A. OVERVIEW

Bullet (BLT): Concentrated CFs, M=6y. Barbell (BBL): Dispersed (2y+10y, wtd avg=6y).
D ≈ [%ΔP / Δr] = slope of price-yield curve. ==Cx = CF dispersion (higher dispersion → higher Cx, more curved). e.g. BBL==

Table Ex (r 0–5%):
BLT: D 5.8→5.58; BBL: D 5.72→4.94.

Benefits High Cx (D fixed): r↑: |ΔV_BBL| < |ΔV_BLT| (lower losses); r↓: |ΔV_BBL| > |ΔV_BLT| (higher gains).
Cost: Lower YTM (expensive).

#EXAMPLE

Portfolio A and Portfolio B have approximately the same CF yield (i.e. their CFs will translate into the same return when measured relative to the inception cost of the portfolio). Portfolio A and Portfolio B also have approximately the same Macaulay Duration, meaning that both portfolios will experience a similar percentage change in value in response to a one-time parallel shift in yields. Furthermore, since the Macaulay Duration matches the liability maturity, the value of the portfolios will change at the same rate as the liabilities, thus remaining immunized against interest rate risk. However, Portfolio B has a higher Convexity (or curvature), ==meaning that its Duration will change at a greater rate in response to changes in yield. As a result, Portfolio B’s duration will be less stable.== Therefore, an immunization strategy relying on Portfolio B is more l==ikely to break down (as the portfolio’s duration match to the liability is less likely to hold moving forward).== ==The structural risk to the immunization strategy is the potential for non-parallel shifts and twists to the yield curve, which lead to changes in the cash flow yield that do not track the change in the yield on the zero-coupon bond. This risk is minimized by selecting the portfolio with the lower convexity (and dispersion of cash flows)== Note that ==default risk is neglected in this discussion because the portfolio consists of government bonds that presumably have default probabilities approaching zero.==

TIP: ==Cx = insurance (cost) vs σ_r↑ (pays off); σ_r↓ → underperforms.==
Logic: Higher Cx → D changes faster.

B. MANAGING A SINGLE LIABILITY

Symbolic Architecture:
==Imz: V_PF ≥ P_Liab, Hedge against i-rate risk== ==(MacD_PF = H (locks return vs parallel Δr)).==
Best: Zero-coupon matching (==no RRR, no price risk==).
==Structural Risk==: ==Non-parallel twist exposure → minimize CF dispersion around Horz (bracket Liab CFs).== #IMP Rebalance: MacD ↓ with time (approach M) & r↑. - ==if Mac D M H, sell low D bonds, buy high D bonds==

Numerical Ex (Twist, 6y Liab, BBL 50/50 2y+10y, MacD=6):
Parallel +1%: %ΔV_PF = +6%, %ΔV_L = +6% → Net 0.
Twist (short –50bp, long +50bp): %ΔV_PF = [0.5×2×(-0.5) + 0.5×10×0.5] = –1.5%, %ΔV_L = 0 → Equity loss.
 #EXAMPLE

Structural risk is reduced by: • ==Minimizing the dispersion of cash flows which is done by minimizing the convexity.== This entails going from a barbell design to a bullet design. 2 Reasons

• We can see that portfolio C has the highest convexity factor at 51.30 of the 3 portfolios leading to the highest dispersion of bond positions.
• We can see that portfolio C has a definite barbell structure with bond maturities of 1 year and 12 year compared with the two other portfolios leading to the ==highest dispersion of bond positions.==

#EXAMPLE

• This portfolio has the ==highest convexity at 140 leading to the highest structural risk==. Or Immunization condition not met as the ==Macaulay duration of 3.50 does not match==.
• This portfolio’s market value of ==$432,000 is less than the liabilities PV of $445,000 therefor does not meet an immunization condition==.

#EXAMPLE #imp

Candidates: When a question asks if ==zero-replication== has been achieved  we’re asking if interest rate immunization has been achieved (does duration match?). Also, recall structural risk leads to ==immunization not being achieved due to cash flow yield not matching YTM of a zero-coupon bond.== In this question, we are given that the cash flow yield of 3.20% matched the required investment return (3.15). We need to discuss both these facts.

