This document discusses various methods for calculating rates of return on investments including holding period return (HPR), arithmetic average return, geometric average return, dollar-weighted average return, and annualized returns. It provides examples of calculating these different rates of return for stocks, mutual funds, and other investments. The document also covers risk measures like standard deviation, variance, value at risk (VaR), scenario analysis, and how diversifying a portfolio across multiple assets with differing correlations can reduce overall portfolio risk for a given expected return.
Hoàn thiện chính sách cổ tức tại công ty cổ phần fptNOT
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Các giải pháp nhằm tăng cường quản trị rủi ro tín dụng của NH Shinhan chi nhánh Trần Thái Tông. Do vậy, tác giả lựa đã lựa chọn đề tài Nâng cao hiệu quả quản trị rủi ro tín dụng tại Ngân hàng Shinhan chi nhánh Trần Thái Tông làm đề tài báo cáo thực tập.
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Nhận viết luận văn đại học, thạc sĩ trọn gói, chất lượng, LH ZALO=>0909232620
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Chuyên đề phân tích báo cáo tài chính ngân hàng, RẤT HAY, ĐIỂM CAO. Chia sẻ cho các bạn sinh viên tài liệu tốt nghiệp ngành tài chính ngân hàng các bạn làm chuyên đề tốt nghiệp tài chính ngân hàng vào tải nhé.
Hoàn thiện chính sách cổ tức tại công ty cổ phần fptNOT
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Các giải pháp nhằm tăng cường quản trị rủi ro tín dụng của NH Shinhan chi nhánh Trần Thái Tông. Do vậy, tác giả lựa đã lựa chọn đề tài Nâng cao hiệu quả quản trị rủi ro tín dụng tại Ngân hàng Shinhan chi nhánh Trần Thái Tông làm đề tài báo cáo thực tập.
Cấu trúc vốn và chi phí vốn tại công ty cổ phần kết cấu thép và cơ khí ht ste...NOT
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Nhận viết luận văn đại học, thạc sĩ trọn gói, chất lượng, LH ZALO=>0909232620
Tham khảo dịch vụ, bảng giá tại: https://vietbaitotnghiep.com/dich-vu-viet-thue-luan-van
Chuyên đề phân tích báo cáo tài chính ngân hàng, RẤT HAY, ĐIỂM CAO. Chia sẻ cho các bạn sinh viên tài liệu tốt nghiệp ngành tài chính ngân hàng các bạn làm chuyên đề tốt nghiệp tài chính ngân hàng vào tải nhé.
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This presentation covers the basics of Dividend Discount Model (DDM). Firstly, fundamental formula for valuing a stock using DDM is discussed. After that, 3 cases i.e DDM for zero growth, constant growth, and variable growth stocks, are discussed.
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Nhận viết luận văn đại học, thạc sĩ trọn gói, chất lượng, LH ZALO=>0909232620
Tham khảo dịch vụ, bảng giá tại: https://baocaothuctap.net
Download luận án tóm tắt ngành tài chính ngân hàng với đề tài: Nghiên cứu các nhân tố tác động đến cơ cấu vốn của các doanh nghiệp niêm yết trên thị trường chứng khoán Việt Nam, cho các bạn làm luận văn tham khảo
Nhận viết luận văn đại học, thạc sĩ trọn gói, chất lượng, LH ZALO=>0909232620
Tham khảo dịch vụ, bảng giá tại: https://vietbaitotnghiep.com/dich-vu-viet-thue-luan-van
Chia sẻ cho các bạn sinh viên tài liệu tốt nghiệp ngành tài chính ngân hàng Chuyên đề phân tích các nhân tố ảnh hưởng đến giá cổ phiếu thường, HOT, ĐIỂM 8 các bạn làm chuyên đề tốt nghiệp tài chính ngân hàng vào tải nhé.
Thực Trạng Huy Động Vốn Tại Ngân Hàng Vietcombank Chi Nhánh Nam Sài Gòn. Kết hợp giữa lý thuyết và thực tế tại Ngân hàng TMCP Ngoại thương VN – CN Nam Sài Gòn, báo cáo này sẽ tìm hiểu về nghiệp vụ huy động vốn tại Ngân hàng. Bằng những phương pháp nào có thể khai thác được nguồn vốn nhàn rỗi từ các cá nhân, tổ chức, doanh nghiệp để thực hiện chức năng luân chuyển tiền tệ đến nơi cần. Đồng thời, báo cáo đưa ra các kiến nghị, biện pháp để ngày càng gia tăng số vốn huy động nhằm đáp ứng nhu cầu sử dụng đầu tư, cho vay … của ngân hàng TMCP Ngoại Thương VN – CN Nam Sài Gòn.
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Nhận viết luận văn Đại học , thạc sĩ - Zalo: 0917.193.864
Tham khảo bảng giá dịch vụ viết bài tại: vietbaocaothuctap.net
Download luận văn đồ án tốt nghiệp ngành ngân hàng với đề tài: Phân tích tình hình tài chính của Ngân hàng Nông nghiệp và Phát triển Nông thôn chi nhánh Nam Hà Nội, cho các bạn làm luận văn tham khảo
Nhận viết luận văn đại học, thạc sĩ trọn gói, chất lượng, LH ZALO=>0909232620
Tham khảo dịch vụ, bảng giá tại: https://vietbaitotnghiep.com/dich-vu-viet-thue-luan-van
Chia sẻ cho các bạn sinh viên tài liệu tốt nghiệp ngành tài chính ngân hàng Chuyên đề phân tích hiệu quả kinh doanh của công ty thương mại và dịch vụ, HAY các bạn làm chuyên đề tốt nghiệp tài chính ngân hàng vào tải nhé.
Nhận viết luận văn Đại học , thạc sĩ - Zalo: 0917.193.864
Tham khảo bảng giá dịch vụ viết bài tại: vietbaocaothuctap.net
Download luận văn đồ án tốt nghiệp với đề tài: Giải pháp nâng cao hiệu quả hoạt động kinh doanh chứng khoán của công ty cổ phần đầu tư chứng khoán Việt Nam, cho các bạn làm luận văn tham khảo
Phát Triển Hoạt Động Chứng Khoán Phái Sinh Tại Công Ty Cổ Phần Chứng Khoán Vps đã chia sẻ đến cho các bạn nguồn tài liệu hoàn toàn hữu ích. Nếu các bạn có nhu cầu cần tải bài mẫu này vui lòng nhắn tin ngay qua zalo/telegram : 0932.091.562 để được hỗ trợ tải nhé!
