Formula Sheet
Performance of Securities
• Future Value FV = PV(1+r)t
• PV = FV/(1+r)t
• r = (FV/PV)1/t -1
• t = log(FV/PV)/log(1+r)
•
A perpetuity or a consol is a bond that pays a fixed payment C forever. No principal
𝐶
1
𝐶
payment. 𝑃𝑉 = 1+𝑟 ×
1 =
𝑟
1−1+𝑟
•
1−
An annuity pays a fixed cash flow C for T periods. 𝑃𝑉 = 𝐶 [
1
(1+𝑟)𝑇
𝑟
]
Remember our convention that no payment is made in the initial period.
Quoted Rates and Compounding
• Annual quoted rate r with N compounding periods per year means in t years:
𝑟
𝐹𝑉 = 𝑃𝑉 × (1 + )𝑁𝑇
𝑁
• Under continuous compounding the formula becomes: FV = PV erT
•
The Effective Annual Rate (EAR) is the annually compounded rate of return that would
result in the future value that a given compounding frequency. If the quoted rate is
compounded m times a year, then
𝐴𝑃𝑅 𝑚
𝐸𝐴𝑅 = (1 +
) −1
𝑚
•
Holding Period Return – The HPR is defined as:
𝑃𝑡+1 + 𝐷𝑡+1
𝐻𝑃𝑅 =
−1
𝑃𝑡
The annualized holding period return (HPRA) is defined as the value that solves:
𝑉0 (1 + 𝑎𝑛𝑛𝐻𝑃𝑅)𝑇 = 𝑉𝑇
•
𝑉
𝑎𝑛𝑛𝐻𝑃𝑅 = ( 𝑉𝑇 )
0
•
1⁄
𝑇
− 1= 𝑎𝑛𝑛𝐻𝑃𝑅 = (1 + 𝐻𝑃𝑅)
1⁄
𝑇
−1
The formula for T periods is simply:
𝑉𝑇
𝑉𝑇 𝑉𝑇−1 𝑉𝑇−2 𝑉1
=
…
𝑉0 𝑉𝑇−1 𝑉𝑇−2 𝑉𝑇−3 𝑉0
= 𝐻𝑃𝑅0,𝑇 + 1 = (1 + 𝐻𝑃𝑅 𝑇 )(1 + 𝐻𝑃𝑅 𝑇−1 ) … (1 + 𝐻𝑃𝑅1 )
𝐻𝑃𝑅0,𝑇 + 1 =
•
And the formula for annualized HPR is:
1
𝑎𝑛𝑛𝐻𝑃𝑅 = [(1 + 𝐻𝑃𝑅1 )(1 + 𝐻𝑃𝑅2 ) … (1 + 𝐻𝑃𝑅 𝑇 )] ⁄𝑇 − 1
•
Internal rate of return is a number, denoted IRR, that solves the equation:
∞
𝐶𝑡
𝑃0 = 𝑃𝑉 = ∑
(1 + 𝐼𝑅𝑅)𝑡
𝑡=1
Portfolio Choice:
• The expected value is the average outcome if the event was repeated infinitely many times.
𝐸(𝑅𝑖 ) = ∑𝑆𝑠=1 𝑅𝑖 (𝑠)𝑝(𝑠)
•
The variance is the average squared deviation from the expected value.
𝑉𝑎𝑟[𝑅𝑖 ] = 𝜎𝑖2 = 𝐸([𝑅𝑖 (𝑠) − 𝐸(𝑅𝑖 )]2)
𝑠
= ∑[𝑅𝑖 (𝑠) − 𝐸(𝑅𝑖 )]2 𝑝(𝑠)
•
𝑠=1
The standard deviation (SD), also know as volatility, is the square root of the variance:
𝜎𝑖 = √𝜎𝑖2
•
The covariance between two random variables is the average of the products of their
respective deviations from the mean.
𝑐𝑜𝑣(𝑅𝑖 , 𝑅𝑗 ) = 𝐸([𝑅𝑖 − 𝐸(𝑅𝑖 )][𝑅𝑗 − 𝐸(𝑅𝑗 )])
𝑆
= ∑[𝑅𝑖 (𝑠) − 𝐸(𝑅𝑖 )][𝑅𝑗 (𝑠) − 𝐸(𝑅𝑗 )]𝑝(𝑠)
•
𝑠=1
Correlation between two variables is defined as the covariance between the two variables,
normalized by the product of the standard deviations:
𝐶𝑜𝑟𝑟[𝑅𝑖 , 𝑅𝑗 ] = 𝜌𝑖𝑗 =
𝐶𝑜𝑣[𝑅𝑖 ,𝑅𝑗 ]
𝜎𝑖 𝜎𝑗
•
Portfolio Math – The expected return on the portfolio is 𝐸(𝑅𝑝 ) = ∑𝑁
𝑖=1 𝑤𝑖 𝐸(𝑅𝑖 )
•
With 2 securities, the variance of portfolio return is:
𝑉𝑎𝑟(𝑅𝑝 ) = 𝑉𝑎𝑟(𝑤1 𝑅1 + 𝑤2 𝑅2 ) = 𝑤12 𝜎12 + 𝑤22 𝜎22 + 2𝑤1 𝑤2 𝐶𝑜𝑣(𝑅1 , 𝑅2 )
= 𝑤12 𝜎12 + 𝑤22 𝜎22 + 2𝑤1 𝑤2 𝜌12 𝜎1 𝜎2
More generally, the portfolio variance is
𝑉𝑎𝑟(𝑅𝑝 ) = 𝑉𝑎𝑟(𝑤1 𝑅1 + ⋯ + 𝑤𝑁 𝑅𝑁 )
𝑁
𝑁
𝑁
= ∑ 𝑤𝑖2 𝜎𝑖2 + 2 ∑ ∑ 𝑤𝑖 𝑤𝑗 𝜌𝑖𝑗 𝜎𝑖 𝜎𝑗
𝑖=1
𝑖=1 𝑗>1
1
•
Mean-variance utility: 𝑈(𝑅𝑝 ) = 𝐸(𝑅𝑝 ) − 2 𝐴𝑉𝑎𝑟(𝑅𝑝 )
•
The Sharpe Ratio (SR) is the risk premium per unit of risk.
•
The Capital Markets Line gives the risk-return combinations achieved by forming portfolios
𝐸[𝑅𝑖 −𝑅𝑓 ]
𝜎𝑖
from the risk-free security and the market portfolio 𝐸[𝑅𝑝 ] = 𝑅𝑓 + (
𝐸(𝑅𝑀 )−𝑅𝑓
𝜎𝑀
) 𝜎𝑃
•
According to the CAPM, the expected return on any asset is given the Security Market Line:
𝐶𝑜𝑣(𝑅𝑖 , 𝑅𝑀)
𝐸[𝑅𝑖 ] = 𝑅𝑓 + 𝛽𝑖 𝐸[𝑅𝑀 − 𝑅𝑓 ], where 𝛽𝑖 = 𝑉𝑎𝑟(𝑅
)
•
Systematic and Idiosyncratic Risk – The total risk of a security can be partitioned into two
components:
𝜎𝑖2 = 𝛽𝑖2 𝜎𝑀2 + 𝜎̅𝑖2
Where, 𝜎𝑖2 =total risk, 𝛽𝑖2 𝜎𝑀2 =systematic or market risk and 𝜎̅𝑖2 =idiosyncratic risk
𝑀