YC strategies ==use Treasury Bonds (pure interest rate play).== This avoids influence from credit and liquidity risk. ==Par yield curve: stylized representation of YTM at various maturities; often extracted from available bond yields via a model (not directly traded Framework: $$ \text{YTM (Corp)} = \text{YTM (Treasury)} + \text{Spread} $$

• Corp Bonds: exposed to Interest rate + Credit + Liquidity risk
• Treasury Bonds: exposed only to interest rate risk

Yield curve strategies structure a portfolio so that it profits from changes in the yield curve (Δyc).

A. Types of Yield Curve Strategies

There are three primary sources of yield curve change:

1. Change in Level (Parallel Shifts)

• Parallel Up: Entire yield curve shifts upward by the same amount.
• Parallel Down: Entire yield curve shifts downward by the same amount.

2. Change in Slope

Slope of the yield curve is typically measured by the 30-year minus 2-year spread: $$ \text{Slope} = \text{YTM}{30} - \text{YTM}{2} $$

• Steepening: Spread increases (long rates rise relative to short rates, or short rates fall relative to long rates).
• Flattening: Spread decreases.
• Inverted: ==When $\text{YTM}{30} < \text{YTM}{2}$ (negative spread).==

3. Change in Curvature (Butterfly)

Curvature is measured using the Butterfly Spread (body vs. wings): $$ \text{Butterfly Spread} = 2 \times \text{YTM}{\text{Middle (Body)}} - (\text{YTM}{\text{Short (Wing)}} + \text{YTM}_{\text{Long (Wing)}}) $$

• When the body rises above the wings (middle yields increase relative to short and long), ==the curve becomes more curved (positive butterfly).==
• The three sources of ΔYC (Level, Slope, Curvature) can occur simultaneously and are uncorrelated.

B. Factors Affecting Fixed Income Portfolio Return

Recall the Taylor expansion for bond price change:

$$ % \Delta P \approx (-D \times \Delta r) + \frac{1}{2} C \times (\Delta r)^2 $$

• Duration (D): Linear (first-order) approximation of %ΔP — larger effect.
• Convexity (C): Curvature (second-order) term — smaller effect but important for large moves or non-parallel shifts.

Example: Bond Characteristics

Strategy Comparison: Bullet vs. Barbell (Duration-Matched)

Strategy I: Bullet
100% in the 10-year bond → Portfolio Duration $D_p = 9.2$, Convexity $C_p = 80$ Strategy II: Barbell (2 yr + 30 yr)
Find weights such that $D_p = 9.2$ (matches bullet, same interest rate risk level).

$$ D_p = w_2 \times 1.8 + w_{30} \times 24.6 = 9.2 $$ Solving: $$ w_2 = 0.675, \quad w_{30} = 0.325 $$ Portfolio Convexity for Barbell: $$ C_p = (0.675 \times 11) + (0.325 \times 400) = 137.425 $$

Parallel Yield Change Scenarios (±50 bps)

Case 1: Parallel shift +50 bps ($\Delta r = +0.005$) $$ %\Delta P \approx (-D_p \times \Delta r) + \frac{1}{2} C_p \times (\Delta r)^2 $$

• Bullet: $(-9.2)(0.005) + \frac{1}{2}(80)(0.005)^2 = -0.046 + 0.001 = -4.5%$
• Barbell: $(-9.2)(0.005) + \frac{1}{2}(137.425)(0.005)^2 = -0.046 + 0.001718 = -4.428%$ Case 2: Parallel shift -50 bps ($\Delta r = -0.005$)
• Bullet: $(-9.2)(-0.005) + \frac{1}{2}(80)(0.005)^2 = +0.046 + 0.001 = +4.70%$
• Barbell: $(-9.2)(-0.005) + \frac{1}{2}(137.425)(0.005)^2 = +0.046 + 0.001718 = +4.772%$

Approach	Best For	Advantages
Portfolio Level (Find $D_p$ and $C_p$ first)	Quick risk assessment & scenario analysis	• Allows you to instantly stress-test the entire portfolio under multiple yield shifts ($\pm 50\text{ bps}$, $\pm 100\text{ bps}$) without re-calculating every single holding. • Standard industry practice for summary risk reporting.
Bond Level (Find individual $\Delta P_i$ first)	Non-parallel shifts & complex portfolios	• Necessary if the yield curve shift is non-parallel (e.g., short-term rates change by $+50\text{ bps}$ while long-term rates change by $+10\text{ bps}$). • Essential when individual bonds have embedded options (like callable or puttable bonds) where duration/convexity changes non-linearly with yield shifts.

