Lecture 2

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What is holding period return?

What is holding period return?

The percentage gain or loss on an investment over a specific period of time.

Formula for holding period return

HPR = (Ending Price + Dividend or Interest - Beginning Price) / Beginning Price

What are the different methods of calculating return over multiple periods?

Time weighting, which has two categories: arithmetic average and geometric average, and Dollar weighting (IRR)

What is time-weighting?

A method of calculating investment returns that ignores the amounts invested each period and is appropriate to use when there is no control over the timing of cash flows.

Each period of return is treated equally, you can change the order/sequence of the returns and it won’t matter

Used mostly by money management industry for performance evaluation

What is the arithmetic average?

The sum of single period returns divided by the number of periods.

Formula for arithmetic average

r = (r1+r2+…+rN) / n

Take the holding period returns for each period, and calculate the arithmetic average

What is the geometric average?

The average per period return that would give the same cumulative performance as the sequence of actual returns

This takes into account compounding, so reinvesting the returns from the previous year. Annual investment returns over the years have an impact on each other. If you lose a substantial amount of money in a particular year, you have that much less capital with which to invest and generate returns in the following years despite the rate being the same.

Formula for geometric average

r = [(1+r1)(1+r2)…(1+rN)]^(1/n) - 1

Take the holding period returns for each period, and calculate the arithmetic average

What is Dollar Weighting?

A method of measuring investment performance that takes into account the timing and amount of cash flows. It focuses on the total amount of money invested at a specific terminal date. Also the internal rate of return on the investment.

Use if the focus is the total amount of money at some terminal date

What is HPR Expected Return?

The probability weighted averages of the returns

Formula for HPR Expected Returns

Expected Return = Sum of (probability of each scenario)*(HPR in each scenario)

What is variance or standard deviation used for?

It is used to measure risk in finance. It helps to quantify the degree of variability or dispersion of returns for an investment or portfolio. It provides an indication of how much an investment's returns are likely to deviate from the average or expected return.

Formula for standard deviation

standard deviation = the square root of variance

Formula for variance

Variance = The sum of: (probability of each scenario)*[(return - expected return)]²

So in other words (p1)*[(r1- E(r))]²+(p2)*[(r2- E(r))]²+(p3)*[(r3- E(r))]²

What is the risk free rate?

The rate of return that can be earned with certainty

The theoretical rate of return that an investor would expect on an investment with zero risk.

T-Bills

What is Risk Premium?

The excess return that investing in the stock market provides over a risk-free rate.

What is risk aversion?

Reluctance to accept risk

What is the price of risk?

Ratio of risk premium to variance.

The amount of return an investor requires in exchange for taking on additional risk. It is a measure of the compensation investors demand for bearing the uncertainty and potential losses associated with an investment

What is Asset Allocation?

The process of distributing an investment portfolio across different asset classes, such as stocks, bonds, cash, and real estate, based on an investor's goals, risk tolerance, and time horizon. It involves determining the optimal mix of assets to achieve a balance between potential returns and risk.

By diversifying investments across various asset classes, investors aim to reduce the overall risk of their portfolio while potentially maximizing returns.

What is Security Selection?

Choosing specific securities within each asset class

What is a complete portfolio?

A portfolio including the risk-free asset and the risky portfolio

How can risk be reduced in a complete portfolio?

Risk can be reduced by allocating more to the risk-free asset. Because the risk-free asset has a guaranteed return and no variability in its returns. By allocating more to the risk-free asset, the overall portfolio becomes less exposed to the fluctuations and uncertainties of the risky asset, thereby reducing risk.

How can expected returns be increased in a complete portfolio?

Expected returns can be increased by investing more on the risky asset. The risk-free asset typically has a lower expected return compared to the risky asset. Therefore, by allocating more to the risky asset, which has a higher expected return, the overall portfolio's expected return increases.

Formula for expected return of complete portfolio

E(rC) = yE(rP)+(1-y)(rf)

Expected Return of Complete Portfolio = % in Risky Asset*(Expected Return on Risky Asset) + % in Risk-Free Asset*(Return on Risk Free Asset)

Formula for risk premium of a complete portfolio

E(rC) - rf = yE[(rP)-rF]

Expected Return of Complete Portfolio - Risk Free Rate = % in Risky Asset*(Expected Return of Risky Asset - Expected Return of Risk Free Asset)

Formula for the complete portfolio variance

VARc= VAR(yrP+(1-y)rF)

= (y²)*(variance of risky asset) + (1-y)²*(variance of risk free asset)

Variance of Complete Portfolio = Variance(% in Risky Asset*(Return on Risky Asset)+% in Risk Free Asset*(Return on Risk Free Asset))

Formula for the complete portfolio standard deviation

st dev = sq rt[VARc] = sq rt [VAR(yrP+(1-y)rF)]

= (y)*(st dev of risky asset) + (1-y)*(st dev of risk free asset)

When is the standard deviation of the complete portfolio proportionate to the standard deviation of the risky component?

When the risk free rate is 0, because that means it has zero variability

What is the Capital Allocation Line?

It’s a line that represents different combinations of risky and risk-free assets in a portfolio. It shows the expected return and standard deviation of these portfolios. The CAL helps investors determine their allocation to the risky asset based on their risk tolerance.

How is the CAL constructed?

The CAL can be constructed using one-month T-bills (risk-free asset) and a market index (risky asset)