ECO204 2022-2023
A STEP-BY-STEP GUIDE TO PROJECT 2(04)
Project 2(04): Construct a one-month-holding-period financial portfolio in February 2023 with:
Fraction 1 of funds invested in a “Risk-Free Asset”
Fraction of funds (with ≥ 0) invested in a “Synthetic
Risky Asset” (SRA) which consists of four risky assets with
investment weights 1, . . , 4 ( ? 0 and σ = 1)
1 in “Equity #1” 2 in “Equity #2”
3 in “Equity #3”
OR “Oil”
4 in a
“Corporate-
Bond”
You are the CIO at a “family-fund”.
This morning (“today” = Jan 31st 2023), the (risk-averse) Board of Directors (Ivey &
Queens graduates) put you in-charge of Project 2(04):
Project 2(04) harkens back to time you took ECO204 and aced the “Shred Ivey-
Queens Project” by constructing a financial portfolio by four different methods.
You’re going to prove your mettle – and once again shred those Queens & Ivey
snobs – by constructing four different models of the Project 2(04) financial portfolio. 2
Project 2(04): “Synopsis”
Ajaz Hussain, Economics, University of Toronto (STG)
ECO204 2022-2023
Lecture & Tutorial Portfolios: Quick Review
4Agent invests $ of her own funds in a financial portfolio consisting of:
Fraction (1 ? ) invested in a Risk Free Asset Fraction invested in ONE actual risky asset
A “Synthetic Risky Asset” (SRA) consisting of ≥ 1 actual risky assets:
Fraction 1 of “SRA” in a
real Risky Asset 1
Fraction 2 of “SRA” in a
real Risky Asset 2
Fraction 3 of “SRA” in a
real Asset 3
Agent invests $ of her own funds in a financial portfolio consisting of:
Fraction (1 ? ) invested in a Risk
Free Asset
Fraction invested in a “SRA”
consisting of ≥ 1 actual risky
assets:
Fraction 1 of
“SRA” in Risky
Asset 1
Fraction 2 of
“SRA” in Risky
Asset 2
Fraction 3 of
“SRA” in Risky
Asset 3
Portfolio “A”
Portfolio “B”
Portfolio “C”
Portfolio A
Portfolio B
Portfolio C
Use Portfolio A algebra by
treating SRA as “the” risky asset
Recall: The “Algebra” of Constructing Portfolio of a Risk Free and Synthetic Risky Asset
For the sake of illustration,
suppose the SRA consists
of 3 risky assets
Calculate optimal investment weights of risky assets in
SRA by maximizing the SRA’s Sharpe Ratio
Calculate optimal fraction invested in “risk free asset”
and “SRA as the risky asset” (with return-risk and
) by maximizing Mean-Variance Utility.
In practice, you’d need the value of parameter [see W12
Lecture, W14 Tutorial]
Result
“Optimal Portfolio” of a Risk-Free-Asset and a SRA
consisting of ≥ 1 risky assets
5
A “Synthetic Risky Asset” (SRA) consisting of ≥ 1 actual risky assets:
Fraction 1 of “SRA” in a
real Risky Asset 1
Fraction 2 of “SRA” in a
real Risky Asset 2
Fraction 3 of “SRA” in a
real Asset 3
Agent invests $ of her own funds in a financial portfolio consisting of:
Fraction (1 ? ) invested in a Risk
Free Asset
Fraction invested in a “SRA”
consisting of ≥ 1 actual risky
assets:
Fraction 1 of
“SRA” in Risky
Asset 1
Fraction 2 of
“SRA” in Risky
Asset 2
Fraction 3 of
“SRA” in Risky
Asset 3
Agent invests $ of her own funds in a financial portfolio consisting of:
Fraction (1 ? ) invested in a Risk Free Asset Fraction invested in SRA
Recall: The “Mechanics” of Constructing Portfolio of Risk Free and Synthetic Risky Asset
Ajaz Hussain, Economics, University of Toronto (STG)
ECO204 2022-2023
“Minimal Data” Required to Construct Portfolio
of a Risk Free and a Synthetic Risky Asset
7“Minimal Data” for Constructing Portfolio with a Risk Free and a Synthetic Risky Asset
A “Synthetic Risky Asset” (SRA) consisting of ≥ 1 actual risky assets:
Fraction 1 of “SRA” in a
real Risky Asset 1
Fraction 2 of “SRA” in a
real Risky Asset 2
Fraction 3 of “SRA” in a
real Asset 3
Agent invests $ of her own funds in a financial portfolio consisting of:
Fraction (1 ? ) invested in a Risk Free Asset Fraction invested in SRA