Conclusion: In both scenarios, the Barbell outperformed the Bullet by ~7 bps.
Higher convexity is beneficial when large yield moves (volatility) are expected.

#EXAMPLE -2.407% = (-4.880 × 0.005) + [0.5 × 26.5 × (0.005²)]
-2.390% = 0.5894 × {[-1.994 × 0.005] + [0.5 × 5 × (0.005²)]} + 0.4106 × {[-9.023 × 0.005] + [0.5 × 90.8 × (0.005²)]}

#EXAMPLE ??

C. The Role of Convexity

Convexity ($C$) captures the curvature in the price-yield relationship. It measures the gap between the actual price and the linear (duration-only) estimate.

Advantages of Higher Convexity #EXAMPLE

• Price rises faster when yields fall.
• Price falls slower when yields rise.
• Acts as ==insurance== against yield volatility. The improved price performance outweighs the higher initial price paid.

Disadvantages of Higher Convexity

• Holding duration constant, ==higher-convexity bonds are more expensive (higher price → lower yield).==
• If yields do not move much, ==you overpaid for convexity you didn’t need.==

High Convexity vs. Low Convexity Strategies (Duration held constant)

Summary — When to prefer higher convexity:

• ==If you expect large yield volatility (Δr ↑ in magnitude).==
• ==Select: Barbell portfolios, longer-maturity bonds, or higher-coupon bonds.==

D. Carry Trade: Inter-Market

Goal: Borrow in a low-yield currency/market and invest in a high-yield currency/market. Expected profit = yield spread (leveraged strategy).

Example:

• UK yields: 6%
• Japan yields: 4%
• FX: ¥150 / £ If no change in yields or FX:
• Borrow in Japan at 4% → Invest in UK at 6% → 2% spread profit. Balance Sheet View:

Risks of Inter-Market Carry Trade

1. Interest rate risk: UK yields rise → UK bond prices fall → Assets ↓
2. FX risk:
   • £ depreciates → Assets (UK bonds in £ terms) ↓
   • Yen appreciates → Liabilities (Japan borrowing) ↑ in £ terms
3. Duration mismatch between assets and liabilities → increases interest rate exposure.
4. Uncovered Interest Rate Parity (UIRP) does not hold in practice. Yield differentials are not always offset by expected FX changes.

Alternative ==via Currency Swap:== Investor effectively:

• Holds UK bond (earns £ fixed)
• Issues Japan debt (pays ¥ floating/fixed) ==Exposes investor to same risk factors as spot carry trade.==

#EXAMPLE MXN carry trade ==90 day bills for 1 yr== 

E. Carry Trade: Intra-Market

Borrow and invest within the same market/currency (typically along the yield curve).

Example:

• Short 4-year bond @ 3% (liability)
• Long 10-year bond @ 5% (asset)
• Net spread/profit = 2%

Assumes no change in the yield curve. Risk = yield curve changes (especially steepening or level rise).