Download luận án tiến sĩ ngành tài chính ngân hàng với đề tài: Hiệu quả hoạt động kinh doanh của các ngân hàng thương mại cổ phần Việt Nam, cho các bạn có thể tham khảo
This presentation covers the basics of Dividend Discount Model (DDM). Firstly, fundamental formula for valuing a stock using DDM is discussed. After that, 3 cases i.e DDM for zero growth, constant growth, and variable growth stocks, are discussed.
Giá 10k/5 lượt download Liên hệ page để mua: https://www.facebook.com/garmentspace
Xin chào, Nếu bạn cần mua tài liệu xin vui lòng liên hệ facebook: https://www.facebook.com/garmentspace Tại sao tài liệu lại có phí ??? Tài liệu một phần do mình bỏ thời gian sưu tầm trên Internet, một số do mình bỏ tiền mua từ các website bán tài liệu, với chi phí chỉ 10k cho 5 lượt download tài liệu bất kỳ bạn sẽ không tìm ra nơi nào cung cấp tài liệu với mức phí như thế, xin hãy ủng hộ Garment Space nhé, đừng ném đá. Xin cảm ơn rất nhiều
Nhận viết luận văn đại học, thạc sĩ trọn gói, chất lượng, LH ZALO=>0909232620
Tham khảo dịch vụ, bảng giá tại: https://baocaothuctap.net
Download luận án tóm tắt ngành tài chính ngân hàng với đề tài: Nghiên cứu các nhân tố tác động đến cơ cấu vốn của các doanh nghiệp niêm yết trên thị trường chứng khoán Việt Nam, cho các bạn làm luận văn tham khảo
Nhận viết luận văn đại học, thạc sĩ trọn gói, chất lượng, LH ZALO=>0909232620
Tham khảo dịch vụ, bảng giá tại: https://vietbaitotnghiep.com/dich-vu-viet-thue-luan-van
Chia sẻ cho các bạn sinh viên tài liệu tốt nghiệp ngành tài chính ngân hàng Chuyên đề phân tích các nhân tố ảnh hưởng đến giá cổ phiếu thường, HOT, ĐIỂM 8 các bạn làm chuyên đề tốt nghiệp tài chính ngân hàng vào tải nhé.
Thực Trạng Huy Động Vốn Tại Ngân Hàng Vietcombank Chi Nhánh Nam Sài Gòn. Kết hợp giữa lý thuyết và thực tế tại Ngân hàng TMCP Ngoại thương VN – CN Nam Sài Gòn, báo cáo này sẽ tìm hiểu về nghiệp vụ huy động vốn tại Ngân hàng. Bằng những phương pháp nào có thể khai thác được nguồn vốn nhàn rỗi từ các cá nhân, tổ chức, doanh nghiệp để thực hiện chức năng luân chuyển tiền tệ đến nơi cần. Đồng thời, báo cáo đưa ra các kiến nghị, biện pháp để ngày càng gia tăng số vốn huy động nhằm đáp ứng nhu cầu sử dụng đầu tư, cho vay … của ngân hàng TMCP Ngoại Thương VN – CN Nam Sài Gòn.
Để xem full tài liệu Xin vui long liên hệ page để được hỗ trợ
:
https://www.facebook.com/garmentspace/
https://www.facebook.com/thuvienluanvan01
HOẶC
https://www.facebook.com/thuvienluanvan01
https://www.facebook.com/thuvienluanvan01
tai lieu tong hop, thu vien luan van, luan van tong hop, do an chuyen nganh
Giá 10k/5 lượt download Liên hệ page để mua: https://www.facebook.com/garmentspace
Xin chào, Nếu bạn cần mua tài liệu xin vui lòng liên hệ facebook: https://www.facebook.com/garmentspace Tại sao tài liệu lại có phí ??? Tài liệu một phần do mình bỏ thời gian sưu tầm trên Internet, một số do mình bỏ tiền mua từ các website bán tài liệu, với chi phí chỉ 10k cho 5 lượt download tài liệu bất kỳ bạn sẽ không tìm ra nơi nào cung cấp tài liệu với mức phí như thế, xin hãy ủng hộ Garment Space nhé, đừng ném đá. Xin cảm ơn rất nhiều
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Nhận viết luận văn Đại học , thạc sĩ - Zalo: 0917.193.864
Tham khảo bảng giá dịch vụ viết bài tại: vietbaocaothuctap.net
Download luận văn đồ án tốt nghiệp ngành ngân hàng với đề tài: Phân tích tình hình tài chính của Ngân hàng Nông nghiệp và Phát triển Nông thôn chi nhánh Nam Hà Nội, cho các bạn làm luận văn tham khảo
Nhận viết luận văn đại học, thạc sĩ trọn gói, chất lượng, LH ZALO=>0909232620
Tham khảo dịch vụ, bảng giá tại: https://vietbaitotnghiep.com/dich-vu-viet-thue-luan-van
Chia sẻ cho các bạn sinh viên tài liệu tốt nghiệp ngành tài chính ngân hàng Chuyên đề phân tích hiệu quả kinh doanh của công ty thương mại và dịch vụ, HAY các bạn làm chuyên đề tốt nghiệp tài chính ngân hàng vào tải nhé.
Nhận viết luận văn Đại học , thạc sĩ - Zalo: 0917.193.864
Tham khảo bảng giá dịch vụ viết bài tại: vietbaocaothuctap.net
Download luận văn đồ án tốt nghiệp với đề tài: Giải pháp nâng cao hiệu quả hoạt động kinh doanh chứng khoán của công ty cổ phần đầu tư chứng khoán Việt Nam, cho các bạn làm luận văn tham khảo
Ch5 Portfolio Theory -Risk and ReturnLiang (Kevin) Guo.docxketurahhazelhurst
Ch5 Portfolio Theory -Risk and Return
Liang (Kevin) Guo
Learning Objectives
Be able to calculate ex post and ex ante risk and return statistical measures, such as holding period return, average returns, expected returns, and standard deviation.