Solution: For zero-replication, we need to determine if the portfolio’s Macaulay duration matches the liabilities duration of 4. Pf Mac D = (0.05 × 1.90) + (0.05 × 3.20) × (. 40 × 3.80) + (. 50 × 4.45) = 𝟒. 𝟎𝟎 We also have return characteristics of 3.20% that meet the liability needs of 3.15% required by the single liability. With the cash flow yield matching the required investment return, it is likely that the portfolio meets the required rate of return. ==Since we have duration and return characteristics that are the same, zero replication is achieved meaning interest rate risk has been immunized==

#EXAMPLE

Considering Ashley expects a greater decline in interest rates to 2.5% compared to Chris who expects the rate only to decline to 2.9%, the hedging ratio should be increased. ==With a decline in interest rates liability BPV >> asset BPV with the disparity growing more for larger declines in interest rates. An increase in hedging ratio will assist in closing the duration gap.== Note: Do not need to show this but the PBO BPV is: 𝑷𝑷𝑩𝑩𝑷𝑷 𝑩𝑩𝑷𝑷𝑽𝑽 = 𝟒𝟒. 𝟐𝟐 × 𝟑𝟑. 𝟒𝟒𝟓𝟓𝑩𝑩 × 𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 = 𝟐𝟐, 𝟐𝟐𝟎𝟎𝟑𝟑, 𝟒𝟒𝟎𝟎𝟎𝟎 We’re given Asset BPV is 675,000 which is much, much less. remember this B/S structure 

C. MANAGING MULTIPLE LIABILITIES USING BONDS

#EXAMPLE

Company wants to use excess cash in order to retire its debt. Strategies available:

1. Tender Offer: Buy back its bonds in the open market. However, this would almost certainly mean having to pay a premium in order to induce all bondholders to relinquish their bonds.
2. Cash Flow Match (Defeasement): Use excess cash to purchase Government bonds whose cash inflow would exactly match (and therefore offset) the CFs due on the firm’s debt. ==This would allow the firm to remove the debt (and the matching portfolio) from its balance sheet.== However, since Government bond yields will be less than corporate yields, the firm would be required to purchase far more Government bonds than the balance outstanding on its debt. ==In effect then, this strategy is similar to paying a premium in order to retire the debt (just like a tender offer).==
3. Duration Matching: Use the excess cash to construct a portfolio with the following parameters:
4. i. ==Basis Point Value (BPV) of the portfolio matches that of the liability.== This will ensure that the portfolio value will change by the same amount as the value of the liabilities in ==response to a parallel change== in interest rates.
5. ii. ==The portfolio’s convexity is slightly higher than that of the liability==. This will ensure that the portfolio will respond relatively better than the liabilities to non-parallel shifts in the yield curve. Part 1: ==Duration matching will require less capital (and therefore is the less expensive option). The manager has the flexibility to purchase any number of bonds, so long as the duration matching parameters are met. For example, the manager can match the BPV of the liability while still purchasing higher yielding bonds (thus reducing the amount of capital investment required)==. However, since the bond CFs would not be an exact match to the liability payments, the ==immunization will not be perfect.== As well, unlike the case of defeasement, ==the firm would still be required to report its debt on its balance sheet== (as the offset between the portfolio and the debt would not be perfect)

Part 2: Portfolio B best suits the duration matching strategy. It meets the 2 conditions:  ==It’s BPV of 64.51 closely matches the liability BPV of 64.47==. In other words, if there was a 1bp increase in interest rates, both the portfolio and liability would decrease by the same amount (approximately ¥64.5M)  ==Portfolio B’s convexity of 50.11 slightly exceeds that of the liability convexity (46.08).==

Strategies: ==CF Matching (exact CF match: no RRR/price risk) / Duration Matching / Contingent Imz.==

CF Matching: Select Bonds whose CFs exactly match timing and amount of liability CFs Duration Matching: ==BPV_PF = BPV_L== (BPV = D × V × 0.0001).
 Numerical Ex (PUL=100M, DL=6; UPF=120M, surplus=20M):
Step 1: D_PF calc → (BPV_PF / UPF) = (BPV_L / PUL) → D_PF = 5.
Step 2: +1bp: ΔUPF = –5 × 120M × 0.0001 = –60k; ΔPUL = –6 × 100M × 0.0001 = –60k → ΔSurplus = 0.
 Contingent Imz: VPF > PVL → active max E(R); = → passive D_PF=DL (==needs liquidity of assets==).

#EXAMPLE

For multiple liability immunization:

1. T==he market value of the assets is greater than or equal to the market value of liabilities.==
2. The money duration or BPV of the assets and liability must match
3. The convexity and dispersion of cash flow statistics for assets are greater than that for liabilities. For all three portfolios, the BPV matches between the assets and liabilities and this criteria is met.