Alternatives to Spot Carry Trade

1. ==Interest Rate Swap==

• Investor receives fixed (long-term ==average yield==) and pays floating (LIBOR).
• Equivalent to: Long a long-term bond + Short (issue) short-term floating debt.
• ==Profit declines if rates rise: (Fixed received - Floating paid) decreases.==

#EXAMPLE #imp valueing swap Say a UK-based manager seeks to extend duration beyond an index by adding 10-year exposure

delta_R = coupon income + delta_P



SWAP OVERLAY #imp #EXAMPLE

In this example, the manager enters into a pay-fixed swap overlay with a notional principal equal to one-half of the size of the total bond portfolio. ==As the pay-fixed swap is a “short” duration position, it is a negative contribution to portfolio duration and therefore subtracted from rather than added to the portfolio.== Recall the $100 million “index” portfolio has a modified duration of 5.299, or (1.994 + 4.88 + 9.023)/3. If the manager enters a $50 million notional 5-year pay-fixed swap with an assumed modified duration of 4.32, the portfolio’s modified duration falls to 3.139, or [(5.299 × ==100==) – (4.32 × ==50==)]/==100==. Stated differently, the bond portfolio BPV falls from $52,990 to $31,390 with the swap. For a 25 bp yield increase, this $21,600 reduction in active portfolio BPV reduces the ==adverse impact of higher rates== by approximately $540,000 versus the “index” portfolio.

2. ==Long Bond Futures== Framework: $$ F = \frac{S_0 (1 + r)^T}{(1 + y)^T} \quad \text{(with income yield } y\text{)} $$

Example: Long 1-year futures on a 10-year deliverable bond.

• Effectively: Paying the short-term rate (cost of carry) and receiving the long-term yield.
• If upward-sloping yield curve and rates unchanged → profit (positive carry).

Summary — Strategies when upward-sloping YC and rates expected unchanged (i.e. carry trade):

• Spot: Short ST bonds → Long LT bonds
• Swap: Receive fixed → Pay floating
• Futures: Long bond futures

F. Strategies if Expect No Change in Yields

Scenario: Upward-sloping yield curve (LT > ST) + Expect yields to remain stable.

I. Buy & Hold Long-Maturity (Long Duration) Bonds

• Over a multi-year horizon, the higher yield of the long bond compounds to higher total return than rolling short-term bonds (which stay at low ST rates).

==More aggressive (levered) version — “Repo Carry”:==

• Short (borrow) 1-year bond @ 3%
• Long (invest) 10-year bond @ 5%
• Earn 2% spread with no net capital (levered carry trade).

#EXAMPLE The fall in long-term yields-to-maturity will lead to price appreciation under the “buy-and-hold” strategy. The difference between long-term and short-term yields-to-maturity in b will fall, ==leading to negative carry if short-term yields-to-maturity rise sharply.== As for c, ==higher short-term yields-to-maturity will enable the manager to reinvest bond coupon payments at a higher rate.==

II. Ride the Yield Curve

Example (1-year investment horizon):

Why it works:

• ==Lock in higher coupon.==
• As bond approaches maturity (“rolls down” the curve), ==its YTM falls → price rises (positive roll-down return).==
• Steeper curve = more roll-down profit.

III. Sell Convexity

• Higher-convexity bonds are expensive (lower yield).
• If you expect low volatility (no large Δr), sell high-Cx bonds and buy low-Cx bonds to capture the convexity premium (higher yield).

Ways to sell convexity:

• Long callable bonds (or short putable bonds)
• 
• Sell options directly (collect premium; expect them to expire worthless if low vol)

==Risk: If yields move sharply or volatility increases, short-convexity positions lose.==

#EXAMPLE inverted YC #imp

Buy and Hold Note – all three profit with upward sloping curve. Not required to explain this correct answer. ==With a buy and hold strategy, we extend our duration beyond our target. By constructing a longer duration portfolio – the fall in longer term YTM will lead to price appreciation.== Rolling down the Yield : With an inverted curve, we lose by selling the security at a higher yield which leads to price depreciation. Repo Carry Trade :The inverted curve leads to a negative carry where the difference between the ST YTM and LT YTM falls.

