Equations

Question 1

nominal interest rate equation?

Answer 1

nominal interest rate (I) =

Expected real interest rate (R)
+
Expected inflation rate (EI)
+
Expected liquidity premium (LP)
+
Expected risk premium (RP)

Question 2

Fisher Equation for nominal interest rate?

Answer 2

Fisher Equation:

   1+𝐼=(1+𝑅)∗((𝑃𝑟𝑖𝑐𝑒 𝑙𝑒𝑣𝑒𝑙 𝑖𝑛 𝑃𝑒𝑟𝑖𝑜𝑑 𝑡)/(𝑃𝑟𝑖𝑐𝑒 𝑙𝑒𝑣𝑒𝑙 𝑖𝑛 𝑃𝑒𝑟𝑖𝑜𝑑 𝑡−1))

1+𝐼=(1+𝑟)∗(1+𝐸𝐼)

1+𝐼=1+𝑅+𝐸𝐼+(𝑅∗𝐸𝐼)

where:
I = nominal interest rate
R = real interest rate
EI = expected inflation rate

Question 3

What is the equation for Macaulay Duration?

Answer 3

D= ∑(t* CF t / (1+y) /P

D = Macaulay Duration (measures the weighted average time until cash flows are received)

∑ = Summation (adds up values over all time periods)

t = Time period (e.g., years)

CF t = Cash flow at time (coupon payments + face value at maturity)

y = Yield to maturity (YTM) per period (expressed as a decimal)

P = Current price of the bond

Question 4

equation for Modified Duration?

Answer 4

D mod = D/ 1+y

D = duration
y = yield to maturity (YTM) per period

Question 5

Equation for current yield?

Answer 5

Current Yield (interest yield, income yield, or flat yield)
The ratio of the annual interest (coupon) payment of the Bond over its current market price.

𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑌𝑖𝑒𝑙𝑑= 𝐶𝑜𝑢𝑝𝑜𝑛 / (𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑚𝑎𝑟𝑘𝑒𝑡 𝑝𝑟𝑖𝑐𝑒)

Question 6

What is the gordan growth model equation?

Answer 6

p = D / R - g

Question 7

What is the ROE equation?

Answer 7

𝑅𝑂𝐸=

𝐸𝑃𝑆 (𝑜𝑟 𝑁𝑒𝑡 𝐼𝑛𝑜𝑚𝑒)
/
𝐵𝑜𝑜𝑘 𝐸𝑞𝑢𝑖𝑡𝑦 𝑃𝑒𝑟 𝑆ℎ𝑎𝑟𝑒 (𝑜𝑟 𝐵𝑜𝑜𝑘 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝐸𝑞𝑢𝑖𝑡𝑦)

Book Value of Equity – net worth of the firm according to the balance sheet (assets minus liabilities)

Book Equity per Share – book value divided by the number of shares (preference and ordinary)

EPS – Total Earnings (Net Income) divided by the number of shares outstanding

Question 8

Holding Period Return (HPR) equation?

Answer 8

𝐻𝑃𝑅=”Capital Gain Yield + Dividend yield

Example
Expected Ending Price at year 1 (𝑷_𝟏) = £110; Beginning Price (𝑷_𝟎) = £100; Expected Dividend at year 1 = £4

HPR =(𝑃_1−𝑃_0)/𝑃_0 +𝐷_1/𝑃_0 =(“£110” −” £100” )/”£100” +(“£” 4)/”£100” =10%+4%=14%

Question 9

what is this equatiion on the sheet 𝐸(𝑅) = ∑𝑝𝑖 × 𝑅𝑖?

Answer 9

Expected Return =
𝐸(𝑅) = ∑𝑝𝑖 × 𝑅𝑖

where 𝑝_𝑖 is the probability of each scenario.

Question 10

what is the Estimated Variance equation?

Answer 10

(𝜎^2 )̂=

1/(𝑛−1) ∑ (𝑡=1)^𝑛 [R𝑡− R̃̅] ^2

Question 11

what is the utility function equation?

