By: Dean Markwick's Blog -- Julia

Re-posted from: https://dm13450.github.io/2024/04/25/Currency-Hedging-and-Principal-Component-Analysis.html

Principal component analysis (PCA) reduces a dataset to its main
components. When we apply it to a dataset of different
currencies it helps us understand how each currency drives the overall
portfolio and what currency might be a common factor.

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This post was inspired by a problem on the r/quant subreddit where someone posted their interview/take-home question.

A client is considering using SGD to (proxy) hedge their exposure to
a basket of other Asian currencies. Is this likely to be effective?
What analysis could you produce that would help inform their
decision? The client is a US Corporate. The client is exposed to
medium-term changes (say monthly) in the currency. The client has equal (USD equivalent) revenues in each Asian currency. We are not considering hedging costs for this analysis (spot-only component). The data for daily close spot values against USD for each pair is provided. Which currency pairs will it work better for? Would it work for an equally weighted currency portfolio? Would another (single) currency work better? Which correlations should we consider and how reliable are these?

This is an interesting question and not too dissimilar to the
occasional question I answer in my day job. So I thought I’d run through how I might answer it.

Getting FX Data

First, we need to get some data and I’ll be using Alphavantage to pull
daily closing prices of the different currencies. I’ll calculate the
log returns and save the data to cache it for future use. Plus
AlphaVantage only lets you make 25 calls a day, so each time I mucked
up I got locked out for the day – delaying the analysis. We have to
start from 2014 as this is the earliest common date across all
currencies.

function _pull_data(ccy)
    println(ccy)
    res = AlphaVantage.fx_daily("USD", ccy, outputsize="full", datatype="csv")
    res = DataFrame(Dict(:Date=>Date.(res[1][:, 1]), :c=>Float64.(res[1][:,5]), :ccy => ccy));
    res = sort(res, :Date)
    res = @transform(res, :LogReturn = [0; diff(log.(:c))])
    res
end

function pull_data(ccy)
    if isfile("$ccy.csv")
        res = CSV.read("$ccy.csv", DataFrame)
    else
        res = _pull_data(ccy)
        CSV.write("$ccy.csv", res)
    end
    res
end

ccys = ["JPY", "CNH", "SGD", "THB", "HKD", "KRW", "TWD"]
res = vcat(pull_data.(ccys)...);
res = sort(res, :Date)
res = @transform(groupby(res, :ccy), :LogReturn = [0; diff(log.(:c))])
res = @subset(res, :Date .>= Date("2014-11-24"))

Like all good blog posts, let’s start with the plot of the
cumulative returns. Only HKD stands out as something different given
its peg to USD.

p = plot(ylabel = "Cummulative Return")
for ccy in ccys
    plot!(p, res[res.ccy .== ccy, :].Date, cumsum(res[res.ccy .== ccy, :].LogReturn), label = ccy, lw = 2)
end
p

According to the problem, our client is long equal amounts of these
Asian currencies, so it makes sense to calculate the market returns by
taking the average return each day.

market = @combine(groupby(res, :Date), :LogReturn = mean(:LogReturn))
market[!, :ccy] .= "Market"
market[!, :c] .= NaN;

Which we add to the original plot.

p = plot!(p, market.Date, cumsum(market.LogReturn)
    label = "Market", color = "black", lw  = 2)

The client thinks that hedging with SGD alone is enough to protect
against the overall market returns. We can see from the graph that
this probably isn’t the case. But how do we recommend a better
approach?

First, we will start with the correlation in returns between the
different currencies. This will shed some light on how linked they
are and is also simple to explain to the client.

cr = cor(Matrix(modelData[:, [:JPY, :CNH, :SGD, :THB, :HKD, :KRW, :TWD]]))
heatmap(ccys, ccys, cr .> 0.5)

We use a heat-map, but only highlight when two currencies have a
correlation > 0.5, otherwise it’s a bit of a psychedelic nightmare.

We can see that HKD has a low correlation with most, KRW and SGD have
a high correlation between each other and KRW has a high correlation with the majority of
these currencies.
However, we will use the covariance matrix to analyse the best hedging portfolio rather than the correlation matrix.

Principal Component Analysis

Principal component analysis (or PCA) is a tool that tries to find a
common basis of variation in a matrix. It’s about transforming the
data into uncorrelated components through linear algebra.

For this we are using the covariance matrix, so now the diagonals are
the individual price series variances and the off-diagonals are the
covariances between two currencies. If this were a different problem
we might rescale the returns so they all had the same volatility but
this would mean applying leverage, which our hypothetical customer
probably wouldn’t be up for it.