G. Strategies if Expect a Change in Yields

I. Duration Management

• ==Expect yields to fall → Lengthen portfolio duration (maximize price gains).==
• ==Expect yields to rise → Shorten portfolio duration (minimize price losses).== Issue: Modified duration assumes parallel yield curve shifts. ==For non-parallel shifts, use Key Rate Duration (KRD).==

Key Rate Duration (KRD)

KRD measures sensitivity to a specific point on the yield curve (non-parallel shifts). $$ % \Delta P_{PF} = w_1 D_1 \Delta r_1 + w_2 D_2 \Delta r_2 + w_3 D_3 \Delta r_3 + \dots $$ $$ \text{KRD}_i = w_i \times D_i $$  LONG SHORT - NON PARALLEL YC / KRD #imp #EXAMPLE 

NOT SURE IF THIS IS CORRECT - maybe just focus on mid and long end  Interpretation:

• Bullet portfolio: High KRD in the middle of the curve.
• Barbell portfolio: High KRD at the short and long ends.

Key Rate PVBP (Price Value of a Basis Point) is an alternative metric (absolute $ change per 1bp move at a specific maturity).

Example: Key Rate PVBP Analysis (per 100 par)

Non-parallel scenarios (focus on mid-to-long end; short-end KRD has low impact due to low duration): 

• ==Flattening → Prefer Barbell==

• ==Steepening → Prefer **Bullet

• A Duration-Neutral (Spread-Only) Steepener sets: $$\text{Money Duration}{\text{Long Front End}} = \text{Money Duration}{\text{Short Back End}}$$

  (Net Money Duration = 0; you only profit from the curve shape changing, not flat rate moves).

• A Directional Bull Steepener sets: $$\text{Money Duration}{\text{Long Front End}} > \text{Money Duration}{\text{Short Back End}}$$

  (Net Money Duration > 0; you profit from both falling rates and curve steepening).

1. The Duration-Neutral Steepener Example

• Goal: Insulate the trade from parallel rate shifts. Profit only if the curve steepens.
• The Sizing Math: You must match total dollar durations.
  • Short $1,000,000 of 10Y = -$70 DV01
  • Buy $3,500,000 of 2Y = +$70 DV01 ($3.5M × $20)
  • Net Portfolio DV01 = $0
• The Profit Mechanics:
  • If Bear Steepening: Overall rates go up. 10Y yield spikes by 10 bps; 2Y yield rises by only 2 bps.
    • 10Y Short makes: 10 bps × $70 = +$700
    • 2Y Long loses: 2 bps × -$70 = -$140
    • Net Profit: +$560
  • If Bull Steepening: Overall rates go down. 2Y yield plunges by 10 bps; 10Y yield drops by only 2 bps.
    • 2Y Long makes: 10 bps × $70 = +$700
    • 10Y Short loses: 2 bps × -$70 = -$140
    • Net Profit: +$560

1. The Duration-Neutral Flattener Example

• Goal: Insulate the trade from parallel rate shifts. Profit only if the curve flattens.
• The Sizing Math: You swap the legs but keep the exact same DV01 weighting.
  • Buy $1,000,000 of 10Y = +$70 DV01
  • Short $3,500,000 of 2Y = -$70 DV01
  • Net Portfolio DV01 = $0
• The Profit Mechanics:
  • If Bear Flattening: Overall rates go up. 2Y yield spikes by 10 bps; 10Y yield rises by only 2 bps.
    • 2Y Short makes: 10 bps × $70 = +$700
    • 10Y Long loses: 2 bps × -$70 = -$140
    • Net Profit: +$560
  • If Bull Flattening: Overall rates go down. 10Y yield plunges by 10 bps; 2Y yield drops by only 2 bps.
    • 10Y Long makes: 10 bps × $70 = +$700
    • 2Y Short loses: 2 bps × -$70 = -$140
    • Net Profit: +$560

#EXAMPLE 

#EXAMPLE

#EXAMPLE 

II. Convexity Management

If expecting large yield moves (high volatility):

• Increase portfolio convexity (more dispersed cash flows).
• Buy options (positive gamma/convexity).
• Go long putable bonds or short callable bonds.  Risk: Overpaying for convexity if volatility does not materialize.