==Common mistakes: • Incorrectly referenced the Macaulay duration justifying the incorrect option. Macaulay duration is used for the single liability case whereas money duration or BPV is used for multiple liability case. BPV matches between A and L.== The convexity of assets is less than the liabilities ==(41.44 < 43.4).==

==Conditions to immunize multiple liabilities:== ==1. MV(Assets) ≥ MV(Liabilities)== ==2. Asset BPV = Liability BPV== ==3. a)Dispersion of CFs & b) convexity of assets > those of liabilities== ==Money duration = Modified duration × Market value== ==BPV = Money duration × 0.0001== ==Contingent Immunization: adds active mgmt of surplus (assets − liabilities) to reduce cost of retiring liabilities. Any asset class usable. Entity can over-hedge or under-hedge vs. passive immunization.==

D. LIABILITY-BASED STRATEGIES USING BOND FUTURES

Framework: ==BPV_PF + (BPV_CTD / CVF) × N_f = BPV_L.==

Numerical Ex (BPV_PF=15k, BPV_L=80k; CTD BPV=50, CVF=0.7 → BPV_F≈71.43):
Step 1: Long futures (BPV_PF < BPV_L).
Step 2: N_f = (80k – 15k) / 71.43 ≈ 910 contracts long. ==(BPV_f = BPV_ctd / cvf = $71.43 per100k contract)==

Notes: Higher BPV_F → fewer N_f. ==Overlay → barbell (dispersed vs Liab).== Daily margin.

#EXAMPLE  #EXAMPLE Overhedge vs Underhedge  #EXAMPLE 

From part C, we calculated a short position of 64 futures contracts to close the duration gap. ==Jay has decided to over-hedge and take an additional position on the short side of 35 contracts (64 to 99 contracts). His belief must be that interest rates are rising to take a larger short position in futures contracts.==

E. LIABILITY-BASED STRATEGIES USING SWAPS

Framework: BPV_PF + BPV_Swap = BPV_L (BPV_Swap ≈ 0.0020 per 100 NP per bp).

Risk: Interest Rate falls   Numerical Ex (BPV_PF=15k, BPV_L=80k):
Step 1: Short swap (gain when r↓ offsets PF loss).
Step 2: NP = (80k – 15k) / 0.0020 = 32.5M short.

Strategic Hedging:
==Expect r↓ → overhedge (e.g., 97.5M = 3×): larger gain.==
==Expect r↑ → underhedge (e.g., 16.25M = ½): smaller loss offset.==

#EXAMPLE Hedging Ratio  Audit:
Deadly Trap: Pure hedge → ΔEquity=0; strategic varies NP for view.
Logic: Always maintain short position base. [600](PMP - Liability Driven Strategies)

==IR swap overlays reduce duration gap (measured by asset & liability BPVs). Large gap common because pension funds hold equities (low/zero duration) while liabilities have high duration.==

#imp ==Hedging ratio = % of duration gap closed w/ derivatives. 0% = no hedging; 100% = immunization (full removal of IR risk).==

F. LIABILITY-BASED STRATEGIES USING SWAPTIONS ##REVIEW

If Hedging using swaps, the entire duration gap can be eliminated - fully hedged. Bit ==what if mgr wants to eliminate downside expsosure to rate changes==  Swaption: Right (pay premium). Receiver SO: gain r↓; Payer SO: gain r↑.

Collar (zero-cost): Long Receiver 4% + Short Payer 6%.
Case r=3% (<4%): Pension loss offset by Receiver gain.
Case r=7% (>6%): Pension gain offset by Payer loss.
Case 4–6%: Exposed (varies).
 #EXAMPLE #imp

Receive-fixed swap With a rate increase up to 3.4%, the receive-fixed swap will begin losing immediately. Above is all that is required for full marks. Additional understanding: • If rates drop to 2.5%, the receive-fixed swap is the most profitable. Receiver swaption : With a rate increase up to 3.4%, ==the receiver swaption is at a loss due to the upfront premium.== Above is all that is required for full marks. Additional understanding: • If rates increase higher than 3.45%, the receiver swaption loses more than then swaption collar. Swaption collar: Additional understanding: • With the forecast, if rates do increase and are capped at 3.4%, the swaption collar will not have losses whereas the receivefixed and receiver swaption will have losses. • The swaption collar profits when rates fall below 3.0

G. EVALUATING THE LIABILITY-BASED STRATEGIES

Choice by View (same 15k/80k gap):
Confident r↓ → full hedge (futures/swap): no exposure, ==no premium.==
Downside only → Long Receiver SO: premium cost.
Reduce distribution (==expected to reach threshold??==)→ Collar: lower cost, narrowed range.

H. RISKS ASSOCIATED WITH LIABILITY-DRIVEN STRATEGIES

LDI Framework: ==BPV_PF + BPV_Hedge = BPV_L.==

#EXAMPLE #imp ==Risks:==

1. Non-parallel (unique Δr per element: PF/H/L).
2. Linearity (==BPV ignores Cx==).
3. D calc (sum parts vs whole PF; ==equal only if flat yc==).
4. Model to calc Pension Liability(PVL assumptions: benefits/timing/life exp).
5. Credit (OTC counterparty). ==Spread risk: assumes equal Δ in asset, liability, & hedging instrument yields, but spreads to benchmark can vary over time.==
6. Futures (daily margin/reinvest).
    #EXAMPLE

Due to the aggressive position in ==equity and alternatives (whose duration is 0)==, a significant duration gap exists between assets and liabilities. ==Only large positions in futures contracts will== ==in affect reduce the gap and these large positions lead to significant daily cash inflows and== ==outflows.== Interest rate swaps are preferable here.

