Module 70 — Fixed Income Mathematics & Bond Markets

Yield curve construction and bootstrapping, duration and convexity in depth, credit spread analysis, term structure models (Vasicek, CIR, Hull-White), mortgage-backed securities, and the fixed income markets of Pakistan and the Gulf — the quantitative foundation for rates trading, credit analysis, and portfolio management.

Learning Objectives

Price bonds and derive yield to maturity for any cash flow structure
Bootstrap a spot rate curve from observable market prices
Calculate and apply modified duration, DV01, and convexity for portfolio risk management
Decompose credit spreads using Z-spread and OAS
Understand the mechanics of mortgage-backed securities and prepayment risk
Navigate Pakistan PIBs, GoP Ijarah Sukuk, and Gulf fixed income markets

1. Bond Pricing Fundamentals

The Bond Pricing Equation

A bond is simply a series of promised cash flows. Its price is the present value of all future cash flows discounted at the yield to maturity (YTM):

P = C/(1+y) + C/(1+y)² + ... + C/(1+y)ⁿ + F/(1+y)ⁿ

Where:

P = bond price
C = coupon payment (= face value × coupon rate / payment frequency)
y = yield to maturity per period
n = number of periods
F = face value (par value)

Key relationships:

Price and yield move in opposite directions (when yields rise, prices fall)
A bond trading at par has coupon rate = YTM
A bond trading at a discount has coupon rate < YTM
A bond trading at a premium has coupon rate > YTM

Day Count Conventions

Bond prices depend on how interest accrues between coupon dates. Different markets use different conventions:

Convention	Used In	Calculation
Actual/365	UK gilts, Pakistan PIBs	Actual days / 365
Actual/360	US money markets, most Libor	Actual days / 360
30/360	US corporate/agency bonds	Each month = 30 days
Actual/Actual (ICMA)	Eurobonds, most government bonds	Actual / actual days in coupon period

Clean vs dirty price:

Dirty price (full price): what you actually pay — includes accrued interest
Clean price (flat price): quoted price — excludes accrued interest
Dirty price = Clean price + Accrued Interest

Accrued Interest = Face Value × Coupon Rate × (Days Since Last Coupon / Days in Coupon Period)

Pakistan PIBs and Government Securities

Pakistan Investment Bonds (PIBs) are the benchmark government securities:

Tenors: 3, 5, 7, 10, 15, 20, and 30 years
Coupon: semi-annual, fixed rate
Issued via SBP auction (primary market)
Secondary market: over-the-counter through SBP-approved primary dealers
Day count: Actual/365

GoP Ijarah Sukuk:

Islamic alternative to PIBs — Shariah-compliant
Structure: government sells state assets to SPV, SPV leases back to government
Return: semi-annual rental payments (equivalent to PIB coupon)
Eligible for SLR (Statutory Liquidity Requirement) compliance by banks

Gulf fixed income:

UAE: T-bills and FGS (Federal Government Securities) — AED-denominated
Saudi Arabia: SAMA bills, government sukuk — SAR-denominated
Qatar: T-bills, government bonds — QAR-denominated
All GCC sovereigns also issue in USD internationally

2. Yield Curve Construction

Types of Yield Curves

Curve Type	Definition	Use
Par yield curve	YTM on par bonds at each maturity	Most quoted in market
Spot (zero) curve	YTM on zero-coupon bonds — no reinvestment risk	Theoretically correct for discounting
Forward curve	Implied rates between two future dates	Pricing forward contracts, FRAs
Discount factor curve	Present value of PKR 1 at each maturity	Derivatives pricing, swap valuation

Bootstrapping the Spot Curve

Bootstrapping derives spot rates from observable par bond prices. Starting from the shortest maturity and working forward:

Example with 3 instruments:

6-month T-bill: Price = 96.12, YTM = 8.00% → 6-month spot rate = 8.00%
1-year bond: Coupon = 8.5%, semi-annual, Price = 100 (at par)
- Uses 6-month spot rate (8.00%) for first coupon
- Solve for 1-year spot rate s₁:
```
100 = 4.25/(1.04) + 104.25/(1 + s₁)²
→ s₁ = 8.52% annualized
```
18-month bond: Coupon = 9%, Price = 100
- Uses 6-month spot (8.00%) and 1-year spot (8.52%) for first two coupons
- Solve for 18-month spot rate s₁₈

Continue until the full curve is bootstrapped.

Pakistan application: SBP publishes a yield curve based on PIB auction results. Primary dealers bootstrap the spot curve from observed PIB prices across tenors.

Yield Curve Shapes and Their Meaning

Shape	Description	Implication
Normal (upward sloping)	Long rates > short rates	Expected growth, inflation rising
Inverted	Short rates > long rates	Expected recession — reliable US recession predictor
Flat	All maturities similar yield	Transition or uncertainty
Humped	Medium term highest	Complex supply/demand dynamics

Pakistan yield curve: The PKR yield curve has been inverted or humped during high-rate IMF program periods (2019, 2023) as SBP raised the policy rate to 22% while the market expected eventual cuts.