#EXAMPLE  The value of a bond with an embedded option is equal to the sum of the value of an option-free bond plus the value to the embedded option. ==The bond investor can be either long or short the embedded option, depending on the type of bond. With a callable bond, the embedded call option is owned by the issuer of the bond, who can exercise this option if yields-to-maturity decrease (the bond investor is short the call option). With a putable bond, the embedded put option is owned by the bond investor, who can exercise the option if yields-to-maturity increase.==

Value of long PB increases less than OFB. ==But for short PB, the price decreases (invert) above pic==

#EXAMPLE Expect shift in YC upwards  a. Long a receiver swaption
b. ==Short== a payer swaption
c. Long a put option on a bond futures contract

C is correct - A put option increases in value as the yield curve shifts upward, while the price of the underlying bond declines below the strike. B is incorrect because a payer swaption is an option to pay-fixed/receive-floating in an interest rate swap. A long, not a short, position in a payer swaption would benefit from higher rates.

#EXAMPLE  For a putable bond, the bond investor has the option to “put” the bond back to the issuer if yields-to-maturity rise. The more rates rise, the more valuable this embedded option becomes. This increasing option value will partially offset the decline in value of the putable bond relative to the option-free bond. This can be seen in the lower panel of Exhibit 22: ==The dotted line for the putable bond has a flatter slope than the solid line for the option-free bond; its price will decrease more slowly as yields-to-maturity increase.==

For a callable bond, the bond issuer has an option to “call” the bond if yields-to-maturity decline; the more rates rise, the lower the call option value. ==Since the bond investor is short the embedded option and the value of the embedded option has fallen, this will partially offset the decline in the value of the callable bond relative to the option-free bond.== The top panel of Exhibit 22 shows that the ==dotted line for the callable bond has a flatter slope than the solid line for the option-free bond.== As rates continue to increase, the embedded option for the putable bond rises in value more quickly at the margin as it shifts toward becoming an inthe-money option. In contrast, the deep out-of-the-money embedded call option moves further out-of-the-money as rates increase and the marginal impact of further rate increases declines.

Exhibit 24: Long Option, Swaption, and Bond Futures Option Strategies

H. Use of Derivatives in Yield Curve Strategies

Derivatives allow overlay adjustments without selling illiquid bonds in the physical portfolio.

Example: Increase Portfolio PVBP (expect yields to fall)


1. Bond Futures Overlay

• Long bond futures (each contract PVBP = $180 per $100k notional).
• Number of contracts: $$ N_f = \frac{\text{Target PVBP} - \text{Current PVBP}}{\text{Futures PVBP per contract}} = \frac{10,000}{180} \approx 56 \text{ (scaled example)} $$

Note: Only hedges parallel shifts. Non-parallel moves create yield curve risk (residual exposure).  Notes:

1. ==Longer the term, greater the D, and hence lower NP==
2. ==Pf and Futures durations may be different - this exposed to YC risk if non-paarallel changes==

2. Interest Rate Swap Overlay

• Short the swap (pay fixed / receive floating) to increase effective duration/PVBP.
• Notional principal sized to achieve target PVBP.
• Longer swap tenor → higher DV01/PVBP per unit notional → smaller notional needed. 

Notes:

1. ==Longer the term, greater the D, and hence lower NP==
2. ==Pf and Swap durations may be different - this exposed to YC risk if non-paarallel changes==
3. ==Swaps OTC - counterparty risk==

3. Leverage

$$ \text{Leverage Ratio} = \frac{\text{Target PVBP}}{\text{Current PVBP}} = 1.25 $$

Balance sheet after leverage:

• Assets: 125% of capital, Duration = 4
• Liabilities (ST borrowing): 25% of capital, Duration ≈ 0
• Equity: 100% of capital
• Post-leverage Duration ≈ 5.0 (achieves target exposure) 

Exhibit 7: Derivatives-Based Static Yield Curve Strategies

I. Strategies for Changes in Yield Curve Slope

Use Key Rate PVBP (focus on mid and long segments; ignore very short end due to low duration impact).

Example:

• If you expect steepening (long rates rise more), reduce Key Rate exposure at the long end (lower long KRD).
• Position the portfolio accordingly using bullets/barbells or derivatives. 