We pull out the covariance matrix

modelData = dropmissing(unstack(res, :Date, :ccy, :LogReturn))
cm = cov(Matrix(modelData[:, [:JPY, :CNH, :SGD, :THB, :HKD, :KRW, :TWD]]))

The MultivariateStats.jl package has the functions for doing PCA and
the appropriate functions for pulling out the right data after fitting the
PCA model.

pcaRes = fit(PCA, cm; maxoutdim=3)

Firstly the weights of all the currencies for the three principal
components.

• PC1 shows the weights for each currency but is unnormalised. The key thing
  we can see here is that HKD is magnitudes smaller than the others.
• PC2 is long JPY and short all the others
• PC3 is long THB and short all the others

Then the explained variance of the three components.

The first component can explain 49% of the variance and then including
the second component 91% of the variance, with the final component
making up 5% to take it to 96% in total. This means that this dataset
can be broken down quite nicely into the two principal components and
this explains most of the variation.

The first principal component is commonly called the
‘market’ portfolio and represents the overall combined market dynamics
of the portfolio. The next portfolio (using the 2nd PC weights) is
uncorrelated to the market and thus more diversified to the overall
market.

In our problem then we can see that we are trying to come up with a
representation of the market and use that to decide how to hedge out
our currencies. So the first principal component is the most relevant.

We take these principal component weights and join them to the original
dataframe to start exploring what the market portfolio looks like.

evFrame = DataFrame(Dict(:ccy => String.([:JPY, :CNH, :SGD, :THB, :HKD, :KRW, :TWD]), 
          :ev1 => eigvecs(pcaRes)[:,1],
          :ev2 => eigvecs(pcaRes)[:,2]))
sort!(evFrame, :ev1)

res = leftjoin(res, dropmissing(evFrame), on = :ccy)

evFrame = sort(evFrame, :ev1);

Then plotting the weights by currency pair

bar(evFrame.ccy, evFrame.ev1 ./ sum(evFrame.ev1), label = "Eigen Weights")

These are the weights of the different currencies of the first eigen
portfolio. This combination of currencies is what we would recommend
if the client was exposed to a similar basket. The key points:

• The client is long these currencies through their business
• They short this portfolio and thus are market-neutral

We now calculate the returns of the eigen portfolios, the portfolio
that only uses the largest 2 (and 3) weights.

evPortfolios = @combine(groupby(res, :Date), 
         :ReturnEV1 = sum(:LogReturn .* :ev1) ./ sum(:ev1), 
         :ReturnEV2 = sum(:LogReturn .* :ev2) ./ sum(:ev2));

ccy2Portfolio = @combine(groupby(res[in.(res.ccy, Ref(["KRW", "JPY"])), :], :Date), 
         :Return2Ccy = sum(:LogReturn .* :ev1) ./ sum(:ev1));

ccy3Portfolio = @combine(groupby(res[in.(res.ccy, Ref(["KRW", "JPY", "SGD"])), :], :Date), 
         :Return3Ccy = sum(:LogReturn .* :ev1) ./ sum(:ev1));

And plotting these returns

plot(market.Date, cumsum(market.LogReturn), label = "Market", color = "black", lw = 2)
plot!(evPortfolios.Date,  cumsum(evPortfolios.ReturnEV1), label = "Eigen Portfolio", lw = 2)
plot!(ccy2Portfolio.Date,  cumsum(ccy2Portfolio.Return2Ccy), label = "2 Ccy", lw =2)
plot!(ccy3Portfolio.Date,  cumsum(ccy3Portfolio.Return3Ccy), label = "3 Ccy", lw = 2)

Then finally, looking at the correlation between these portfolios

• The Eigen Portfolio 1 is most correlated with the equal-weighted portfolio.
• With just KRW and JPY you get to a 93% correlation with the market.
• KRW, JPY and SGD gets you to a 95% with the market.

As expected Eigen portfolio 2 is the most uncorrelated with the
market.

Summary

So our final answer to the client would be:

• We have a proprietary portfolio (the market eigen portfolio) that you
  should hedge with – this will give you the best outcome.
• If you don’t want the full portfolio use a 60/40 ratio of KRW and
  JPY.
• SGD probably isn’t a great idea and will leave you exposed.

Now, we are assuming that these weightings are stable through time and
haven’t changed recently and are therefore valid for the future
returns too. We are ignoring transaction costs, KRW being an NDF and
more expensive to trade compared to a spot currency (like JPY) means
that this approach will break down if the client needs to hedge a
significant amount.

By: Staging Team

Re-posted from: https://sciml.ai/news/2024/04/16/numfocus_sdg/index.html

NumFOCUS small development grant: Collocation Methods for Boundary Value Differential-Algebraic Equations