Summary COMM 221 FINAL FORMULA SHEET

𝑅 or 𝑘 = discount rate (𝑃1 − 𝑃0 ) + Σ𝐶𝐹 𝑩𝒆𝒕𝒂 𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒕𝒊𝒐𝒏𝒔:
Total Return =
𝑃0
𝐺 = Growth rate 1 𝐶𝑂𝑉 σstock
Periodic Average Return = (1 + 𝑇𝑅) 𝑡 − 1 β= = ρmarket,stock ×
σ2market σmarket
𝑁𝑃𝑉 = Inflows − Outflows 1 Beta of Portfolio = 𝑊𝑎 β𝑎 + 𝑊𝑏 β𝑏 + ⋯
Geometric Mean = [(1 + 𝑟1 )(1 + 𝑟2)(1 + 𝑟3 ) … ]𝑛 − 1
+ 𝑊𝑛 β𝑛
Net profit Ending Equity 𝑡 𝑹𝒊𝒔𝒌 𝑴𝒆𝒕𝒓𝒊𝒄𝒔: 𝑫𝒊𝒗𝒊𝒅𝒆𝒏𝒅 𝒂𝒏𝒅 𝑹𝒆𝒒𝒖𝒊𝒓𝒆𝒅 𝑹𝒆𝒕𝒖𝒓𝒏:
𝑅𝑂𝐸 = =( ) −1
Equity 𝐵𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝐸𝑞𝑢𝑖𝑡𝑦
oldf
Effective to effective = (1 + effective rate)newf − 1 Ex Post Standard Deviation = √Variance 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑
𝑃𝑉 =
𝐾
Corporate bond = Risk free + Risk premium Expected Return = Σ[𝑥 × 𝑝(𝑥)] 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑅𝑎𝑡𝑒 𝑜𝑛 𝑅𝑒𝑡𝑢𝑟𝑛 𝐾
= 𝑅𝑖𝑠𝑘𝐹𝑟𝑒𝑒 + β(𝐸𝑅𝑚 − 𝑅𝑖𝑠𝑘𝐹𝑟𝑒𝑒)
Face Value × Coupon Rate Variance = 𝐸(𝑥 2 ) − [𝐸(𝑥)]2 = [𝑝(𝑥) × (𝑥 − 𝐸𝑅(𝑥)) ] +
2 𝐴𝑙𝑝ℎ𝑎:
Coupon Payment =
Payments per year
𝐼 = 𝑃𝑉 × 𝑟 × 𝑔 𝑪𝒐𝒗𝒂𝒓𝒊𝒂𝒏𝒄𝒆 𝒂𝒏𝒅 𝑪𝒐𝒓𝒓𝒆𝒍𝒂𝒕𝒊𝒐𝒏: α = (𝑅𝑖 − 𝑅𝑓 ) − β(𝑅𝑚 − 𝑅𝑓 )