#EXAMPLE


J. Strategies for Curvature Changes in Yield Curve

I. Expect Yield Curve to Become More Curved (-ve Butterfly)


• Long Butterfly (also called “long wings, short body”):
  • Long Barbell (short + long maturities — the “wings”)
  • Short Bullet (intermediate maturity — the “body”)  Parameters: Levered Strategy
• ==Overall portfolio duration = 0 (market-neutral to parallel shifts).==
• ==Net long convexity (barbell has higher Cx than bullet).==

Benefits:

• Profits if curvature increases (body underperforms wings).
• ==Also profits if curve flattens.==
• Benefits from ==rising volatility (higher Cx).==

Risks: Costs more (lower yield); underperforms if curvature does not increase or volatility falls.

II. Expect Yield Curve to ==Become Less Curved== (+ve Butterfly)

• Short Butterfly:

  • Long Bullet (body)
  • Short Barbell (wings)

• Net ==short convexity.==

• Profits if ==curvature decreases or volatility falls.==

• ==Also benefis if YC steepens==

III. Condor (Variation of Butterfly)


• Spreads the “body” across two intermediate points (e.g., 5yr + 20yr) instead of concentrating on one (e.g., 10yr).
• More ==conservative than butterfly: smaller profit/loss if curvature changes, because long and short cash flows are better matched.==
• ==Lower dispersion → lower net convexity vs. pure butterfly.==

K. Yield Curve Strategies Across Currencies

International investing offers diversification and higher return potential, but introduces FX risk.

Example: UK investor buys 2-year zero-coupon US Treasury (currently trading at 95, par $100m, net value $95m).

• Spot FX: $1.20 / £ (or £0.8333 / $)
• Cost in £ today: £79.167m

After 1 year (assume):

• US Treasury value: $97.2m
• Expected FX: $1.14 / £ (£0.8772 / $)
• Proceeds in £: £85.263m
• Total Return in £: +7.7%  Return Breakdown:
• US Treasury local return: +2.315%
• FX return (£ appreciation): +5.268%
• Approximate: $R_{DC} \approx R_{FC} + R_{FX} = 7.583%$
• Exact: $(1 + R_{DC}) = (1 + R_{FC})(1 + R_{FX})$ → $R_{DC} = 7.7%$

Risks (Unhedged):

• ==US Treasury price risk (sold before maturity).==
• ==FX risk on principal and any coupons.==

Hedging:

• ==Sell FX forward==
• Use currency swap (==synthetic foreign bond exposure==)

Hedging with Currency Swap:

• Creates synthetic exposure.
• In practice, a ==basis spread exists between direct and synthetic (swap) foreign yields.==
• If basis is positive, synthetic may be cheaper or more expensive depending on sign. 

Interest Rate Parity (IRP) implies that risk-free rates across currencies, once hedged for FX, should be equal. Deviations create arbitrage or carry opportunities (subject to basis risk). Exhibit 30: Active Cross-Currency Strategies

L. Evaluating Expected Return of Yield Curve Strategies

Example: 3-year, 5% coupon bond trading at par (YTM = 5%). 1-year investment horizon.

Expected yield changes over next year:

• Roll-down (if curve shape unchanged): Δr (roll) = -10 bps (YTM falls as bond shortens).
• Constant-maturity change: Δr (CM) = +30 bps (yields at the remaining maturity expected to rise).

Total expected yield change: $$ E(\Delta r) = -0.10% + 0.30% = +0.20% $$

==Framework== for Expected Return

$$ E(R) = \text{Income} + % \Delta P_{\text{roll}} + % \Delta P_{\text{CM}} $$

Step-by-step calculation (detailed in notes):

• Income (coupon): +5%
• Roll-down price change: +0.186%
• Constant-maturity price change (using duration ≈ 1.9 at t=2): -0.57%
• Total E(R) ≈ 4.616%  Alternative (total ΔP approach) yields similar result (~4.63%).

Note on FX: ==For foreign bonds, further adjust E(R) by expected %ΔFX (appreciation adds, depreciation subtracts).== #EXAMPLE 

#EXAMPLE 

Quick Reference Summary

==Higher yielding currency will trade at forward discount==