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Multiple risk-free interest rates

Financial markets produce more than one risk-free interest rate. This is because there are several separate market segments where structured trades replicate such a rate. Differences in remuneration arise for two reasons. First, financial frictions can prevent arbitrage. Second, some risk-free assets pay additional convenient yields, typically by virtue of their liquidity and suitability as collateral. Put simply some “safe assets” have value beyond return. U.S. government bonds, in particular, seem to provide a sizable consistent convenience yield that tends to soar in crisis. This suggests that there are arbitrage opportunities for investors that are flexible, impervious to convenience yields and tolerant towards temporary mark-to-market losses.

van Binsbergen Jules, William Diamond Marco Grotteria (2018), “Risk Free Interest Rates”, September 1, 2018

The post ties in with SRSV’s lecture on implicit subsidies.
The below are excerpts from the paper. Emphasis and cursive text have been added. Some formulas have been expressed as text.

Why there is more than one risk-free interest rate

“We provide evidence that risk-free interest rates can vary substantially across different asset markets, in contrast to the unique rate implied by the neo-classical asset pricing literature…Arbitrage spreads are due to constraints on the trading of financial intermediaries.”

“In frictionless asset pricing models, [the risk-free rate] is determined by the investors’ time preference. However, recent literature has questioned whether the time preference of money is the only determinant…providing evidence that the scarcity of safe assets drives up their price and lowers the corresponding interest rate.”

“The time preference of investors can only be inferred by measuring the risk-free rate implied by the prices of risky assets, where the spread between this implied risk-free rate and the observed return on safe assets measures the convenience yield that these safe assets provide… The risk-free rate implied by the pricing of risky assets lies strictly above the rate earned on safe assets, where the difference is often interpreted as a measure of the severity of financial frictions.”

“We use a large panel of risky assets to estimate a convenience-yield and essentially credit-risk-free measure of risk free interest rates…We focus on a large cross-section of assets that all provide a risk free payoff, which by their very nature do not require a compensation for risk.”

Different risk-free rates in the fixed income market

“We consider four distinct categories of arbitrage spreads [difference in risk free rates] using government bond data.

  • We consider the spread between [U.S. government] notes/bonds that mature within the next 6 months and yields on treasury bills that mature on the exact same date. Treasury bills are more liquid and therefore tend to have lower yields
  • We consider the spread between two types of STRIPS (Separate Trading of Registered Interest and Principal of Securities) constructed respectively from interest and principal payments on U.S. government debt. These securities pay identical cash flows and are backed by the full faith of the U.S. government, so any difference between the yields on coupon versus principal STRIPS…Whichever of the principal or interest STRIP has a higher supply outstanding tends to have a lower yield
  • We use the [fitted] yield curve to compute an implied yield for the most recently issued bond of each maturity, called the on-the-run bond, and take its difference from the true yield on that bond. On-the-run bonds tend to be more liquid than off-the-run bonds and therefore trade at a lower yield…Spreads between repo rates makes it difficult for a levered investor to profit from the spread between on- and off-the-run bonds…
  • We also use the [fitted] yield curve to consider the relative spread between treasury notes (which by definition have a maturity less than 10 years after issuance) and bonds (which mature more than 10 years after issuance), that have less than 10 years of maturity left…We then take the median of this spread across all notes, and the median of the spread across all bonds on each day and compute a daily difference between these two medians…This spread is small in normal times but spikes during the financial crisis.”

“Several patterns appear across all of these arbitrage spreads. First, both their level and volatility generally increase during the financial crisis period of late 2008 and early 2009. Second, most spreads are smallest in the later part of our sample, suggesting that government bond markets are now even more integrated than they were before the crisis.”

Risk-free rates in the equity options market

“We infer the risk-free rate from a number of option markets… Option-market-implied risk-free rates provide a convenience-yield-free and effectively credit-risk-free measure of time preference measured accurately at a minutely frequency.”

“The starting point of our analysis here is the put-call parity relationship for European options…buying the put of the high strike and the call of the low strike while writing the call of the high strike and the put of the low strike. If one holds positions till maturity, then the payoff is risk free and equal to the difference between high and low strike…This trading strategy earns exactly the risk-free rate.”

“Before 2008 the S&P500 options-implied yields are above the corresponding government bond yield, and closely follow LIBOR. Between 2008 and 2017 a substantial deviation from LIBOR occurs and the S&P500 options-implied yields are in between the LIBOR rate and the government bond yield. This suggests that between 2009 and 2017 banks faced substantial credit risk, as measured by the spread between LIBOR and the S&P500 options-implied zero-coupon yield…both spreads exhibit large variation, and they both go up during the crisis and have since been reduced to levels closer to zero.”

Risk free rates in the FX market

“We test the no-arbitrage hypothesis in the FX markets. To construct a risk-free strategy for this market, we use the well-known covered interest parity relationship. We compare for a US-based agent two alternative strategies. The first alternative is to invest in a riskless asset denominated in dollars. The other is to exchange money into foreign currency, invest into the riskless asset denominated in that foreign currency (with the same time to maturity) and buy a promise to exchange the money back into dollars at a predetermined rate at maturity. Covered Interest Parity (CIP) is a no-arbitrage relationship that states that both strategies should earn the same return.”

We construct the cross-currency basis…as the difference between domestic and foreign risk-free rate minus the annualized log ratio of spot to forward exchange rate…[with the latter representing] the continuously-compounded forward premium.

On average the cross-currency basis with respect to the US dollar [has been] close to zero [2010-2018]…Cross-country heterogeneity is visible. Canadian dollar, Euro, and British pound have an average basis deviation between 0.7 and 2.1 basis points. Their volatility is also very low. On the other hand, Japanese yen, Swiss Franc, and New Zealand dollar have witnessed a cross-currency basis which is both larger on average (from 10 to 19 basis points) and more volatile.

Risk free rates in the commodities market

To construct a risk free asset in commodity markets we use the no-arbitrage relationship between the futures price and the spot price…To derive estimates of the risk free interest rate, we focus on futures contracts on underlying assets that are very cheap to store relative to their underlying value [implying that the difference between futures and spot prices approximately represents a risk free rate]…As such we focus on precious metals: gold, silver and platinum.

We summarize the key statistics for gold and silver implied interest rates…[and] compare these rates to the government bond yields…The estimated convenience yield for government bonds relative to metal-implied interest rates is the same as for our previous estimates and equal to about 40 basis points for gold [and] silver.


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