Forward Rates

The forward rate f(t₁, t₂) is the rate implied by spot rates for borrowing/lending between times t₁ and t₂:

(1 + s₂)^t₂ = (1 + s₁)^t₁ × (1 + f(t₁,t₂))^(t₂-t₁)

Intuition: If you can lock in 1-year money at 8% and 2-year money at 9%, the 1-year forward rate one year from now is approximately 10% — the market's best estimate of where 1-year rates will be in 12 months.

3. Duration & Convexity

Macaulay Duration

Macaulay Duration (D_mac) is the weighted average time to receive the bond's cash flows, where weights are the present values of each cash flow as a share of the total bond price:

D_mac = (1/P) × Σ [t × CFₜ / (1+y)ᵗ]

Intuition: If a bond has Macaulay Duration of 4.5 years, the "average" investor gets their money back in 4.5 years. Zero-coupon bonds have duration equal to maturity.

Modified Duration

Modified Duration (D_mod) measures the percentage price change for a 1% change in yield:

D_mod = D_mac / (1 + y/m)

Where m = payment frequency per year.

%ΔP ≈ −D_mod × Δy

Example: A bond with modified duration 6.5 and yield rising 25 bps (0.25%) falls approximately 6.5 × 0.25% = 1.625% in price.

DV01 (Dollar Value of a Basis Point / PVBP)

DV01 = the dollar change in bond value for a 1 basis point (0.01%) change in yield:

DV01 = D_mod × P × 0.0001

DV01 is the primary risk metric for bond traders. A portfolio with DV01 of PKR 500,000 gains/loses PKR 500,000 for every 1bp move in rates.

Portfolio duration: Portfolio DV01 = sum of individual bond DV01s.

Convexity

Duration is a linear approximation. For large yield changes, convexity corrects the estimate:

%ΔP ≈ −D_mod × Δy + (1/2) × Convexity × (Δy)²

Convexity formula:

C = (1/P) × Σ [t(t+1) × CFₜ / (1+y)^(t+2)]

Why convexity matters:

Positive convexity: bond price rises more (when yields fall) than it falls (when yields rise) — bonds with positive convexity are preferable
For a given duration, higher convexity is always better — the price behavior is more favorable in both directions
Callable bonds can have negative convexity at low yields (issuer calls when rates fall, capping upside)

Bond Type	Convexity	Reason
Non-callable bond	Positive	Standard price-yield relationship
Callable bond	Can be negative	Call option limits price appreciation
MBS pass-through	Negative (at low rates)	Prepayment increases as rates fall
Zero coupon bond	High positive	All cash flow at maturity, high duration

4. Credit Spreads

Spread Measures

G-spread (Government spread): The difference between a corporate bond's YTM and the on-the-run government bond YTM at the same maturity. Simple but ignores yield curve shape.

I-spread (Interpolated spread): Spread over the interpolated swap rate (not government). More precise than G-spread for corporate bonds in markets where swaps are the benchmark.

Z-spread (Zero-volatility spread): The constant spread added to every spot rate on the benchmark curve such that the present value of the bond's cash flows equals the market price:

P = Σ CFₜ / (1 + sₜ + Z)ᵗ

Z-spread is better than G-spread because it uses the full spot curve, not just one point. Z-spread = 0 for an on-the-run government bond.

OAS (Option-Adjusted Spread): For bonds with embedded options (callable, putable, MBS), OAS strips out the value of the option:

OAS = Z-spread − Option Cost

For a callable bond: Z-spread > OAS (you're getting paid extra for giving away the call option to the issuer). For a putable bond: OAS > Z-spread (you're paying for the put option).

OAS is the "true" credit spread — what you earn after adjusting for optionality.

Asset swap spread: Spread in an asset swap where the bond's coupons are exchanged for Libor/SOFR + spread. Widely used by credit traders.

CDS (Credit Default Swap) Spreads

A CDS spread is the annual premium paid for protection against default:

Protection buyer: pays CDS spread annually, receives par in case of default
Protection seller: receives spread, pays par at default

CDS-bond basis: CDS spread − Bond spread (usually Z-spread or asset swap spread). In theory, the basis should be near zero (cash bond and CDS should be equivalent credit protection). In practice:

Positive basis: CDS > cash bond — often in stressed markets
Negative basis: CDS < cash bond — opportunity for negative basis trades

Pakistan credit: Pakistan's CDS spread (USD sovereign) has ranged from 200bps (stable) to 4,000bps (2022 crisis, pre-IMF deal). Corporate CDS does not exist in Pakistan — credit analysis is bilateral with banks.

5. Term Structure Models

Term structure models attempt to describe how short-term interest rates evolve over time.

Vasicek Model

The first analytically tractable model (1977):

dr = κ(θ − r)dt + σdW

Mean-reverting: rate pulls toward long-run mean θ at speed κ
Constant volatility σ
Problem: allows negative interest rates (not realistic in most environments)
Advantage: closed-form bond pricing formula exists

Calibration to Pakistan data: High mean reversion (κ) and high long-run mean (θ ≈ 12–15%) are needed to fit the PKR curve.