Understand the difference between time-weighted and dollar-weighted returns, geometric and arithmetic averages.
Be able to construct portfolios of different risk levels, given information about risk free rates and returns on risky assets.
Be able to explain the CML theory.
Table of Contents
5.1 Rates of Return
5.2 Risk and Risk Premiums
5.3 Inflation and Real Rates of Return
5.4 Asset Allocation Across Risky and Risk Free Portfolios
5.5 Passive Strategies and The Capital Market Line (CML)
5.1 Rates of Return
Considering one-single period investment: regardless of the length of the period.
Holding period return (HPR): measuring Ex-Post (Past) Returns over one-single period.
HPR = [PS - PB + CF] / PB
where
PS = Sale price (or P1)
PB = Buy price ($ you put up) (or P0)
CF = Cash flow during holding period ( Such as dividend, interest)
Example: You put up $50 at the beginning of the year for an investment. The value of the investment grows 4% and you earn a dividend of $3.50. What is your HPR?
Annualizing HPRs
Annualize a holding period return: translate it into percentage per year.
(1) Without compounding (Simple or APR):
HPRann = HPR/n
(2) With compounding: EAR
HPRann = [(1+HPR)1/n ]-1
where n = number of years held
Annualizing HPRs for holding periods of greater than one year
Example: Suppose you buy one share of a stock today for $45 and you hold it for two years and sell it for $52. You also received $8 in dividends at the end of the two years. What is the annual rate of return with and without compounding?
HPR =
(1) Annualized w/out compounding
(2) The annualized HPR assuming annual compounding is (n =2 ):
(
Annualizing HPRs for holding periods of less than one year
Example: Suppose you buy one share of a stock today for $45 and you hold it for 3 months and sell it for $52. You also received $8 in dividends at the end of the two years. What is the annual rate of return with and without compounding?
HPR =
(1) Annualized w/out compounding
(2) The annualized HPR assuming annual compounding is (n =0.25 ):
Investment Returns over multiple periods
The holding period return (HPR) is a simple measure of investment return over a single period.
But how to measure the performance of a mutual fund over the last ten-year period?
Several measures to find the average investment return for a time series of returns .
(a) Arithmetic average return (simple Time-weighted average)
(b) Geometric average return (Geometric time-weighted average)
(c) Dollar-weighted return
(a) Arithmetic Average Return (AAR)
(a) Arithmetic average (simple Time-weighted average)
Arithmetic means are the sum of ...
how can I sell my pi coins for cash in a pi APPDOT TECH
You can't sell your pi coins in the pi network app. because it is not listed yet on any exchange.
The only way you can sell is by trading your pi coins with an investor (a person looking forward to hold massive amounts of pi coins before mainnet launch) .
You don't need to meet the investor directly all the trades are done with a pi vendor/merchant (a person that buys the pi coins from miners and resell it to investors)
I Will leave The telegram contact of my personal pi vendor, if you are finding a legitimate one.
@Pi_vendor_247
#pi network
#pi coins
#money
how to sell pi coins in South Korea profitably.DOT TECH
Yes. You can sell your pi network coins in South Korea or any other country, by finding a verified pi merchant
What is a verified pi merchant?
Since pi network is not launched yet on any exchange, the only way you can sell pi coins is by selling to a verified pi merchant, and this is because pi network is not launched yet on any exchange and no pre-sale or ico offerings Is done on pi.
Since there is no pre-sale, the only way exchanges can get pi is by buying from miners. So a pi merchant facilitates these transactions by acting as a bridge for both transactions.
How can i find a pi vendor/merchant?
Well for those who haven't traded with a pi merchant or who don't already have one. I will leave the telegram id of my personal pi merchant who i trade pi with.
Tele gram: @Pi_vendor_247
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The European Unemployment Puzzle: implications from population agingGRAPE
We study the link between the evolving age structure of the working population and unemployment. We build a large new Keynesian OLG model with a realistic age structure, labor market frictions, sticky prices, and aggregate shocks. Once calibrated to the European economy, we quantify the extent to which demographic changes over the last three decades have contributed to the decline of the unemployment rate. Our findings yield important implications for the future evolution of unemployment given the anticipated further aging of the working population in Europe. We also quantify the implications for optimal monetary policy: lowering inflation volatility becomes less costly in terms of GDP and unemployment volatility, which hints that optimal monetary policy may be more hawkish in an aging society. Finally, our results also propose a partial reversal of the European-US unemployment puzzle due to the fact that the share of young workers is expected to remain robust in the US.
what is the future of Pi Network currency.DOT TECH
The future of the Pi cryptocurrency is uncertain, and its success will depend on several factors. Pi is a relatively new cryptocurrency that aims to be user-friendly and accessible to a wide audience. Here are a few key considerations for its future:
Message: @Pi_vendor_247 on telegram if u want to sell PI COINS.
1. Mainnet Launch: As of my last knowledge update in January 2022, Pi was still in the testnet phase. Its success will depend on a successful transition to a mainnet, where actual transactions can take place.
2. User Adoption: Pi's success will be closely tied to user adoption. The more users who join the network and actively participate, the stronger the ecosystem can become.
3. Utility and Use Cases: For a cryptocurrency to thrive, it must offer utility and practical use cases. The Pi team has talked about various applications, including peer-to-peer transactions, smart contracts, and more. The development and implementation of these features will be essential.
4. Regulatory Environment: The regulatory environment for cryptocurrencies is evolving globally. How Pi navigates and complies with regulations in various jurisdictions will significantly impact its future.
5. Technology Development: The Pi network must continue to develop and improve its technology, security, and scalability to compete with established cryptocurrencies.
6. Community Engagement: The Pi community plays a critical role in its future. Engaged users can help build trust and grow the network.
7. Monetization and Sustainability: The Pi team's monetization strategy, such as fees, partnerships, or other revenue sources, will affect its long-term sustainability.
It's essential to approach Pi or any new cryptocurrency with caution and conduct due diligence. Cryptocurrency investments involve risks, and potential rewards can be uncertain. The success and future of Pi will depend on the collective efforts of its team, community, and the broader cryptocurrency market dynamics. It's advisable to stay updated on Pi's development and follow any updates from the official Pi Network website or announcements from the team.
how to sell pi coins in all Africa Countries.DOT TECH
Yes. You can sell your pi network for other cryptocurrencies like Bitcoin, usdt , Ethereum and other currencies And this is done easily with the help from a pi merchant.
