FORMULAS FAIB
Inflation-linked bonds
Nominal coupon: 100× 1 + 5% ×3% = 3.15
100× 1 + 5% * ×3% = 3.31, etc.
+,,× +-.% ×/%
Real coupon: =3
+-.%
+,,× +-.% 0 ×/%
= 3, etc.
+-.%
𝐵𝑟𝑒𝑎𝑘𝑒𝑣𝑒𝑛 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 = 𝑇𝑏𝑜𝑛𝑑 𝑌𝑇𝑀 − 𝑇𝐼𝑃𝑆 𝑌𝑇𝑀
𝑅𝑒𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝑇𝑟𝑒𝑎𝑠𝑢𝑟𝑦 − 𝑅𝑒𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝐶𝑎𝑠ℎ ≈ 𝑇𝑒𝑟𝑚 𝑃𝑟𝑒𝑚𝑖𝑢𝑚 +
𝐼𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝑅𝑖𝑠𝑘 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
𝑇𝐼𝑃𝑆 𝑅𝑒𝑡𝑢𝑟𝑛 − 𝑅𝑒𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝐶𝑎𝑠ℎ ≈ 𝑅𝑒𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
𝑅𝑒𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝑇𝑟𝑒𝑎𝑠𝑢𝑟𝑦 − 𝑇𝐼𝑃𝑆 𝑅𝑒𝑡𝑢𝑟𝑛 ≈ 𝐼𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝑅𝑖𝑠𝑘 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
𝜇 PQR = 2% + 𝛾Σ𝑤WXYZ[\,PQR
𝜇 W^^_` = 2% + 𝛾Σ𝑤WXYZ[\,W^^_`
Harmonized Savings Plan
1. Invest in own company’s stock:
a. Companies have incentives (increase employee effort, financing, shareholder base).
b. Risks for employees:
i. Idiosyncratic risk: 𝑉𝑎𝑟 𝑟Z,b = 𝛽Z* 𝑉𝑎𝑟 𝑟d,b + 𝑉𝑎𝑟 𝜀Z,b
Idiosyncratic Variance
Total Variance Systematic Variance:
compensated for with higher
expected return
ii. Underdiversification.
2. Active versus passive: performance active management measured by alpha in four factor
model: 𝑟Zb = 𝛼Z + 𝛽Z,+ 𝑀𝑎𝑟𝑘𝑒𝑡b + 𝛽Z,* 𝑆𝑀𝐵b + 𝛽Z,/ 𝐻𝑀𝐿b + 𝛽Z,i 𝑀𝑂𝑀b + 𝜀Z,b , corrected for
fees and expenses.
a. Expense ratios: low-cost funds beat high-cost funds.
i. Dollar invested: 1 − 𝑥 1 + 𝑟
ii. Funds final value: 1 − 𝑥 l 𝐺l
+opq r srq +opq l srq
b. Terminal wealth ratio: 𝑇𝑊𝑅 = =
+op0 r sr0 +op0 sr0
srq
i. = 1 if investments have same returns.
sr0
+opq
c. Retention ratio:
+op0
3. Assets to include: equities, money market (no inflation risk, reinvestment risk), long-term
bond (lower reinvestment risk, inflation risk) à inflation-linked bond.
4. Flexibility for participants:
a. Default allocation, participants can deviate (limited flexibility, full flexibility).
b. Individuals in homogeneous groups, default allocation for groups.
5. Allocation changes over time:
a. Market timing: avoid bear markets and exploit bull markets.
i. Efficient market hypothesis: security prices rapidly reflect new information.
Wachter and Warusawitharana: predictive information contained in
dividend yields and term spread can be beneficially used to optimally vary
stock-bond allocation.
Inflation-linked bonds
Nominal coupon: 100× 1 + 5% ×3% = 3.15
100× 1 + 5% * ×3% = 3.31, etc.
+,,× +-.% ×/%
Real coupon: =3
+-.%
+,,× +-.% 0 ×/%
= 3, etc.
+-.%
𝐵𝑟𝑒𝑎𝑘𝑒𝑣𝑒𝑛 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 = 𝑇𝑏𝑜𝑛𝑑 𝑌𝑇𝑀 − 𝑇𝐼𝑃𝑆 𝑌𝑇𝑀
𝑅𝑒𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝑇𝑟𝑒𝑎𝑠𝑢𝑟𝑦 − 𝑅𝑒𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝐶𝑎𝑠ℎ ≈ 𝑇𝑒𝑟𝑚 𝑃𝑟𝑒𝑚𝑖𝑢𝑚 +
𝐼𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝑅𝑖𝑠𝑘 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
𝑇𝐼𝑃𝑆 𝑅𝑒𝑡𝑢𝑟𝑛 − 𝑅𝑒𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝐶𝑎𝑠ℎ ≈ 𝑅𝑒𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
𝑅𝑒𝑎𝑙 𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝑇𝑟𝑒𝑎𝑠𝑢𝑟𝑦 − 𝑇𝐼𝑃𝑆 𝑅𝑒𝑡𝑢𝑟𝑛 ≈ 𝐼𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝑅𝑖𝑠𝑘 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
𝜇 PQR = 2% + 𝛾Σ𝑤WXYZ[\,PQR
𝜇 W^^_` = 2% + 𝛾Σ𝑤WXYZ[\,W^^_`
Harmonized Savings Plan
1. Invest in own company’s stock:
a. Companies have incentives (increase employee effort, financing, shareholder base).
b. Risks for employees:
i. Idiosyncratic risk: 𝑉𝑎𝑟 𝑟Z,b = 𝛽Z* 𝑉𝑎𝑟 𝑟d,b + 𝑉𝑎𝑟 𝜀Z,b
Idiosyncratic Variance
Total Variance Systematic Variance:
compensated for with higher
expected return
ii. Underdiversification.
2. Active versus passive: performance active management measured by alpha in four factor
model: 𝑟Zb = 𝛼Z + 𝛽Z,+ 𝑀𝑎𝑟𝑘𝑒𝑡b + 𝛽Z,* 𝑆𝑀𝐵b + 𝛽Z,/ 𝐻𝑀𝐿b + 𝛽Z,i 𝑀𝑂𝑀b + 𝜀Z,b , corrected for
fees and expenses.
a. Expense ratios: low-cost funds beat high-cost funds.
i. Dollar invested: 1 − 𝑥 1 + 𝑟
ii. Funds final value: 1 − 𝑥 l 𝐺l
+opq r srq +opq l srq
b. Terminal wealth ratio: 𝑇𝑊𝑅 = =
+op0 r sr0 +op0 sr0
srq
i. = 1 if investments have same returns.
sr0
+opq
c. Retention ratio:
+op0
3. Assets to include: equities, money market (no inflation risk, reinvestment risk), long-term
bond (lower reinvestment risk, inflation risk) à inflation-linked bond.
4. Flexibility for participants:
a. Default allocation, participants can deviate (limited flexibility, full flexibility).
b. Individuals in homogeneous groups, default allocation for groups.
5. Allocation changes over time:
a. Market timing: avoid bear markets and exploit bull markets.
i. Efficient market hypothesis: security prices rapidly reflect new information.
Wachter and Warusawitharana: predictive information contained in
dividend yields and term spread can be beneficially used to optimally vary
stock-bond allocation.
