Page **1** of **2**

### risk-free rates and effects on quant finance

Posted: **January 19th, 2011, 6:37 pm**

by **armchairquant**

Another newbie question I have is on the risk-free rates and their effects on finance.Everything I've been reading uses the US Treasuries as the risk-free rates because of their nature of practically never defaulting (since they can always print more money).1) This is a purely hypothetical question that borders on science fiction, but I've been curious and had no one to ask. If we ever got into a situation where the US treasuries weren't considered risk-free and there turned out to be a discernible risk of default, what sort of ripple effect would there be in finance? (I realize that the situation is extremely unlikely, the only way I could see this happening is if politicians created a law that prevented the Fed from printing more money to pay off debt, or maybe if we moved back to the gold standard again a la Ron Paul.) Regardless, if that happened and the UST no longer were risk-free, I imagine this would complicate how things were calculated... what would happen? Would the prices of financial instruments change drastically overnight because the calculation of risk-free rates have to change? Or would the US use another government's bonds as a risk-free rate?2) Do other countries calculate their risk-free rates using the US treasuries, or do they use their own government bonds as their measure of risk-free rate?

### risk-free rates and effects on quant finance

Posted: **January 19th, 2011, 6:48 pm**

by **Traden4Alpha**

QuoteOriginally posted by: armchairquantEverything I've been reading uses the US Treasuries as the risk-free rates because of their nature of practically never defaulting (since they can always print more money).Printing more money, and the inflation that it brings, is a type of default. If someone promises to give you 100 "Dollars" in 30 years and those 100 "Dollars" don't buy as much wheat, copper, labor, rent, etc. when the 30 years are up, is that any different than if the borrower defaulted but paid you in "Dollars" that still bought the same amounts of wheat, copper, labor, rent, etc. as you expected?In practice, no one thinks of the UST as strictly risk-free but also factors in CDS prices on UST (which mostly corrects for the chance of default) and UST-TIPs spread (which mostly corrects for the expected level of inflation)

### risk-free rates and effects on quant finance

Posted: **January 19th, 2011, 7:52 pm**

by **TinMan**

There is no point in considering the pricing consequences of the US defaulting, since if that happens the financial system collapses anyway.It's like asking what a giant asteroid wiping out the earth would do to bond yields, it's irrelevant.The idea of buying protection on the US seems ridiculous to me too.If they default, who are you going to collect from when all the banks are bust?The US is default risk free in the context of the system we're in.Besides which, the 'risk free rate' is irrelevant in valuing derivatives.

### risk-free rates and effects on quant finance

Posted: **January 19th, 2011, 8:25 pm**

by **Traden4Alpha**

QuoteOriginally posted by: TinManThere is no point in considering the pricing consequences of the US defaulting, since if that happens the financial system collapses anyway.It's like asking what a giant asteroid wiping out the earth would do to bond yields, it's irrelevant.The idea of buying protection on the US seems ridiculous to me too.If they default, who are you going to collect from when all the banks are bust?That seems a bit extreme. Is the probability of collapse conditional on U.S default really exactly 100.00000%? Or is it 99% or maybe 95%? And what about in 5 years time if the BRIC++ countries are 50% larger and more of the world's banks have diversified away from the USD?

### risk-free rates and effects on quant finance

Posted: **January 19th, 2011, 8:29 pm**

by **Hansi**

QuoteOriginally posted by: TinManIt's like asking what a giant asteroid wiping out the earth would do to bond yields, it's irrelevant.Some people don't even understand that. I remember my old boss telling me about one road show Q&A session where a fairly senior investor for a large fund was really mad about an asteroid hitting the earth being marked as an unmodelled risk in an excess mortality bond OC.

### risk-free rates and effects on quant finance

Posted: **January 19th, 2011, 8:37 pm**

by **Martinghoul**

The risk-free rate is a convenient simplification, no more. In reality, things are a lot more complicated and there's hardly ever any use for the concept. So I am in agreement with T4A and TinMan.As to US defaulting, that is a quaint notion. For me, one of the bigger ironies of the mkt is that the conventional US sov CDS contract is denominated in EUR, if I am not mistaken.

### risk-free rates and effects on quant finance

Posted: **January 19th, 2011, 8:49 pm**

by **acastaldo**

In the 1970s Fischer Black did a lot of thinking on how portfolio theory would change if there is no risk free asset (partly motivated by the high inflation in those years).He concluded that if there is no risk free asset, then there is a combination of securities which plays a similar role. He called it the zero beta portfolio. In the ordinary pfolio theory everyone chooses a combination of the market portfolio and the risk free asset, in Black's theory everyone chooses a combination of the market portfolio and the zero beta portfolio.Black had one of his students test empirically how a minimum risk portfolio could be constructed. Various ingenious ideas were tested including long/short portfolios, commodities, combinations of govt bonds of different maturities etc. But IIRC none of these gave a better (lower volatility) solution than the straightforward approach of using T bills.An interesting academic exercise but not much practical payoff.

### risk-free rates and effects on quant finance

Posted: **January 19th, 2011, 9:00 pm**

by **Traden4Alpha**

QuoteOriginally posted by: acastaldoIn the 1970s Fischer Black did a lot of thinking on how portfolio theory would change if there is no risk free asset (partly motivated by the high inflation in those years).He concluded that if there is no risk free asset, then there is a combination of securities which plays a similar role. He called it the zero beta portfolio. In the ordinary pfolio theory everyone chooses a combination of the market portfolio and the risk free asset, in Black's theory everyone chooses a combination of the market portfolio and the zero beta portfolio.Black had one of his students test empirically how a minimum risk portfolio could be constructed. Various ingenious ideas were tested including long/short portfolios, commodities, combinations of govt bonds of different maturities etc. But IIRC none of these gave a better (lower volatility) solution than the straightforward approach of using T bills.An interesting academic exercise but not much practical payoff.Very interesting!And yet, with no clear objective method to synthesize a true risk-free asset, any supposed "risk-free" asset will, in fact, have risks.