I. BOND INDEXES

Adv: Low fees, diversification, low underperf risk.
The primary advantages of ==running a portfolio to mimic an index are as== follows: i) Lower management fees (due to the absence of active analysis). ii) Diversification (as most indexes provide broad coverage). iii) Reduced risk of underperformance (as the aim is to remain at par with the index)

==Approaches: Pure (all bonds exact weights) / Enhanced (sample matching risk factors) / Active (deviate from index primary risk factors)==.

Issues: ==OTC illiquid, valuation (matrix), rebalance (maturing), bums problem (value-wtd large debtors).==

Pure Indexing impractical (tx/illiquidity) → better to use Enhanced (match D, KRD, sector/quality/spread D, embedded options, issuer).

#EXAMPLE Risk Factors

The Paisley Fixed-Interest Fund represents the lowest risk of the three fund choices, ==given both its choice of underlying bond index (half of which is in short-term securities) and lack of active management strategies==. The Paisley Fund could be a suitable choice for an investor near retirement who is seeking income with a ==minimum risk profile==.

The Dundee Fund represents a ==medium risk profile given the choice of the composite benchmark and suggests an enhanced approach to indexing.== This fund may be the best choice for a middle-aged worker seeking to add a fixed-income component with ==moderate risk to his portfolio.==

The Newcastleton Fund has the highest risk of the three choices and is an example of an ==actively managed fund that has a mandate to take positions in primary risk factors, such as duration and credit, that deviate== from those of the of the index to generate excess return. This fund could be an ==appropriate choice for a younger worker who is seeking exposure to fixed income but willing to accommodate higher risk==

J. METHODS FOR ESTABLISHING BOND MARKET EXPOSURE

Index is hypothetical,but portfolio needs to purchase securities. ==To offset costs:== Yield Enhance: Algos, ==tilts (undervalued, sector, callable if vol↓)==, ESG.
Tools:
==Index replication: buy/sell bonds only when index changes. Full replication can be expensive & infeasible for broad indexes w/ illiquid bonds.== ==Enhancement strategies to ↓ tracking costs: ↓ trading costs, use models to identify undervalued bonds & gauge relative value along yield curve, over/underweight credit sectors over business cycle, evaluate call features for value given large yield changes.== ==Passive exposure via ETFs (trade on exchange throughout day) or mutual funds.== ==TRS: OTC derivative allowing institutional investor to transform asset/liability from one category to another (e.g., variable-rate CFs referencing MRR → total return on a bond index). Requires less initial cash outlay for similar performance. Carries counterparty credit risk.==

K. BENCHMARK SELECTION

Qualities: ==Clear inclusion/removal, weighting, investable, daily value + history.==
Issues: Maturity drag (D shortens), sector shortages, bums problem.
Solution: ==Multi-index blend for target D.==
#EXAMPLE Index Selection

Given this investor’s ==high degree of risk aversion, an index with short or== ==intermediate duration with limited credit risk would be most appropriate to== ==limit market value risk.== Of the alternatives listed, the S&P Eurozone Sovereign Bond 1–3 Years Index or the FTSE 1–3 Years Pfandbrief Index (given the high credit quality of covered bonds) would be appropriate choices.

This investor’s ==very long investment horizon== suggests that the BERC is an appropriate index, ==because it has the longest duration== of the indexes given. In addition, the long-term S&P Eurozone Sovereign Bond or FTSE Pfandbrief indexes (10+ years) could be appropriate choices as well. Next Europe should ==consider the trade-off between duration and risk in its discussion== ==with the endowment.==

The Danish life insurer faces two types of future obligation, ==namely a shortterm outlay for expected claims and a long-term horizon for future obligations==. For the ==short-term exposure, stability of market value is a primary consideration, and the insurer would seek an index with low market risk.== Of the above alternatives, the 1–3 Years S&P Sovereign Bond or the FTSE Pfandbrief 1–3 Years would be the best choices. The longer-term alternatives in the Solution to 2 would be most appropriate for the long-term future obligations.

Audit:
Deadly Trap: Single index operational flaws.

==Key decision for FI investment manager. Guided by goals & objectives.== ==Features to recognize:== ==1. Finite bond maturities → index duration drifts down over time== ==2. Index composition changes over time w/ business cycle & issuer maturity preferences==