be operating.
Under the current agreement between the U.S., Israel and Egypt, the convoy will raise the U.N. flag. International inspectors will make sure no weapons are on board the vehicles before crossing over from Arish, Egypt, to Rafah, a city located on the Gaza Strip’s border with Egypt.
The aid convoy will likely cross without militarized security. This puts it at some danger of diversion once inside Gaza. But whether the aid convoy is attacked, seized or left alone, the Biden administration will have demonstrated its willingness to attempt a humanitarian relief operation. In this sense, a relatively small first convoy bearing water, medical supplies and food, among other items, serves as a test balloon for a sustained operation to follow soon after.
In that case, the presence of U.S. armed forces might provoke attacks on Gaza-bound aid convoys by Hamas and Islamic jihad fighters that otherwise would not have occurred. Combined with the mobilization of two U.S. Navy carrier groups in the eastern Mediterranean Sea, I’d be concerned that such a move might also stoke regional anger. It would undermine the Biden administration’s attempts to cool the situation.
On U.N.-approved missions, aid delivery may be secured by third-party peacekeepers – meaning, in this case, personnel who are neither Israeli nor Palestinian – with the U.N. Security Council’s blessing. In this case, tragically, it’s unlikely that such a resolution could conceivably pass such a vote, much less quickly enough to make a difference.
Topher L. McDougal does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.
“The majority of wound infections often manifest themselves immediately postoperatively, so close followup should take place […]”
Credit: 2023 Barbarewicz et al.
“The majority of wound infections often manifest themselves immediately postoperatively, so close followup should take place […]”
BUFFALO, NY- October 20, 2023 – A new research perspective was published in Oncoscience (Volume 10) on October 4, 2023, entitled, “Diagnosis and management of postoperative wound infections in the head and neck region.”
In everyday clinical practice at a department for oral and maxillofacial surgery, a large number of surgical procedures in the head and neck region take place under both outpatient and inpatient conditions. The basis of every surgical intervention is the patient’s consent to the respective procedure. Particular attention is drawn to the general and operation-specific risks.
Particularly in the case of soft tissue procedures in the facial region, bleeding, secondary bleeding, scarring and infection of the surgical area are among the most common complications/risks, depending on the respective procedure. In their new perspective, researchers Filip Barbarewicz, Kai-Olaf Henkel and Florian Dudde from Army Hospital Hamburg in Germany discuss the diagnosis and management of postoperative infections in the head and neck region.
“In order to minimize the wound infections/surgical site infections, aseptic operating conditions with maximum sterility are required.”
Furthermore, depending on the extent of the surgical procedure and the patient‘s previous illnesses, peri- and/or postoperative antibiotics should be considered in order to avoid postoperative surgical site infection. Abscesses, cellulitis, phlegmone and (depending on the location of the procedure) empyema are among the most common postoperative infections in the respective surgical area. The main pathogens of these infections are staphylococci, although mixed (germ) patterns are also possible.
“Risk factors for the development of a postoperative surgical site infection include, in particular, increased age, smoking, multiple comorbidities and/or systemic diseases (e.g., diabetes mellitus type II) as well as congenital and/ or acquired immune deficiency [10, 11].”
Continue reading the paper: DOI:https://doi.org/10.18632/oncoscience.589
Correspondence to: Florian Dudde
Keywords: surgical site infection, head and neck surgery
Oncoscience is a peer-reviewed, open-access, traditional journal covering the rapidly growing field of cancer research, especially emergent topics not currently covered by other journals. This journal has a special mission: Freeing oncology from publication cost. It is free for the readers and the authors.
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G77 Nations, China, Push Back On U.S. "Loss And Damage" Climate Fund In Days Leading Up To UN Summit
As was the case in primary school with bringing in presents, make sure you bring enough for the rest of the class, otherwise people get ornery...
This age old rule looks like it could be rearing its head in the days leading up to the UN COP 28 climate summit, set to take place in the United Arab Emirates in about six weeks.
At the prior UN COP 27, which took place in Egypt last year, the U.S. pushed an idea for a new World Bank "loss and damage" climate slush fund to help poor countries with climate change. But the G77 nations plus China, including many developing countries, are pushing back on the idea, according to a new report from the Financial Times.
The goal was to arrange how the fund would operate and where the money would come from for the "particularly vulnerable" nations who would have access to it prior to the upcoming summit in UAE.
But as FT notes, Pedro Luis Pedroso Cuesta, the Cuban chair of the G77 plus China group, has said that talks about these details were instead "deadlocked" over issues of - you guessed it - where the money is going and the governance of the fund.
The U.S.'s proposal for the fund to be governed by the World Bank has been rejected by the G77 after "extensive" discussions, the report says. Cuesta has said that the nations seek to have the fund managed elsewhere, but that the U.S. wasn't open to such arrangements.
Cuesta said: “We have been confronted with an elephant in the room, and that elephant is the US. We have been faced with a very closed position that it is [the World Bank] or nothing.”
Christina Chan, a senior adviser to US climate envoy John Kerry, responded: “We have been working diligently at every turn to address concerns, problem-solve, and find landing zones.” She said the U.S. has been "clear and consistent" in their messaging on the need for the fund.
Cuesta contends that the World Bank, known for lending to less affluent nations, lacks a "climate culture" and often delays decision-making, hindering quick responses to climate emergencies like Pakistan's recent severe flooding.
The G77 coalition voiced concerns about the World Bank's legal framework potentially limiting the fund's ability to accept diverse funding sources like philanthropic donations or to access capital markets.
With just days left before the UN COP 28 summit, the World Bank insists that combating climate change is integral to its mission and vows to collaborate on structuring the fund.