# What are the implication of a negative risk-free rate on SML?

What happens to the Security market line (within the CAPM model) when the risk-free rate turns negative?

• First Comment: Are negative rates really consistent with the assumptions underlying CAPM? For example why would anyone invest risk-free instead of consuming wealth immediately?
– g g
Jul 10 '14 at 15:17
• Second comment: Does CAPM still make any sense as a model if risk free rates are negative? One could hold cash in physical notes in a safe vault earning zero which means negative risk free rates create arbitrage opportunities. So the fact that negative risk free rates are possible at all can only be explained with features not contained in the CAPM model, such as the cost of safe bank vaults or the fact that there is no risk-free asset in the first place.
– g g
Jul 10 '14 at 15:18
• Yes to comments above. The Europeans, Swiss and Japanese have yet to consume their wealth, despite the repeated efforts of the ECB, SNB & BOJ to incentivise precisely that! Parking the bank-vault arb as impractical, people HAVE to invest risk-free. The cash exists, which co-exists as bank reserves at the central bank. These can move around within the system, but cannot enter or leave it. Somebody owns it at the end of every day. The fact it's -0.1% versus +0.1% or +3.5% is incidental. Mar 27 '20 at 10:33

The risk-free rate is the y-intercept of the Security market line. If the risk free rate goes negative the y-intercept of the Security market line would simply be below the x-axis. So if the risk-free rate decreases the whole line shifts down. This just means people are willing to pay for safety. According to the formula for the SML:

• E(Ri) : expected return of a security
• E(Rm) : expected return of the market
• B : systematic risk
• Rm : market risk
• Rf : risk-free rate

E(Ri) = Rf + B(E(Rm) - Rf)

When we discuss CAPM it assumes a lot of things about the market and investor behaviour. There is enough literature on "CAPM doesn't hold". In fact most low beta stocks plot above the security market line (SML). So it would be a mistake to take CAPM so seriously in practice and I would cross question if CAPM works as it is?

In theory, if there are negative interest rates then according to the CAPM equation you would have a negative intercept. Also the market premium would go up by the amount of risk free rate. But this is pure mathematics and one should ask the question, is there value in holding risk-free asset?

In practice, it is wrong to assume CAPM holds even when risk-free rate is positive. Empirically the SML plots with a negative slope, i.e. low beta stocks (value) have higher expected return than high beta stocks (growth). One argument in favour for this is if investors believe in CAPM blindly then they overprice high beta stocks (everyone assume high beta = high return), depleting the "expected return" and under price the low beta stocks increasing the "expected return".

This paper is a good reference to read about low beta anomaly.

I would like to note one consequence of negative riskfree rate:

When the riskfree rate becomes more negative, the Market Portfolio (red) converges to the Global Minimum Variance Portfolio (blue).

The Market Portfolio does never equal the GMV Portfolio though, since the slope is infinite.

Implications of negative yields on the short end of the US treasury securities yield curve on the pricing of risk assets using the CAPM:

• Required returns on the SML just move down across all investment volatility choices. The SML now intersects both the Y and X axes.
• Risk tolerance adaptation: When positive investment return choices exist, rational investors will choose the investment on the SML which intersects with the X axis; i.e. cash/gold or other 0%/positive yield investment choice which will replace a negative yielding US government security, or negative return bank deposit, as the "risk free" investment.

• Rf in the CAPM model therefore changes to 0%, and in a rational world, will never be below 0% as long as non-negative investment choices exist.

• Highly negative consequences for the banking industry, as deposits are withdrawn and held in bank notes or positive yielding investments...a precursor to bank runs and, or price bubbles in higher risk asset classes including non-US markets.
• Irrational investors may choose to accept negative investment returns but are highly incentivised to spend rather than save...significantly fueling general price inflation.

Three things happen:

1- the riskless point on the origin can sink and stay negative.

2- this should steepen the SML, ie marginal return per unit risk increases (assuming no change in the efficient frontier). Ultimately MaxSharpe will roll down the efficient frontier towards the MinVol point. Ultimately, this juicing up of risk-reward is the point of loose monetary policy in the first place!

3- this pushes the whole efficient frontier down (lower rates -> lower yields, how much impact open to debate). The greater the bias down, there is a bit of associated shift to the right (ie higher vol). This in turn might then flatten the whole efficient frontier a bit.

