Benchmarking in CAD and risk-adjusting

The S&P 500 is not in your currency

Your baskets are Canadian shares priced in Canadian dollars. The S&P/TSX Composite is too — compare it directly. The S&P 500 is priced in US dollars, and a Canadian investor who held it would have earned its US-dollar return and whatever the US dollar did in Canadian terms. Both matter, and they multiply:

$$1+r_{\text{CAD}}=(1+r_{\text{USD}})(1+r_{\text{FX}})$$

where $r_{\text{FX}}$ is the change in the CAD per USD rate — the US dollar's return, measured in Canadian dollars. Over the term the S&P 500 returned 6 % in USD while the rate moved from 1.36 to 1.33 CAD/USD. Lesson 8.8's rule: a falling CAD/USD number is a stronger Canadian dollar, so the US dollar fell:

=1.33/1.36-1 = −0.022059 — −2.21 %

=1.06*(1.33/1.36)-1 = 0.036618 — 3.66 % in Canadian dollars. The currency ate more than a third of the index's return. Had the rate gone the other way — 1.36 → 1.39, a weaker Canadian dollar — the US dollar's return would be =1.39/1.36-1 = +2.21 % and the index =1.06*(1.39/1.36)-1 = 8.34 % in CAD. Same index, same term, a five-point swing from the exchange rate alone. A group that pastes 6 % beside its Canadian baskets is comparing two currencies and calling it a benchmark.

Return per unit of risk

Suppose the chosen basket finished ahead of the random one. Did it earn more, or did it just risk more? Lesson 13.2's tools answer that from the same weekly return columns. For each basket, take the mean weekly return and the weekly standard deviation; subtract a weekly risk-free rate (a GoC T-bill yield of about 3.6 % a year is roughly 0.07 % a week); divide:

$$\text{Sharpe-style ratio}=\frac{\bar{r}-r_f}{\sigma }$$

The chosen basket: mean 0.45 % a week, σ 2.1 % → =(0.0045-0.0007)/0.021 = 0.180952 — 0.181 units of excess return per unit of risk. The random basket: mean 0.60 %, σ 3.4 % → =(0.0060-0.0007)/0.034 = 0.155882. The random basket earned more per week and yet ranks lower — its extra return came with proportionally more volatility. This is the number to present, because it is the number the instructor will ask for when a group claims its basket "won".

In Excel the whole ratio is one line on the weekly column: =(AVERAGE(r)-rf)/STDEV.S(r). Sample standard deviation, because twelve weeks are a sample of what the basket could have done (Lesson 13.2).

Beta — how much was just the market?

Lesson 13.4's =SLOPE(basket_weekly, tsx_weekly) gives each basket's beta against the S&P/TSX Composite over the same weeks. A basket with β = 1.3 that beat the index in a rising term was, in part, just a leveraged bet on the index rising; a basket with β = 0.7 that kept pace did something harder. Report the beta beside the Sharpe-style ratio, and say what the index did.

A small sample — say so

Twelve weekly observations make a mean with a wide standard error: with σ = 2.1 % a week, the mean weekly return of 0.45 % is measured to within roughly ±0.6 % (2.1 ÷ √12), which is larger than the mean itself. The Sharpe-style ratio and the beta are honest descriptive statistics of one term; they are not estimates of the basket's true risk-adjusted return, and the presentation should say so in one sentence.

Cold

USD return to CAD: multiply the growth factors, with r_FX the USD's change in CAD terms; Sharpe-style = (mean − r_f) ÷ STDEV.S; beta by SLOPE against the TSX; twelve weeks is a small sample.