Answer 11

𝑈=𝐸(𝑟) − 0.5 𝐴𝜎²

where U = Utility
E(r) = Expected return on the asset or portfolio
A = Coefficient of risk aversion
σ2 = Variance of return
½ = A scaling factor

Question 12

what is the sharpe ratio and Sortino measure ?

Answer 12

Sharpe ratio: ( ¯r𝑝− ¯𝑟f) / 𝜎𝑝

Portfolio average excess return over a sample period, divided by the standard deviation of excess returns over the same period

Sortino measure is almost identical except that the denominator is downside risk (i.e., the standard deviation of negative returns).

Question 13

what is the Treynor Measure ?

Answer 13

Treynor Measure: ( ¯𝑟𝑝− ¯𝑟𝑓) /𝛽𝑝

Similar to Sharpe, gives the portfolio’s excess return per unit of risk, but it uses systematic risk instead of total risk (𝜎_𝑝)

Question 14

what is

𝛼𝑝 = 𝑟𝑝 − [ 𝑟𝑓 + 𝛽𝑝 ( 𝑟𝑀 − 𝑟𝑓) ]

Answer 14

Jensen’s (1968) Measure is the average return on the portfolio over and above that is predicted by the CAPM, given the portfolio’s beta and the average market return.

𝛼𝑝 = 𝑟𝑝 − [ 𝑟𝑓 + 𝛽𝑝 ( 𝑟𝑀 − 𝑟𝑓) ]

α p = Jensen’sAlpha
(themeasureofexcessreturn)

R p = Actualreturnoftheportfolio

R f =Risk-freerateofreturn

𝛽𝑝 = Portfoliobeta (systematicriskmeasure)

𝑅𝑚 =Marketreturn

Question 15

what is the Information Ratio ?

Answer 15

InformationRatio=
– 𝛼𝑝 ∕ 𝜎 (𝑒𝑝)

Where:

𝛼𝑝 = Jensen’s Alpha
(the measure of excess return)
​
𝜎 (𝑒𝑝) = Standard deviation of the tracking error
(i.e., the standard deviation of the portfolio’s excess return over the benchmark)

Question 16

What is S= (E(rp) - rf) / 𝜎𝑝 on the formula sheet?

Answer 16

The slope of CAL is the reward-to-volatility ratio, i.e., the Sharpe ratio (S):

S= (E(rp) - rf) / 𝜎𝑝

S = Sharpe Ratio — It measures the risk-adjusted return of an investment. A higher Sharpe ratio indicates a better risk-adjusted performance.

E(r p) = Expected Return of the Portfolio — This is the anticipated average return of the investment portfolio over a certain period.

r f = Risk-Free Rate — This is the return on a risk-free investment, typically represented by government bonds like U.S. Treasury bills.

σ p = Standard Deviation of the Portfolio’s Returns — It measures the volatility or total risk of the portfolio’s returns.

Question 17

What is the single factor model equation?

Answer 17

**𝑅𝑖= E(𝑟𝑖)+𝛽𝑖⋅𝐹+𝑒i
**
Ri = return of security i.
E(ri) = expected return of security i.
βi = sensitivity to common macro factors.
F = impact of unanticipated macro events.
ei = impact of unanticipated firm-specific events.

𝜎𝑖^2=𝜎𝑚^2+𝜎^2 (𝑒𝑖)

Question 18

What is the equation for the single index model?

Answer 18

𝑟𝑖 − 𝑟𝑓 = 𝛼𝑖 + 𝛽𝑖 (𝑟𝑚 − 𝑟𝑓 ) + 𝑒i

Ri = return of security i.

According to the Single Index Model (SIM), the excess returns (ri) depends on;

αi – stock i’s expected return if the market’s excess return is zero

βi(rm- rf) – component due to fluctuations in overall market

ei – component due to unexpected firm-specific events (zero mean)

Question 19

What is the covariance inthe single index model (SIM)?

Answer 19

Cov(ri, rm) = 𝛽𝑖 𝛽𝑗 𝜎𝑚^2