What is a pi merchant ?
Since pi is not launched yet in any exchange. The only way you can sell right now is through merchants.
A verified Pi merchant is someone who buys pi network coins from miners and resell them to investors looking forward to hold massive quantities of pi coins before mainnet launch in 2026.
I will leave the telegram contact of my personal pi merchant to trade with.
@Pi_vendor_247
how can i use my minded pi coins I need some funds.DOT TECH
If you are interested in selling your pi coins, i have a verified pi merchant, who buys pi coins and resell them to exchanges looking forward to hold till mainnet launch.
Because the core team has announced that pi network will not be doing any pre-sale. The only way exchanges like huobi, bitmart and hotbit can get pi is by buying from miners.
Now a merchant stands in between these exchanges and the miners. As a link to make transactions smooth. Because right now in the enclosed mainnet you can't sell pi coins your self. You need the help of a merchant,
i will leave the telegram contact of my personal pi merchant below. 👇 I and my friends has traded more than 3000pi coins with him successfully.
@Pi_vendor_247
US Economic Outlook - Being Decided - M Capital Group August 2021.pdfpchutichetpong
The U.S. economy is continuing its impressive recovery from the COVID-19 pandemic and not slowing down despite re-occurring bumps. The U.S. savings rate reached its highest ever recorded level at 34% in April 2020 and Americans seem ready to spend. The sectors that had been hurt the most by the pandemic specifically reduced consumer spending, like retail, leisure, hospitality, and travel, are now experiencing massive growth in revenue and job openings.
Could this growth lead to a “Roaring Twenties”? As quickly as the U.S. economy contracted, experiencing a 9.1% drop in economic output relative to the business cycle in Q2 2020, the largest in recorded history, it has rebounded beyond expectations. This surprising growth seems to be fueled by the U.S. government’s aggressive fiscal and monetary policies, and an increase in consumer spending as mobility restrictions are lifted. Unemployment rates between June 2020 and June 2021 decreased by 5.2%, while the demand for labor is increasing, coupled with increasing wages to incentivize Americans to rejoin the labor force. Schools and businesses are expected to fully reopen soon. In parallel, vaccination rates across the country and the world continue to rise, with full vaccination rates of 50% and 14.8% respectively.
However, it is not completely smooth sailing from here. According to M Capital Group, the main risks that threaten the continued growth of the U.S. economy are inflation, unsettled trade relations, and another wave of Covid-19 mutations that could shut down the world again. Have we learned from the past year of COVID-19 and adapted our economy accordingly?
“In order for the U.S. economy to continue growing, whether there is another wave or not, the U.S. needs to focus on diversifying supply chains, supporting business investment, and maintaining consumer spending,” says Grace Feeley, a research analyst at M Capital Group.
While the economic indicators are positive, the risks are coming closer to manifesting and threatening such growth. The new variants spreading throughout the world, Delta, Lambda, and Gamma, are vaccine-resistant and muddy the predictions made about the economy and health of the country. These variants bring back the feeling of uncertainty that has wreaked havoc not only on the stock market but the mindset of people around the world. MCG provides unique insight on how to mitigate these risks to possibly ensure a bright economic future.
Falcon stands out as a top-tier P2P Invoice Discounting platform in India, bridging esteemed blue-chip companies and eager investors. Our goal is to transform the investment landscape in India by establishing a comprehensive destination for borrowers and investors with diverse profiles and needs, all while minimizing risk. What sets Falcon apart is the elimination of intermediaries such as commercial banks and depository institutions, allowing investors to enjoy higher yields.
when will pi network coin be available on crypto exchange.DOT TECH
There is no set date for when Pi coins will enter the market.
However, the developers are working hard to get them released as soon as possible.
Once they are available, users will be able to exchange other cryptocurrencies for Pi coins on designated exchanges.
But for now the only way to sell your pi coins is through verified pi vendor.
Here is the telegram contact of my personal pi vendor
@Pi_vendor_247
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how to sell pi coins at high rate quickly.DOT TECH
Where can I sell my pi coins at a high rate.
Pi is not launched yet on any exchange. But one can easily sell his or her pi coins to investors who want to hold pi till mainnet launch.
This means crypto whales want to hold pi. And you can get a good rate for selling pi to them. I will leave the telegram contact of my personal pi vendor below.
A vendor is someone who buys from a miner and resell it to a holder or crypto whale.
Here is the telegram contact of my vendor:
@Pi_vendor_247
how to swap pi coins to foreign currency withdrawable.DOT TECH
As of my last update, Pi is still in the testing phase and is not tradable on any exchanges.
However, Pi Network has announced plans to launch its Testnet and Mainnet in the future, which may include listing Pi on exchanges.
The current method for selling pi coins involves exchanging them with a pi vendor who purchases pi coins for investment reasons.
If you want to sell your pi coins, reach out to a pi vendor and sell them to anyone looking to sell pi coins from any country around the globe.
Below is the contact information for my personal pi vendor.
Telegram: @Pi_vendor_247
2. 5-2
Rates of Return
• Holding-Period Return (HPR)
• HPR of a share of stock depends on the increase (or decrease) in
the price of the share over the investment period as well as on any
dividend income the share has provided.
• The rate of return is defines as dollars earned over the investment
period (price appreciation as well as dividends) per dollar invested:
HPR=
𝐸𝑛𝑑𝑖𝑛𝑔 𝑝𝑟𝑖𝑐𝑒 −𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑝𝑟𝑖𝑐𝑒+𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑
𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒
This definition of HPR assumes that the dividend
is paid at the end of the holding period.