𝐹𝑉 = 𝑃𝑉(1 + 𝑟)𝑛 Covariance𝐴𝐵 = probability × (𝑅𝑎 − 𝐸𝑅𝑎 )(𝑅𝑏 − 𝐸𝑅𝑏 ) α > 0 ⇒ Outperformed market
= 𝐸(𝑅𝑎 𝑅𝑏 ) − (𝐸𝑅𝑎 × 𝐸𝑅𝑏 )
Total Interest = 𝐼 = 𝐹𝑉 − 𝑃𝑉 𝐶𝑂𝑉𝑎𝑏 α < 0 ⇒ Underperformed market
Correlation Coefficient =
𝜎𝑎 𝜎𝑏
𝐹𝑉 𝑷𝒐𝒓𝒕𝒇𝒐𝒍𝒊𝒐 𝑴𝒆𝒕𝒓𝒊𝒄𝒔: 𝑬𝒙𝒑𝒆𝒄𝒕𝒆𝒅 𝑼𝒕𝒊𝒍𝒊𝒕𝒚:
𝑃𝑉 = 𝐹𝑉(1 + 𝑟)−𝑛 =
(1 + 𝑟)𝑛
𝐴𝑃𝑅 𝑚 Expected Return of Portfolio = (𝑊𝑖 × 𝐸𝑅𝑖 ) Expected Utility = 𝑃𝑟1 𝑙𝑛1 + 𝑃𝑟2𝑙𝑛2 ,
𝐸𝐴𝑅 = (1 +
) −1 𝑤 = 𝑤𝑒𝑖𝑔ℎ𝑡 Then do 𝑒 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑢𝑡𝑖𝑙𝑖𝑡𝑦
𝑚
𝑚 = number of compound periods per year 𝐶𝐸𝑄: 𝑈 = 𝑊 𝑥
𝑚
𝐴𝑃𝑅 𝑓 Standard Deviation of 2-Asset Portfolio 𝑬𝒍𝒂𝒔𝒕𝒊𝒄𝒊𝒕𝒚:
𝑅periodic or 𝐸𝑄𝑅 or 𝐸𝑀𝑅 = (1 + ) −1 𝑄1 − 𝑄0/𝑄0 Δ𝑄 𝑃old
𝑚 = √𝑊𝑎2 𝜎𝑎2 + 𝑊𝑏2 𝜎𝑏2 + 2𝑊𝑎 𝑊𝑏 𝐶𝑂𝑉𝑎𝑏 η= ×
𝑓 = number of payments per year 𝑃1 − 𝑃0/𝑃0 Δ𝑃 𝑄old
𝑟old
Standard Deviation with perfect correlations Δ𝑄 𝑃avg
𝑅new = (1 + 𝑟old )𝑟new − 1 Arc Elasticity = ×
𝜌𝑎𝑏 = 1 ⇒ 𝜎𝑝 = 𝑊𝑎 𝜎𝑎 + 𝑊𝑏 𝜎𝑏 Δ𝑃 𝑄avg
𝑟old can also be 𝑘 = effective rate
𝜌𝑎𝑏 = −1 ⇒ 𝜎𝑝 = |𝑊𝑎 𝜎𝑎 − 𝑊𝑏 𝜎𝑏 | Δ𝑄𝐷
𝐸𝐴𝑌 = (1 + 𝐸𝑄𝑅 )𝑚 − 1
𝑄avg
𝜌𝑎𝑏 = 0 ⇒ 𝜎𝑝 = √𝑊𝑎2 𝜎𝑎2 + 𝑊𝑏2 𝜎𝑏2 η𝑑 =
Δ𝑃
𝑃avg
Continuous Compounding = 𝑒 apr×𝑛 − 1 Standard Deviation of a portfolio with a risk free asset 𝑬𝒍𝒂𝒔𝒕𝒊𝒄𝒊𝒕𝒚 𝑻𝒚𝒑𝒆𝒔:
1 σ𝑝 = 𝑊risky × σrisky
𝑛 = years, if days =
365
𝑹𝒊𝒔𝒌 𝑭𝒓𝒆𝒆 𝑹𝒂𝒕𝒆 𝒂𝒏𝒅 𝑪𝑨𝑷𝑴: Elastic (η > 1) 𝑃 ↑, 𝐸 ↓
𝑮𝒓𝒐𝒘𝒊𝒏𝒈 𝑨𝒏𝒏𝒖𝒊𝒕𝒚: Inelastic (η < 1) 𝑃 ↑, 𝐸 ↑
Unit Elastic (η = 1) Revenue Maximized
𝑃𝑀𝑇1 1+𝑔 𝑛 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑅𝑒𝑡𝑢𝑟𝑛 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜𝑝 = 𝑊𝑎 𝐸𝑅𝑎 + 𝑊𝑟𝑓 𝑅𝐹 𝑺𝒖𝒑𝒑𝒍𝒚 & 𝑫𝒆𝒎𝒂𝒏𝒅 𝑷𝒐𝒍𝒊𝒄𝒊𝒆𝒔:
𝑃𝑉growing annuity = 𝑃𝑉0 = (1 − ( ) )
𝑟−𝑔 1+𝑟 𝑆𝑡 𝑑𝑒𝑣 𝑜𝑓 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 σ𝑝 = 𝑊𝑎 σ𝑎
𝑃𝑀𝑇1 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑅𝑎𝑡𝑒 𝑜𝑓 𝑅𝑒𝑡𝑢𝑟𝑛 Price Ceiling: (Below Equilibrium)
𝑃𝑉growing annuity when 𝑟=𝑔 = 𝑃𝑉0 = ×𝑛 𝐸𝑅𝑚 − 𝑅𝐹
1+𝑟 Δ𝐶𝑆 = +𝐵 − 𝐷 Δ𝑃𝑆 = −𝐵 − 𝐸
𝐾 = 𝑅𝐹 + ( ) σ𝑝
σ𝑚 𝐷𝑊𝐿 = 𝐷 + 𝐸
𝐹𝑉growing annuity = 𝑃𝑉𝑡 = 𝑃𝑉0(1 + 𝑟)𝑡 𝑺𝒉𝒂𝒓𝒑𝒆 𝑹𝒂𝒕𝒊𝒐 𝑨𝒏𝒂𝒍𝒚𝒔𝒊𝒔: Price Floor: (Above Equilibrium)
Δ𝐶𝑆 = −𝐵 − 𝐷 Δ𝑃𝑆 = +𝐵 − 𝐸 𝐷𝑊𝐿
= 𝐷+𝐸
𝑮𝒓𝒐𝒘𝒊𝒏𝒈 𝑷𝒆𝒓𝒑𝒆𝒕𝒖𝒊𝒕𝒚: If Sharpe Stock > Sharpe Market Price Quota: (Before Equilibrium)
⇒ Under-priced (good buy) Δ𝐶𝑆 = −𝐵 − 𝐷 Δ𝑃𝑆 = +𝐵 − 𝐸
𝐷𝑊𝐿 = 𝐷 + 𝐸
𝑃𝑀𝑇 If Sharpe Stock < Sharpe Market Price Tax:
𝑃𝑉 =
𝑟−𝑔 ⇒ Over-priced (bad buy)
𝑫𝒊𝒗𝒊𝒅𝒆𝒏𝒅 𝑫𝒊𝒔𝒄𝒐𝒖𝒏𝒕 𝑴𝒐𝒅𝒆𝒍: If Sharpe Stock = Sharpe Market ⇒ Fairly priced Adjust to solve 𝑃 = 40 + 0.5𝑄, then add tax

𝐷1 𝒀𝒊𝒆𝒍𝒅𝒔 𝒂𝒏𝒅 𝑹𝒆𝒕𝒖𝒓𝒏𝒔: Find new S=D with tax; new Equilibrium P,Q
𝑃𝑉0 =
𝑟−𝑔 Consumer Price = 𝑃new
𝐷1 = 𝐷0(1 + 𝑔)𝑛 Σ𝐶𝐹 Producer Price = 𝑃new − Tax
Income Yield =
𝑃0 Gov Revenue = 𝐵 + 𝐶 = Quantity × Tax
Sharpe Ratio 𝑃1 − 𝑃0 𝐷𝑊𝐿 = 𝐸 + 𝐹
Expected Return − Risk Free Rate Capital Gain Yield = 𝐶𝐺𝑌 =
𝑃0
=
𝜎
𝐸𝑅 − 𝑅𝐹
=
𝜎