max
1,2,3
Sharpe Ratio of SRA =
Risk Premium of SRA
Risk of SRA
“Data” for Constructing Portfolio of a Risk Free Asset and a Synthetic Risky Asset (with = , . . , risky assets)
for = , . . , for , = , . . ,
“You”
Risk Free:
FRED Series
DGS3MO
Randomly
assigned
“Equity #1”
from CRSP
Randomly
assigned
“Equity #2”
from CRSP
Randomly
assigned
“Equity #3”
from CRSP
Randomly
assigned a
Corporate
Bond from a
Blackrock ETF
“Commodity”:
Oil FRED
Series
DCOILWTICO
8
Project 2(04) Assets & “Project 2-Excel-Template”
Project 2(04): Construct a one-month-holding-period financial portfolio in February 2023 with:
Fraction 1 ? of funds invested in
a “Risk-Free Asset”
Fraction of funds (with ≥ 0) invested in a “Synthetic Risky
Asset” (SRA) consisting of four risky assets with investment
weights , = 1, . . , 4
1 in
“Equity #1”
2 in
“Equity #2”
3 in
“Equity #3”
OR “Oil”
4 in a
“Corporate-
Bond”
Project 2(04) Excel-Template-Demo-Portfolio: Construct a one-month-holding-period financial portfolio in February 2023 with:
Fraction 1 ? of funds invested in
a “Risk-Free Asset” FRED Series DGS3MO
Fraction of funds (with ≥ 0) invested in a “Synthetic Risky Asset” (SRA)
consisting of five risky assets with investment weights , = 1, . . , 5
1 in Equity #1:
APTV
2 in “Equity
#2”: CCL
3 in “Equity
#3”: MTN
4 in a “Equity
#4”: EXAS
5 in a
“Corporate-
Bond”: SPRINT
To help you, I’ve posted an “Excel-Template” for the following “DEMO” portfolio ?
9“Asset” Comment
“Risk Free Asset”
FRED Series DGS3MO
Download monthly data via FRED Excel Add-in or FRED website. See FRED-DGS3MO tab and W14 Tutorial on how to
transform FRED data to obtain monthly (by “compounded method”) in Feb 2023.
By definition: = 0, = 0 for all = 1, . . .
CRSP “Equities”
See Equity-Bond-Assign tab for CRSP equities assigned to you and download data from CRSP.
Currently, CRSP reports “returns” through June 2022. See Lecture Delivered W11 on “cleaning” and “pruning” CRSP
data for a common “date range” (see Equities-CRSP-Jun22 tab for demo example).
For July 2022 – Jan 2023 “data”, use Google Sheets in Equities-Google-Jan23 tab to download “daily prices”
[Urge you to look at the Excel “setup” and “formulas”]
Combine CRSP and Google equities data sets (see Equities-Data tab).
S&P500
For reasons to be explained later, we will need data on SP500 returns.
Best option is to download SP500 Nominal Index from multpl.com and compute returns for the date range of the “final
equities” data set (See Lecture Delivered W11 and SP500 tab)
“Corporate-Bond”
Download the Blackrock-IBHC-ETF file (updated daily) from here? “clean/format” the data (see Blackrock-IBHC-
ETF tab) ? Extract the row corresponding to your bond ? copy & paste & transpose that row in new worksheet
(see DEMO-Sprint tab)
Use the bond’s ISIN # to find “today’s price” (be creative and put some elbow grease).
Use “Solver” to compute a “finely-tuned” YTM monthly rate (see Table in Columns H-L in DEMO-Sprint tab).
Research your corporate-bond rating (AAA, AA, A, B, BB, BBB, CCC) ? Go to Corp-Bonds-Quandl tab and take
note of data-series (monthly yields) corresponding to your bond rating
“Commodity -- Oil”:
FRED Series
DCOILWTICO
NOT in Excel-Template. Download monthly data of Oil prices ($/barrel) via FRED Excel Add-in or FRED website ?
Forecast next month’s Oil price by assuming growth rate of Oil prices is Normally distributed and “picking” a growth
rate at random from this distribution (see Gold Claims Case)
Combine the final equities returns data set, SP500 returns data set, Quandl-Bond monthly yield data series corresponding to your bond’s rating,
and the Oil returns data set, over the date range of the “final equities” data set to obtain a “unified” data set (see ALL-DATA tab).