3. 5-3
• You put up $50 at the beginning of the year
for an investment. The value of the
investment grows 4% and you earn a
dividend of $3.50. Your HPR was…
a) 4%
b) 3.5%
c) 7%
d) 11%
Example: Holding Period Return
4. 5-4
• Solution:
4% + $3.50/$50 = 11%
Answer (d) is correct
Example: Holding Period Return
5. 5-5
Rates of Return
• Measuring Investment Returns over
Multiple Periods
• Arithmetic average
• Sum of returns in each period divided by number of periods
• Geometric average
• Single per-period return; gives same cumulative performance as
sequence of actual returns
• Compound period-by-period returns; find per-period rate that
compounds to same final value
• Dollar-weighted average return
• Internal rate of return on investment
6. 5-6
Quarterly Cash Flows/Rates of Return of a Mutual Fund
1st
Quarter
2nd
Quarter
3rd
Quarter
4th
Quarter
Assets under management at start of
quarter ($ million)
1 1.2 2 0.8
Holding-period return (%) 10 25 −20 20
Total assets before net inflows 1.1 1.5 1.6 0.96
Net inflow ($ million) 0.1 0.5 −0.8 0.6
Assets under management at end of
quarter ($ million)
1.2 2 0.8 1.56
7. 5-7
Example
Calculate arithmetic average, geometric average and
dollar-weighted return.
Arithmetic average = (10+25-20+20)/4 = 8.75%
Geometric average =
(1+0.10)(1+0.25)(1-0.20)(1+0.20)=(1+r)^4
r = 7.19%
Dollar-Weighted Average Return
0 = −1.0 +
−0.1
1+𝐼𝑅𝑅 1 +
−0.5
(1+𝐼𝑅𝑅)2 +
0.8
(1+𝐼𝑅𝑅)3 +
−0.6+1.56
(1+𝐼𝑅𝑅)4
IRR = 3.38%
8. 5-8
Example: AR, GR & DWR
• A fund begins with $10 million and reports
the following three-month results
• Compute the arithmetic, geometric and
dollar-weighted average return.
Months 1 2 3
Net inflows (end of month, $
million)
3 5 0
HPR(%) 2 8 (4)
9. 5-9
Example: AR, GR & DWR
• Solution:
a. The arithmetic average is (2+8-4)/3 = 2%
per month
b. The geometric average is:
[(1+0.02)(1+0.08)(1-0.04)]^1/3 -1 = 0.0188
or 1.88% per month
10. 5-10
Example: AR, GR & DWR
• Solution: Dollar-Weighted Average (IRR)
Months
1 2 3
AUM at beginning of month 10.0 13.2 19.256
Investment profits during the month
(HPR*Assets)
0.2 1.056 (0.77)
Net inflows during the month 3.0 5.0 0.0
Assets under management at end of
month
13.2 19.256 18.486
Time
0 1 2 3
Net Cash Flow -10 -3.0 -5.0 +18.486
IRR is 1.17% per month
11. 5-11
Rates of Return
• Conventions for Annualizing Rates of
Return
• APR = Per-period rate × Periods per year
• 1 + EAR = (1 + Rate per period)
• 1 + EAR = (1 + Rate per period)n = (1 + )n
• APR = [(1 + EAR)1/n – 1]n
• Continuous compounding: 1 + EAR = eAPR
APR
n
12. 5-12
Rates of Return
Q1. If weekly return is 0.2%, then calculate
the compound annual return.
Q2. If the return for 15 days is 0.4%, the
annualized return is?
Q3. What is the annualized return for an 18-
month return of 20%?
14. 5-14
Example: EAR & APR
• Suppose you buy a $10,000 face value
Treasury bill maturing in one month for
$9900. On the bills maturity date, you
collect the face value. Since there are no
other interest payment, calculate the
holding-period return, APR and EAR.
16. 5-16
Other Return Measures
• Gross Return: is the return earned by an asset manager prior to deductions for
management expenses, custodial fees, taxes or any other expenses that are not directly
related to the generation of returns but rather related to the management and
administration of an investment.
• Net Return: is a measure of what the investment vehicle (mutual fund etc.) has earned
for the investor and accounts for all managerial and administrative expenses that reduce
an investor’s return.
• Pre-tax and After-Tax Nominal Return: Returns computed prior to and after adjustment
of taxes on realized gain.
• Real Return & Nominal Return: A nominal return (r) consist of three components: a real
risk-free return as compensation for postponing consumption (rf), inflation as
compensation for loss of purchasing power (π) and a risk premium for assuming risk (rp).
Thus, nominal return and real return can be expressed as:
1 + 𝑟𝑛𝑜𝑚𝑖𝑛𝑎𝑙 = 1 + 𝑟𝑓 ∗ 1 + 𝜋 ∗ 1 + 𝑟𝑝
(1+𝑟𝑟𝑒𝑎𝑙) = (1 + 𝑟𝑓)* 1 + 𝑟𝑝
1 + 𝑟𝑟𝑒𝑎𝑙 = 1 + 𝑟𝑛𝑜𝑚𝑖𝑛𝑎𝑙 / 1 + 𝜋
17. 5-17
Risk and Risk Premiums
• Scenario Analysis and Probability
Distributions
• Scenario analysis: Possible economic scenarios;
specify likelihood and HPR
• Probability distribution: Possible outcomes with
probabilities
• Expected return: Mean value of distribution of
HPR
• Variance: Expected value of squared deviation
from mean
• Standard deviation: Square root of variance
19. 5-19
Risk and Risk Premiums
• Deviation from Normality and Value at Risk
• Kurtosis: Measure of fatness of tails of probability
distribution; indicates likelihood of extreme outcomes
• Skew: Measure of asymmetry of probability distribution
• Using Time Series of Return
• Scenario analysis derived from sample history of returns
• Variance and standard deviation estimates from time
series of returns:
21. 5-21
Value at Risk
• It is defined as the maximum dollar amount
expected to be lost over a given time horizon, at a
pre-defined confidence level.
• For example, if the 95% one-month VAR is $1
million, there is 95% confidence that over the next
month the portfolio will not lose more than $1
million.
22. 5-22
Var Calculation
• Suppose you worry about large investment losses in worst-case scenario for
your portfolio. You might ask: “How much would I lose in a fairly extreme
outcome, for example, if my return were in the fifth percentile of the
distribution?”
• You can expect your investment experience to be worse than this value only
5% of the time and better than this value 95% of the time.
• In investment parlance, this cutoff is called the value at risk (VaR). A loss-
averse investor might desire to limit portfolio VaR, i.e. limit the loss
corresponding to a probability of 5%.
• For normally distributed returns, VaR can be derived from the mean and
standard deviation of the distribution.
• Excel’s standard normal function =NORMSINV(0.05) computes the fifth
percentile of a normal distribution with a mean of zero and a variance of 1,
which turns out to be -1.64485.