(1) the "no-investment" option that suggests that risk-free cannot really go negative is not an option. For two reasons:

(1a) there is a theoretical arbitrage to be had putting banknotes in a vault, assuming said vault was perfectly secure and cheaper to keep secure than the interest burden on your cash. Except just try it. I take my life-savings in paper and hoard it. How do I then buy a car or house with those \$100/CHF1,000 beauties? The alarm bells will go off, faster than I can say "Anti Money Laundering Officer". The costs that drug dealers and corrupt officials habitually bear turning banknotes into bank deposits is ample evidence that the arb doesn't really exist.

(1b) even if you could manage the "no-investment" trick, someone else somewhere else will have to do it in your absence. Whatever you bought to get rid of that cash, it becomes their cash. And so on. The cash can move around the banking system, but it will be in someone's account at the end of every day; matched by a bank reserve at the central bank. The money does not simply disappear (unless the central bank elects to take it back).

So at the end of every day, someone is looking at a figurative cash versus securities option, with a negative rate on the cash. If they manage to dodge that choice by investing in securities, they are still making that choice by sitting in securities, because they know what the opportunity cost of switching back to cash would be. And if (a big if) they did manage to "not invest", they would simply force the dilemma onto someone else who must. The system doesn't care whether it's you or him/her making the decisions here.

(2) If you take the classic fish-hook profile of a textbook efficient frontier, cash rates don't change this curve. They just move the vertical reference point - the return for zero vol - on the Y-axis. This simply steepens/flattens the slope of the SML to the tangency portfolio. Which in turn moves the tangency portfolio up or down the frontier.

So imagine cash rates equal to the highest-returning asset in your portfolio. The SML is flat, and your tangency portfolio is limit-long Argentina.

Cut rates, and the tangency portfolio rolls down the efficient frontier, until at some point, it will hit (and stay) at the Minimum Variance Portfolio (ie the leftmost point on the EF). At this point, the SML will be so steep that investors are paid so much to tolerate volatility, that it will be more rational to lever up on MinVol than to asset-allocate to any other asset mix. Cash is simply so dear relative to "Assets", borrowing Cash to buy more Assets is a superior option to differentiating between different kinds of Asset.

Note that this process applies for any change in interest rates, positive or negative. There is nothing special about the difference between +0.1% and -0.1% whatsoever.

(3) Except of course, you cannot hold-constant the efficient frontier thus in reality! It's messy; and different asset allocators hold very different views about the impact of interest rates on asset yields, even on volatilities, maybe even on cross-asset correlations.

All that really be said is that:

• people tend to argue about how much lower rates drag the EF down in sympathy with the lower cash returns. Some say a lot down. Most say a little down. A few (mostly for ideological reasons) say no impact. Nobody says up. The bias is down.

• if one thinks that rates do impact yields more, then lower rates can have a volatility-amplifying effect that pushes the EF out to the right. Albeit this is dependant on the curve falling! In its simplest terms, consider a perpetual bond with a running yield of Y%. Its price will be 1/Y. Which means that its derivative, ie the change in price per change in yield, is -1/Y^2. IE halve interest rates and prices (for perpetual duration) will fluctuate four times faster. Obviously less for sub-infinite maturity and assets with risk-premia baked in, etc. But hopefully you see the basic argument.

• if the volaility effect happens, this could in turn flatten out the efficient frontier itself. Lots of economist-types start spitting dummies about this; and the argument is more common from risk-management types.

Consider a very simple, classical, hypothetical market construct. You have a long-term government bond, with duration risk as above. And you have stocks that embed an equity risk premium on top of this bond yield. Assume for simplicity's sake that the ERP fluctuates independently of interest rates and bond yields. It is equity-specific. Lower rates -> lower yields -> greater convexity -> higher bond vols will then increase stock:correlations. Because the BY variance is up and the ERP variance is not, the former represents more of the aggregate equity variance mix. Stocks "look more like" bonds. Higher Correlations flatten the efficient frontier.

There is a LOT of "debate" about these effects. So you can't take them for granted, let alone with a hard-and-fast rule of thumb as to the size of the effects. But we can say that nobody argues for higher yields, lower vols and curvier frontiers. The debate is all about if and how much the opposites happen.

hope this helps.