“Minimal Data” for Constructing Portfolio with a Risk Free and a Synthetic Risky Asset
10
“Minimal Data” for Constructing Portfolio with a Risk Free and a Synthetic
Risky Asset
(Shown for illustrative purposes) The “unified data set of monthly returns” (see ALL-DATA tab)
“Month” Equity 1 monthly returns Equity 2 monthly returns Equity 3 monthly returns SP500 monthly returns Oil Price monthly returns
Monthly Yields on Quandl-
bond with same rating as
your corporate bond
Month-1 # # # # # #
…
Month-N # # # # # #
See ALL-DATA tab.
Ajaz Hussain, Economics, University of Toronto (STG)
ECO204 2022-2023
Four Methods for “Estimating” the “Data Required to
Construct Portfolio of a Risk Free Asset and a Synthetic
Risky Asset (with = , . . , Risky Assets)
Four Methods for “Estimating” the “Data Required to Construct Portfolio of a Risk Free
Asset and a Synthetic Risky Asset (with = , . . , Risky Assets)
Stats Stats-Shrinkage
Monte-Carlo
Simulations Wiener
Process
Monte-Carlo
Simulations Ito
Process
Calculate fraction of funds allocated to SRA
Four Methods for “Estimating” the “Data Required to Construct Portfolio of a Risk Free
Asset and a Synthetic Risky Asset”
13
Constructing Portfolio with a Risk Free and a Synthetic Risky Asset: “Stats Approach”
“Data” for Constructing Portfolio of a Risk Free Asset and a Synthetic Risky Asset (with = , . . , risky assets)
Asset Expected Return: Future Risk: Future Covariance
Risk Free Asset 0 , = 0
Equities ,?
Corporate-Bond (Monthly) YTM of Quandl Bond with the
same rating
Where +1 estimated
by technique used in “Gold
Claims” case
oil ,?
See Portfolio-Stats tab.
Note: Using “baseline ” value such that 100% invested in SRA
14
Constructing Portfolio with a Risk Free and a Synthetic Risky Asset: “Stats-Shrinkage Approach”
“Data” for Constructing Portfolio of a Risk Free Asset and a Synthetic Risky Asset (with = , . . , risky assets)
Asset Expected Return: Future Risk: Future Covariance
Risk Free Asset 0 , = 0
Equities + 500
Corporate-Bond (Monthly) YTM of Quandl Bond with the
same rating
Where +1 estimated
by technique used in “Gold
Claims” case
oil ,?
See Portfolio-Stats-Shrinkage tab.
Note: Using “baseline ” value such that 100% invested in SRA
15
Constructing Portfolio with a Risk Free and a Synthetic Risky Asset: “Monte-Carlo Simulations Wiener
Process” Approach”
“Data” for Constructing Portfolio of a Risk Free Asset and a Synthetic Risky Asset (with = , . . , risky assets)
Asset Expected Return: Future Risk: Future Covariance
Risk Free Asset 0 , = 0
Equities of the simulations of the simulations ,?
Using equity simulations
Corporate-Bond (Monthly) YTM of Quandl Bond with the same
rating
,?
Using equity simulations
Oil Prices +1 ?
Where +1 estimated by
technique used in “Gold Claims”
case
oil ,?
Using equity simulations
Assume equity ~ ,
2
Simulate (say) 1,000 rounds of
Randomly select simulations
See RET-Wiener-Process tab and Portfolio-Wiener-Process tab.
Note: Using “baseline ” value such that 100% invested in SRA
16
Constructing Portfolio with a Risk Free and a Synthetic Risky Asset: “Monte-Carlo Simulations Wiener
Process” Approach”
“Data” for Constructing Portfolio of a Risk Free Asset and a Synthetic Risky Asset (with = , . . , risky assets)
Asset Expected Return: Future Risk: Future Covariance
Risk Free Asset 0 , = 0
Equities of the simulations of the simulations ,?
Using equity simulations
Corporate-Bond (Monthly) YTM of Quandl Bond with the same
rating
Using equity simulations
Oil Prices +1
Where +1 estimated by
technique used in “Gold Claims”
case
oil ,
Using equity simulations
Assume equity ~
2
2
, 2
Simulate (say) 1,000 rounds of
Randomly select simulations
See RET-Ito-Process tab and Portfolio-Ito-Process tab.
Note: Using “baseline ” value such that 100% invested in SRA