• In other words, a value that is 1.64485 standard deviations below the mean
would correspond to a VaR of 5%
23. 5-23
Var Calculation (Continued)
• Thus, VaR = E(r) + (-1.64485)σ
• We can obtain this value directly from Excel’s nonstandard normal function =
NORMINV(0.05, E(r),σ)
• If the given sample of returns is not normally distributed then 5% VaR is estimated
directly as the fifth percentile rate of return.
• For instance, for a sample of 100 returns, first the returns are ordered from high to low
and then count the fifth observation from the bottom, that will give you the value of
VaR.
• If the 5% of the observations, don’t make an integer; then interpolation is required.
Suppose we have 72 monthly observations so that 5% of the sample is 3.6
observations.
• Then we approximate the VaR by going 0.6 of the distance from the third to the fourth
rate from the bottom.
• Further assume, the third and fourth observations are -42% and -37%, then the
interpolated value for VaR is -42 + 0.6(42-37) = -39%
25. 5-25
Normal Distribution Properties Relevant for
Investment Management
Two special properties of the normal distribution lead to
critical simplifications of investment management when
returns are normally distributed:
1. The return on a portfolio comprising two or more assets
whose returns are normally distributed also will be
normally distributed
2. The normal distribution is completely described by its
mean and standard deviation. No other statistic is
needed to learn about the behaviour of normally
distributed return.
26. 5-26
Example: VaR
Suppose the current value of a stock portfolio is $23 million.
A financial analyst summarizes the uncertainty about next
year’s holding-period return using the scenario analysis in
the following table. What are the annual holding-period
returns of the portfolio in each scenario?
a) Calculate the expected holding-period return and the
standard deviation of returns. Also, calculate the VaR of
the portfolio with normally distributed returns with the
same mean and standard deviation as this stock?
b) Suppose that the worst three rates of return in a sample
of 36 monthly observations are -17%, -5% and 2%.
Estimate the VaR.
29. 5-29
Example Solution:VaR
a) Expected HPR = 0.308 or 30.8%
• Variance = 0.134
• Standard Deviation = √0.134 = 36.6%
• For the corresponding normal distribution, VaR
would be 30.8% - 1.645*36.6% = -29.43%
b) With 36 returns, 5% of the sample would be
0.05*36 = 1.8 observations. The worst return is
-17% and second worst is -5%. Using interpolation,
we estimate the fifth percentile return as:
-17% + 0.8(17-5) = -7.4%
30. 5-30
Portfolio Risk
• Portfolio return : when two individual assets are combines in a
portfolio, we can compute the portfolio return as a weighted average
of the return of the two assets. The equation is given as :
• 𝑅𝑝 = 𝑤1𝑅1 + 𝑤2𝑅2
• Portfolio Risk: can be calculated by taking the variance of both assets
and including the covariance between the assets. The equation is
given by:
• 𝜎𝑝 = 𝑤1
2
𝜎1
2
+ 𝑤2
2
𝜎2
2
+ 2𝑤1𝑤2𝐶𝑜𝑣 𝑅1, 𝑅2
• OR
• 𝜎𝑝 = 𝑤1
2
𝜎1
2
+ 𝑤2
2
𝜎2
2
+ 2𝑤1𝑤2𝜌12𝜎1𝜎2
• Wherein , 𝐶𝑜𝑣 𝑅1, 𝑅2 = 𝜌12𝜎1 𝜎2
31. 5-31
Relationship between Return & Risk
Compute the expected return and standard deviation for a
portfolio comprising of two assets- Asset 1 with an annual
return of 7% and annualized risk of 12%;
Asset 2 has an annual return of 15% and annualized risk of
25%. s
If the correlation between the two assets is +1, 0.5, 0.2 and
-1.0.
(Assume different combinations of weights
(0%-100%, 10%-90%, 20%-80%, 30%-70%..etc) between
the two assets and plot the relationship on a graph)
32. 5-32
Relationship between Return & Risk
The figure shows the portfolio return for four correlation coefficients ranging
from -1 to +1 and weights ranging between two assets from 0% to 100%.
Portfolio risk becomes smaller with each successive decrease in the
correlation coefficient with the smallest risk when 𝜌12=-1.
(refer to excel for computations)
33. 5-33
Portfolio of Many Risky Assets
• If in the portfolio, the number of risky assets
are ‘N’, then expected return and standard
deviation of the portfolio can be written as:
34. 5-34
Avenues for Diversification
a. Diversify with asset classes or among countries
b. Diversify with index funds
c. Diversify by not owning your employer’s stock
d. Buy insurance for risky portfolios (e.g. buying put option or any commodity having
negative correlation with equity)
e. Evaluate each asset before adding to a portfolio : A general rule to evaluate whether a
new asset should be included to an existing portfolio is based on the following return-
risk trade-off relationship:
E(𝑅𝑛𝑒𝑤) = 𝑅𝑓 +
𝜎𝑛𝑒𝑤𝜌𝑛𝑒𝑤,𝑝
𝜎𝑝
∗ [E 𝑅𝑝 − 𝑅𝑓]
Where E(𝑅𝑛𝑒𝑤) 𝑖𝑠 𝑡ℎ𝑒 𝑟𝑒𝑡𝑢𝑟𝑛 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑎𝑠𝑠𝑒𝑡, 𝑅𝑓 is the return on the risk-free asset, σ is the standard
deviation, ρ is the correlation coefficient and the subscripts ‘new’ and ‘p’ refers to the new and existing
portfolio.
If the new asset’s risk-adjusted return benefits the portfolio, then the asset should be included. The
condition can be rewritten using the Sharpe ratio on both sides of the equation as:
𝐸 𝑅𝑛𝑒𝑤 − 𝑅𝑓
𝜎𝑛𝑒𝑤
>
𝐸 𝑅𝑝 −𝑅𝑓
𝜎𝑝
∗ 𝜌𝑛𝑒𝑤,𝑝
35. 5-35
Concept of Risk Aversion & Portfolio Selection
• Risk aversion is related to the behaviour of
individuals under uncertainty.
• Assume that an individual is offered two
alternatives: one where he will get $50 for
sure and other is a gamble with a 50%
chance that he gets $100 and 50% chance
that he gets nothing.
• The expected value in both cases is $50.
