How did cash perform in developed countries since 1694

Summary

  • Retrieved short-term interest rates in developed countries since 1694
  • Display how cash would have earned historically assuming short-term interest rates were deposit rates

Short-term interest rates

Generally speaking short-term interest rates mean interest rates whose borrowing term is shorter than 1 year. In this article I used 3-month LIBOR rates as short-term rates unless mentioned otherwise. Prior to LIBOR I extrapolated by using T-bill rates, central bank policy rates etc. (Price sources are listed below.)

I assume in this post that short-term interest rates are the rates investors would have got as deposit rates. Of course LIBOR is estimated borrowing rates and therefore it is not guaranteed that investors could get those returns. (Technically speaking LIBID should be used rather than LIBOR since it is offered rates.) From retail investors point of view, they would get much lower returns due to fees etc.

Why bother looking at cash rates?

It is important to see how cash performed when thinking about other asset classes. For example suppose you earned 10% by investing equities. It is not meaningful if cash rates were also 10% because in that situation you would be better off to deposit cash to the bank and you could get 10% without any risk. (Assuming the bank is risk-free.)

In a nutshell, excess returns over cash are the measure you should look when evaluating the performance of investments.

Historical short-term rates

Here I display rates in US, Europe (and Germany), Japan and UK.

Since 1694

UK has the longest historical data among these countries and it starts from 1694 when BOE was established.
You can see that:

  • Rates are evolving around 5% in the very long run
  • After WWI, interest rates in Germany soured as they experienced hyperinflation in 1920s. Effectively Deutsche Mark became worthless at that time. (The graph is trimmed for easier comparison.)
  • During WWII, US kept the interest rates low and stayed lowest among these countries
  • Inflation picked up since 1960s and CBs jacked up the rates to normalise it. Interest rates peaked in 1980s
  • Rates are in the historical low range after the financial crisis in 2008

Since 1900

This is the same data as above but since 1900.
In Japan, it has been in the lowest range for nearly 20 years after the collapse of the bubble and Asian crisis in 1997.

Historical performance

Here I display historical performance of cash using above interest rates. Data in US starts since 1914 and it is flat prior to the inception.
One can see that cash performed relatively well during 60s thanks to the high interest rates especially in the US and the UK. This suggests that T-bill is actually a good way to hedge inflation as Ang[1] says.

The surge in 1920s in Germany is again due to the hyperinflation. So effectively cash performance became zero at that moment.

Overall the wealth curve is very smooth because this is risk-free (ignoring sovereign risk and credit risk). And this should be the base performance to compare to when dealing with investment strategies in other asset classes.

Price source

US

EU (Germany)

Japan

UK

Reference

Andrew Ang: Asset Management: A Systematic Approach to Factor Investing (Financial Management Association Survey and Synthesis)
http://amzn.eu/dTt7yXy

Greeks under normal model

Thinking, Fast and Slow

Illusions you cannot avoid

Probably mentioning this book in 2017 is a bit out of fashion, but there are yet loads of takeaways for understanding systematic biases people inevitably have. The author – Daniel Kahneman, who won Nobel Prize in 2002 for his work on human’s judgement and decision making – introduces those biases with realistic examples throughout the book (some of which are deliberately designed to fool you and you cannot escape even if you know it!).

Systematic biases and risk premia

For financial practitioners, those biases are sometimes keys to understand the nature of risk premia because often they are the very reasons why it exists in the first place. Furthermore in this way, they can provide us some guideline to believe whether the premia is persistence or not in the foreseeable future.

There are currently various ways to access to “risk premia strategy” or “smart beta strategy” via ETFs or other formats especially for institutional investors. Asking if the premia is coming from systematic biases or not could be a reasonable indicator to distinguish between genuine premia and transitory market anomaly. If that is a mere market anomaly, and you believe that a systematic bias is not the one causing it, then it may be arbitraged away anytime soon when large money were put into that factor. Or even worse the factor you rode on might be just the result of the statistical trap like the size factor.

Among these biases, what I found interesting were probability effect and certainty effect.

Probability effect

Probability effect is a bias where people put higher “probability” to rare events than they deserve. This is why so many people keep buying lottery tickets and the reason why stocks with higher skewness underperform the peers with lower skewness.

Certainty effect

This is where people value sure things too much than they should from the statistical point of view. People pay more money to shift the probability to 100% from 95% to get relieved. Effectively insurance companies make money from this bias by providing clients the sure protection. Similarly this can explain why put options on S&P500 is far too expensive compared to realised volatility, why there exists the volatility risk premia, and why long VIX futures is always bleeding.

These two are just examples of the fallacy Daniel Kahneman revealed. And by the end of the book, you would realise how stupid we are!

Thinking, Fast and Slow by Daniel Kahneman
http://amzn.eu/8BvvNZd

PutWrite versus BuyWrite: Yes, Put-Call Parity Holds Here Too

Source

PutWrite versus BuyWrite: Yes, Put-Call Parity Holds Here Too
https://papers.ssrn.com/sol3/Papers.cfm?abstract_id=2894610

Author

Roni Israelov: AQR Capital Management, LLC

Abstract

The CBOE PutWrite Index has outperformed the BuyWrite Index by approximately 1.1 percent per year between 1986 and 2015. That is pretty impressive. But troubling. Yes – troubling – because the theory of put-call parity tells us that such outperformance should be almost impossible via a compelling no-arbitrage restriction. This paper explains the mystery of this outperformance, which has implications for portfolio construction.

Summary

Historically PutWrite index outperformed BuyWrite index even though the put-call parity suggests that such a discrepancy should not exist.
The author lay outs three constructional differences that could have caused the outperformance.

  1. Naked beta position: By construction, BuyWrite index is a portfolio of short call option and long S&P500. Once the option expires, BuyWrite index still has a naked beta position which PutWrite does not. The author shows that this transitory equity exposure which lasts 4 hours explains most of the mismatches.
  2. Delta: BuyWrite sells the most closest strike higher than the spot while PutWrite sells the lower strike. The author estimates the two nearest strikes have delta difference of 0.03.
  3. Cash position: Collateral of BuyWrite is invested in 1 month T-bill while PutWrite invests in both 1 month and 3 month T-bill.

任意の共分散行列に従う正規乱数の生成

概要

  • 任意の共分散行列に従う正規乱数の生成
  • 共分散行列の固有値と固有ベクトルから求められる
  • pythonで乱数が生成されることを確認する

数式

線形代数のおさらいという感じですが…(― ―)
各要素が実数の対称行列\( A\) (n行n列)は
$$
A = U \cdot \Sigma \cdot U^T
$$
と対角化可能(diasonalisable)である。ただし\(U\)は各列が固有ベクトル、\(\Sigma\)は対角行列で対角要素が固有値。
このとき\(R\)を、各要素がiid標準正規分布に従う1行n列のベクトルとした場合、
$$
R \cdot \Sigma^{1/2} \cdot U^T \sim N \left( 0, A \right)
$$
となる。

pythonで確認

データの用意

今回はSP500, EURO STOXX 50, FTSEの日次リターンの共分散行列を使用します。オリジナルデータはこんな感じ↓
original_data

元の共分散行列、相関行列はこんな感じ↓
original_cov

共分散行列に従うサンプルの生成

lo_returnという変数に日次変数が格納されていると仮定しています。

correlated_sampleという変数に、もとの共分散行列に従う(とされる)サンプルが格納されています。実際に確認したのが以下。どうやら近いっぽい。
sample_cov

ちなみに生成されたサンプルは以下のような感じ
sample_data