### risk-free rates and effects on quant finance

Posted: **January 19th, 2011, 9:03 pm**

by **TinMan**

QuoteOriginally posted by: acastaldoIn the 1970s Fischer Black did a lot of thinking on how portfolio theory would change if there is no risk free asset (partly motivated by the high inflation in those years).He concluded that if there is no risk free asset, then there is a combination of securities which plays a similar role. He called it the zero beta portfolio. In the ordinary pfolio theory everyone chooses a combination of the market portfolio and the risk free asset, in Black's theory everyone chooses a combination of the market portfolio and the zero beta portfolio.Black had one of his students test empirically how a minimum risk portfolio could be constructed. Various ingenious ideas were tested including long/short portfolios, commodities, combinations of govt bonds of different maturities etc. But IIRC none of these gave a better (lower volatility) solution than the straightforward approach of using T bills.An interesting academic exercise but not much practical payoff.In a nutshell that's all it is, the risk free rate is an economists construct in order that they can have their equilibrium models.

### risk-free rates and effects on quant finance

Posted: **January 19th, 2011, 9:20 pm**

by **armchairquant**

QuoteOriginally posted by: acastaldoIn the 1970s Fischer Black did a lot of thinking on how portfolio theory would change if there is no risk free asset (partly motivated by the high inflation in those years).He concluded that if there is no risk free asset, then there is a combination of securities which plays a similar role. He called it the zero beta portfolio. In the ordinary pfolio theory everyone chooses a combination of the market portfolio and the risk free asset, in Black's theory everyone chooses a combination of the market portfolio and the zero beta portfolio.Black had one of his students test empirically how a minimum risk portfolio could be constructed. Various ingenious ideas were tested including long/short portfolios, commodities, combinations of govt bonds of different maturities etc. But IIRC none of these gave a better (lower volatility) solution than the straightforward approach of using T bills.An interesting academic exercise but not much practical payoff.Thanks!! This was exactly the type of answer I was hoping for!

### risk-free rates and effects on quant finance

Posted: **January 20th, 2011, 7:14 am**

by **daveangel**

Quote That seems a bit extreme. Is the probability of collapse conditional on U.S default really exactly 100.00000%? Or is it 99% or maybe 95%? And what about in 5 years time if the BRIC++ countries are 50% larger and more of the world's banks have diversified away from the USD? look what happened when one US bank defaulted.

### risk-free rates and effects on quant finance

Posted: **January 20th, 2011, 8:30 am**

by **TinMan**

QuoteOriginally posted by: Traden4Alpha That seems a bit extreme. Is the probability of collapse conditional on U.S default really exactly 100.00000%? Or is it 99% or maybe 95%? And what about in 5 years time if the BRIC++ countries are 50% larger and more of the world's banks have diversified away from the USD?Yes it's 100%If the US defaults, what's the financial position of the US banks going to be?How much US debt is in the system as collateral? That'll all be worthless.You can buy all the CDS protection you want, it will never pay out, the bank you bought it from will be gone.Like Dave says, one small american bank going under caused chaos.If it happens the game is up.

### risk-free rates and effects on quant finance

Posted: **January 20th, 2011, 11:33 am**

by **eh**

Remember that the "risk-free" rate is only the overnight borrowing rate. Inflation risk is minimal overnight. Longer dated instruments (such as bonds) are certainly not risk free, regardless of the issuer, not least because we do not know the overnight rate tomorrow, next week, next year etc.If you want a pricing model that does not rely too heavily on fixed income and short rates, look at the benchmark approach of Eckhard Platen. He uses the growth optimal portfolio as numeraire to price instruments. It is a good idea if your investment horizon is long.

### risk-free rates and effects on quant finance

Posted: **January 21st, 2011, 9:50 pm**

by **Gmike2000**

QuoteOriginally posted by: TinManQuoteOriginally posted by: Traden4Alpha That seems a bit extreme. Is the probability of collapse conditional on U.S default really exactly 100.00000%? Or is it 99% or maybe 95%? And what about in 5 years time if the BRIC++ countries are 50% larger and more of the world's banks have diversified away from the USD?Yes it's 100%If the US defaults, what's the financial position of the US banks going to be?How much US debt is in the system as collateral? That'll all be worthless.You can buy all the CDS protection you want, it will never pay out, the bank you bought it from will be gone.Like Dave says, one small american bank going under caused chaos.If it happens the game is up.The swiss banks may actually survive. They will take a hit on tsy holdings etc etc etc. But money (what is left of it anyways) would flood into switzerland from all over the world.

### risk-free rates and effects on quant finance

Posted: **January 21st, 2011, 10:44 pm**

by **Martinghoul**

QuoteOriginally posted by: Gmike2000The swiss banks may actually survive. They will take a hit on tsy holdings etc etc etc. But money (what is left of it anyways) would flood into switzerland from all over the world.Indeed. After all, as everyone knows, the vaults of the SNB (as well those of every Swiss bank and household) are full of gold, chocolate (gold-wrapped chocolate is very convenient) and guns.