• What will an investor choose?
36. 5-36
Concept of Risk Aversion
• So, there are three possibilities in front of investor: to gamble, not to gamble
or be indifferent.
• If the investor chooses to gamble, then the investor is said to be risk loving
or risk seeking.
• If the investor is indifferent about the gamble or the guaranteed outcome,
then the investor may be risk neutral. Risk neutrality means that the investor
cares only about the return and not about risk, so higher return investments
are more desirable even if they come with higher risk.
• If the investor chooses the guaranteed outcome, he/she is said to be risk
averse because the investor does not want to take the chance of not getting
anything at all.
• In general, investors are likely to shy away from risky investments for a lower
but guaranteed return. A risk neutral investor would maximize return
irrespective of risk and a risk-seeking investor would maximize both risk and
return.
37. 5-37
Utility Theory & Indifference Curves
• Utility is a measure of relative satisfaction from consumption of
various goods and services or in the case of investments, the
satisfaction that an investor derives from different portfolios.
• Utility theory allows us to quantify the rankings of investment choices
using risk and return.
• Utility function is given by:
U= 𝐸 𝑟 −
1
2
𝐴𝜎2
Where ‘U’ is the utility of an investment, E(r) is the expected return, 𝜎2
is the variance of the investment and ‘A’ is the measure of risk aversion,
which is measured as the marginal reward that an investor requires to
accept additional risk.
Thus, ‘A’ is higher for more risk-averse individuals.
38. 5-38
Key Takeaways from Utility Function
a) Utility is unbounded on both sides. It can be highly positive or
highly negative.
b) Higher return contributes to higher utility.
c) Higher variance reduces the utility but the reduction in utility gets
amplified by the risk aversion coefficient.
d) Utility does not indicate or measure satisfaction. It can be useful
only in ranking various investments. For example, a portfolio with
utility of 4 is not necessarily two times better than a portfolio with a
utility of 2.
e) The risk aversion coefficient ‘A’ is greater than zero for a risk-
averse investor, zero for risk neutral investor and negative for risk
loving investor
f) A risk-free investor (𝜎2 =0) generates the same utility for all
individuals.
39. 5-39
Concept & Application of Indifference Curves(IC)
in Portfolio Management
• An indifference curve plots the combination of risk-return
pairs that an investor would accept to maintain a given
level of utility.
• IC are thus defined in terms of a trade-off between
expected rate of return and variance of the rate of return.
• Because an infinite number of combinations of risk and
return can generate the same utility for the same investor,
indifference curves are continuous at all points.
• By definition, all pints on any one of the three curves have
the same utility. An
40. 5-40
Indifference Curves for Risk Averse Investors
• The utility of risk-averse investor always increases as one moves from
northwest – higher return with lower risk.
• The IC are convex because of diminishing marginal utility of return.
• The upward-sloping convex IC has a slope coefficient closely related to the
risk aversion coefficient.
• The greater the slope, the higher is the risk aversion of the investor as a
greater increment in return is required to accept a given increase in risk
41. 5-41
Indifference Curves for Various Types of Investors
The most risk-averse investor has an indifference curve with the greatest slope
(in this case it is indifference curve labelled – ‘Ip’).
The risk-loving investor’s indifference curve, however, exhibits a negative slope,
implying that the risk-lover is happy to substitute risk for return
The indifference curves of risk-neutral investors are horizontal because the utility
is invariant with risk.
42. 5-42
Application of Utility Theory to Portfolio Selection
• The simplest application of utility theory and risk aversion is to a portfolio of two
assets, a risk-free asset and a risky asset.
• The risk-free asset has zero risk and a return of Rf.
• The risky asset has a risk of σi(>0) and an expected return of E(Ri).
• Further, E(Ri)>Rf
• Construct a portfolio of these two assets with expected return of E(Rp) and
standard deviation of σp. Further, give ‘w1’ weight to risk-free asset and (1-w1)
weight to risky asset.
• Thus, the E(Rp) = w1*Rf + (1-w1)*E(Ri)
• 𝜎2
𝑝 = 𝑤12∗
𝜎𝑓2
+ (1 − 𝑤1)2
∗ 𝜎𝑖2
+ 2 ∗ 𝑤1 ∗ 1 − 𝑤1 ∗ 𝜌𝑓𝑖 ∗ 𝜎𝑓 ∗ 𝜎𝑖
• As the σf =0 (i.e. std. deviation of risk-free asset is zero), the first and third term in
the above formula for variance are zero leaving only the second term.
• Hence, the std. deviation of the portfolio is given by σ𝑝 = (1-w1)*σi
43. 5-43
Application of Utility Theory to Portfolio Selection
Assuming only two assets are available in the economy and the risky asset
represents the market, the line in above exhibit is called the ‘CAPITAL
ALLOCATION LINE (CAL)’.
The CAL represents the portfolios available to an investor.
CAL has an intercept of Rf and a slope of
(𝐸 𝑅𝑖 −𝑅𝑓)
𝜎𝑖
, which is the additional;
required return for every increment in risk and is sometimes referred to as the
market price of risk.
45. 5-45
Indifference Curve & CAL
• Overlaying each individual’s indifference curves on the CAL will provide us with the optimal portfolio
for that investor.
• Points under the CAL may be attainable but are not preferred by any investor because the investor
can get a higher return for the same risk by moving up to the allocation line.
• Points above the CAL are desirable but not attainable with available assets.
• In the exhibit, Curve1 is above the capital allocation line, Curve 2 is tangential to the line and Curve 3
intersects the line at two points.
• Curve 1 has the highest utility and Curve 3 has the lowest utility. Because Curve 1 lies completely
above the CAL, points on Curve 1 are not achievable with the available assets on the CAL.
• Point m is clearly superior to point n as the investor is able to earn more return at the same level of
risk.
• At point a and b, the IC curve intersects the CAL, but the investor can invest at point a or b to derive
the risk-return trade-off and utility associated with Curve 3.
• Thus, point ‘m’ and the utility associated with curve 2 is the best that the investor can do because
he/she cannot move to a higher utility IC.
• Optimal portfolio, essentially is the point of tangency between the CAL and the indifference
curve. Further, the optimal portfolio maximizes per unit of risk and gives maximum
satisfaction to the investor.
