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I saw a question the other day that said

Assume you have only two assets to build a portfolio. Name and explain three scenarios under which a completely risk-free portfolio can be formed?

I have a few questions:

  1. Is this poorly worded? Surely there are no assets that are completely risk-free. Should this be written as "virtually risk-free"? If not, how can an asset have absolutely zero risk?

  2. The only assets I can think of that would be virtually risk-free are AAA-rated corporate bonds and low-yield western government bonds, i.e. US treasuries. So from my knowledge of only two asset-types that are virtually risk-free, you could only construct one scenario - a portfolio consisting of both US treasuries and AAA corporate bonds. Where is my knowledge lacking?

  3. What is the correct answer?

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    $\begingroup$ Assuming continuous trading is possible and no transaction costs, a delta hedged option portfolio is also risk-free, although an option on its own can be seen as a risky asset. It really depends on the context. Is it a static portfolio you are asked to form, or can you manage it dynamically. What are the assets available to you and what are the simplifying assumptions made. As such, the question is very broad and can be interpreted in many different ways. But what you claim is perfectly right if no simplifying assumptions are made $\endgroup$
    – Quantuple
    Mar 25, 2016 at 12:14
  • $\begingroup$ Does he/she mean 'conditions' rather than 'scenarios'? I.e. one of the two has zero volatility or the two are perfectly correlated (pos or neg) or one is driven by a single BM (no stch vol, jump diff or Levy/ yes Black Scholes, local vol) and the other is a derivative (subj to regularity conditions). $\endgroup$
    – Kiwiakos
    Mar 26, 2016 at 8:22

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Note the words "Assume" and "Scenarios". These words imply that you do not need to concern yourself with any assets that actually exist. A simple model of a market may have only one asset...clearly a vastly simplifying assumption and scenario. In this case we only have two assets. Again, this is a vastly simplifying scenario. This is a toy model which can help us understand the messiness that is real life financial markets.

Here is one of your three scenarios: you have two assets,one of which is risk free (ie, has deterministic outcome in all states of the world relative to buying power): simply buy the risk free asset and don't buy the other (presumably not risk free) asset.

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Risk-free assets refer to assets with a definite rate of return and no risk of default. 　　From the perspective of mathematical statistics, risk-free assets refer to assets with zero variance or standard deviation of investment returns. Of course, the covariance and correlation coefficient between the rate of return of risk-free assets and the rate of return of risky assets are also zero. 　　 From a theoretical point of view, only fully indexed bonds issued by the central government with a maturity matching the length of the investor’s investment period can be regarded as risk-free assets. In the real economy, there are very few securities in circulation that fully meet the above conditions. Therefore, in investment practice, risk-free assets are generally regarded as money market instruments, such as the Treasury bill interest rate LIBOR.

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The trick to this question is that you are asked to build a risk-free portfolio, so its individual assets can be risky assets.

  1. Imperfect markets - If there are price inconsistencies on different markets, say market A is high and market B is low, you can build a risk-free portfolio which nets to a profit by buying a security on market B for a low price and selling to market A for a higher price.
  2. Perfect negative correlations - If two assets have a perfect negative correlations such that any price movement in security A is perfectly offset by price movements in security B, then a portfolio of equals parts A and B is risk-free as it is perfectly hedged.
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Many have quoted a delta hedged option to be a risk free asset. However, I totally beg to differ here because by risk free asset , we mean a guaranteed return in future ( which is not the case with dh option). Also , a delta hedged option still has vega risk.

However, In my opinion , other than treasury , you can consider your savings bank account as a risk free asset !

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    $\begingroup$ The question is not about risk free assets but rather about risk free portfolios. $\endgroup$
    – ApplePie
    Mar 26, 2016 at 19:47
  • $\begingroup$ It is merely a theoretical claim assuming continuous delta hedging, no friction costs, and a single factor pure diffusion model for the underlying (plus considering that the market will also behaves that way). After all this is only the replication argument which got Merton and Scholes the Nobel prize. Of course it has a domain of validity and it is imperfect, but I think you are missing the point by comparing theory and the real world. Do you think it is easy in practice to find 2 perfectly correlated assets to form a riskless portfolio? And what about negative savings rate? $\endgroup$
    – Quantuple
    Mar 27, 2016 at 10:20
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    $\begingroup$ Savings account is NOT risk free because inflation is stochastic. The deterministic dollar amount that is returned will not buy the same goods in every state of the future. $\endgroup$
    – user9403
    Mar 27, 2016 at 15:22
  • $\begingroup$ @AlexandreP.Levasseur yes I agree , but the query about risk free assets was was raised in point 1, hence this answer. $\endgroup$
    – HyperVol
    Mar 28, 2016 at 11:43
  • $\begingroup$ @Quantuple yes I agree , theoreotically - you're correct regarding delta hedging. But i was thinking more on practical lines , where this portfolio, sure , is not risk free ! $\endgroup$
    – HyperVol
    Mar 28, 2016 at 11:50
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  1. Either or both assets have are "risk-free" (in the sense of zero volatility, or guaranteed short-term returns). One could then build a portfolio using it/these, ignoring risky assets.

  2. If the assets are perfectly correlated, or perfectly negatively correlated, AND you have confidence in their volatilities, then one could construct a zero vol portfolio hedging the two assets. Which is only as "risk-free" as one's confidence in assumed vol, and the ability to frictionlessly trade without cost or market impact etc. (to rebalance the portfolio).

  3. If both assets are risky but either has liquid futures (or options), then one can sell these (or short-call, long-put overlay) these to convert a synthetic risk-free asset to invest in.

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