47. 5-47
Indifference Curve & CAL
• The above exhibit shows two ICs for two different
investors: Suppose Kate has a risk aversion coefficient of
2 and John has a risk coefficient of 4.
• The IC for Kate is to the right of the IC for John because
Kate is less risk averse than Jane and accept a higher
amount of risk.
• Accordingly, their optimal portfolios are different: point K
is the optimal portfolio for Kate and Point J is the optimal
portfolio for John.
• Further, the slope of John’s curve is higher than Kate’s
suggesting that John needs greater incremental return as
compensation for accepting an additional amount of risk
compared with Kate.
48. 5-48
Efficient Frontier & Investor’s Optimal Portfolio
• If two assets are perfectly correlated, the risk-return opportunity set is
represented by a straight line connecting those two assets.
• If the two assets are not perfectly correlated, the portfolio’s risk is less
than the weighted average risk of the components, and the portfolio
formed from the two assets bulges on the left as a curve with ρ less
than 1.
• The addition of new assets to this portfolio creates more and more
portfolios that are either a linear combination of the existing portfolio
and the new asset or a curvilinear combination, depending on the
correlation between the existing portfolio and the new asset.
• As the number of available assets increases, the number of possible
combinations increases rapidly.
• When all investible assets are considered, and there are hundreds
and thousands of them, we can construct an ‘opportunity set of
investments’.
50. 5-50
Efficient Frontier & Investor’s Optimal Portfolio
• The ‘investment opportunity set’ as shown in previous slide
shows the effect of adding a new asset class, such as
international assets.
• As long as the new asset is not perfectly correlated with the
existing asset class, the investment opportunity set will expand
out further to the northwest, providing a superior risk-return
trade-off.
• The investment opportunity set with international assets
dominates the opportunity set that includes only domestic
assets.
• Adding other asset classes will have the same impact on the
opportunity set. Thus, we should continue to add asset classes
until they do not further improve the risk-return trade-off.
52. 5-52
Minimum-Variance Portfolios
• Consider points A, B and X in exhibit (shown in the previous slide) and assume
that they are on the same horizontal line by construction.
• Thus, the three points have the same expected return E(Rp) as do all other
points on the imaginary line connecting A,B and X.
• Given a choice, a, investor will choose the point with the minimum risk, which is
Point X. Point X, however, is unattainable because it does not lie within the
investment opportunity set.
• Thus, the minimum risk that we can attain for E(Rp) is at point A and point B and
all points to the right of point A are feasible but they have higher risk.
• Similarly, point C is the minimum variance point for the return earned at C and
points to the right of C have higher risk.
• In all cases, the ‘minimum variance portfolio’ is the one that lies on the solid
curve and the entire collection of these minimum-variance portfolios is referred to
as the ‘minimum-variance frontier’.
• The minimum variance frontier defines the smaller set of portfolios in which
investors would want to invest.
53. 5-53
Global Minimum-Variance Portfolio
• The left-most point on the minimum-
variance frontier is the portfolio with the
minimum variance among all portfolios of
risky assets and is referred to as the
‘global minimum-variance portfolio’.
• An investor cannot hold a portfolio
consisting of risky assets that has less risk
than that of the global minimum-variance
portfolio.
54. 5-54
Minimum-Variance Portfolio: Markowitz Efficient Frontier
• Consider point A and C on the minimum-variance frontier, both of them have the
same risk.
• Given a choice, an investor will choose Portfolio A because it has a higher
return.
• This applies to all the points on the minimum-variance frontier that lie below the
global minimum-variance portfolio.
• The curve that lies below and to the right of the global minimum-variance
portfolio is referred to as the ‘Markowitz Efficient Frontier’ because it contains all
portfolios of risky assets that rational, risk-averse investors will choose.
• As we move right from the global minimum-variance portfolio, there is an
increase in risk with a concurrent increase in return.
• The increase in return with every unit increase in risk, however, keeps
decreasing as we move from left to the right because the slope continues to
decrease.
• Thus, investors obtain decreasing increase in returns as they assume more risk.
56. 5-56
Capital Allocation Line & Optimal Risky Portfolios
• All portfolios on the efficient frontier are candidates for being combined with
the risk-free asset. Two combinations are shown in the exhibit and presented
by point P and A.
• Comparing CAL –A and CAL-P, reveals that there is a point on CAL(P) with a
higher return and same risk for each point on CAL(A).
• Thus, the portfolios on CAL(P) dominate the portfolios on CAL(A).
• Lets compare point X and Y, it is quite clear that pint X is on the efficient
frontier and has the highest return of all risky portfolios for its risk. However,
point Y on CAL-P is achievable by leveraging portfolio P and it lies above
point X and has the same risk but higher return.
• CAL-P dominates both CAL-A and Markowitz efficient frontier of risky assets.
• Thus, CAL-P is the optimal capital allocation line and Portfolio P is the optimal
risky portfolio.
58. 5-58
Optimal Investor Portfolio
• The location of an optimal investor portfolio depends on the investor’s
risk preferences.
• Moving from the risk-free asset along the capital allocation line, we
encounter investors who are willing to accept more risk.
• At point P, the investor is 100% invested in the optimal risky portfolio.
• Beyond point P, the investor accepts even more risk by borrowing
money and investing in the optimal risky portfolio.
• Portfolio P is the optimal risky portfolio that is selected without regard
to investor preferences.
• To identify the optimal portfolio, we overlay the indifference curve on
the capital allocation line and find that point C on CAL-P is the
optimal portfolio.
59. 5-59
Two-Fund Separation Theorem
• The two-fund separation theorem states that all investors regardless of taste,
risk preferences and initial wealth will hold a combination of two portfolios or
funds: a risk-free asset and an optimal portfolio of risky assets.
• The separation theorem allows us to divide an investor’s investment problem
into two distinct steps: the investment decision and the financing decision.
• In the first step, the investor identifies the optimal risky portfolio from
numerous risky portfolios without considering the investor’s preferences.
Further, the CAL connects the optimal risky portfolio and the risk-free assets.
• In the second step, depending on each investor’s risk preference, using
indifference curves, determines the investor’s allocation to the risk-free asset
(lending) and to the optimal risky portfolio.
• Portfolios beyond the optimal risky portfolio are obtained by borrowing at the
risk-free rate.