CIR Model (Cox-Ingersoll-Ross, 1985)

dr = κ(θ − r)dt + σ√r dW

Mean-reverting (same as Vasicek)
Volatility scales with √r — prevents negative rates (when rates are low, volatility falls)
Still has closed-form bond pricing
More realistic than Vasicek for non-negative rate environments

Hull-White Model

dr = [θ(t) − κr]dt + σdW

θ(t) is a time-varying drift — calibrated to exactly fit the observed yield curve
Widely used in practice for derivatives pricing because it matches today's term structure perfectly
Allows negative rates (same as Vasicek)

HJM Framework (Heath-Jarrow-Morton)

Models the entire forward rate curve directly:

df(t,T) = μ(t,T)dt + σ(t,T)dW

HJM shows that under no-arbitrage, the drift μ is determined by the volatility structure σ. The Libor Market Model (LMM/BGM) is the most important HJM model in practice — it models forward rates (caplets) directly and is used to price interest rate exotics (caps, floors, swaptions, Bermudans).

6. Interest Rate Derivatives

Caps and Floors

Cap: A series of call options on the floating rate (caplets). If the floating rate (e.g., 3-month KIBOR) exceeds the strike K, the cap buyer receives the difference. CFOs use caps to limit floating-rate borrowing costs.

Floor: A series of put options on the floating rate (floorlets). If the rate falls below K, the floor buyer receives the difference. Investors holding floating-rate bonds buy floors to protect minimum income.

Collar: Buy a cap + sell a floor — limits the range of floating rate exposure. Reduces hedging cost to zero if cap strike = floor strike (zero-cost collar).

Pricing (Black's formula for caplets):

Caplet value = P(0,T₂) × τ × [F × N(d₁) − K × N(d₂)]

Where F is the forward rate, τ is the accrual period, P(0,T₂) is the discount factor.

Pakistan application: SBP policy rate-linked floating debt (KIBOR + spread). FERROQUANT hedges a PKR 2B floating-rate TFC by buying a KIBOR cap at 15% — if KIBOR exceeds 15%, FERROQUANT receives the excess (conceptually; OTC market in Pakistan is nascent).

Swaptions

A swaption is the option to enter a swap at a future date:

Payer swaption: Option to PAY fixed / receive floating (equivalent to a call on rates)
Receiver swaption: Option to RECEIVE fixed / pay floating (equivalent to a put on rates)

Pricing: Black's formula applied to the forward swap rate.

Uses:

Liability management: locking in the right to issue debt at today's rate environment in the future
Cancellable swaps: embedded swaption allows early termination

7. Mortgage-Backed Securities

Pass-Through Structure

A mortgage pass-through pools mortgage loans and passes the monthly payments (principal + interest) to investors:

Originators (banks) create mortgage loans
Loans are pooled and sold to a GSE (Freddie Mac, Fannie Mae) or private entity
GSE issues MBS certificates backed by the pool
Monthly payments from homeowners pass through to MBS holders

Key metrics:

WAC (Weighted Average Coupon): average coupon of the underlying mortgages
WAM (Weighted Average Maturity): average remaining term
MBS coupon = WAC − servicing fee − guarantee fee

Prepayment Risk

Homeowners can prepay their mortgage at any time (refinancing, home sale, extra payments). Prepayment creates reinvestment risk for MBS holders:

When rates fall, prepayment increases — investors receive principal back when they least want it (must reinvest at lower rates)
This creates negative convexity for MBS

PSA benchmark: The Public Securities Association prepayment model assumes prepayment rates ramp up over 30 months to 100% PSA (= 6% CPR):

200% PSA = twice the standard prepayment speed
50% PSA = half the standard speed

CMOs (Collateralized Mortgage Obligations)

CMOs redistribute cash flows from MBS pools into different tranches with different risk/return profiles:

Sequential pay tranches: Front-pay tranches get all principal first — shorter effective duration
PAC (Planned Amortization Class): Principal payments according to a fixed schedule within a prepayment band — most predictable, tightest OAS
Support/companion tranche: Absorbs prepayment variability to protect PAC — wider OAS for more risk
Interest-only (IO) strip: Receives only interest — value falls when prepayments rise (less interest paid on smaller balance)
Principal-only (PO) strip: Receives only principal — value rises when prepayments rise (principal returned sooner, discounted at lower rates)

WAL (Weighted Average Life): The weighted average time to return a dollar of principal. More practical than maturity for MBS because principal is returned continuously.

OAS for MBS: OAS strips out the negative convexity (prepayment option value). An MBS with Z-spread of 150bps and prepayment option cost of 60bps has OAS = 90bps.

8. Fixed Income in Pakistan & Gulf Markets

Pakistan Fixed Income Market

Instrument	Issuer	Maturity	Market
T-bills (MTBs)	Government	3, 6, 12 months	SBP auction
PIBs	Government	3–30 years	SBP auction + OTC
GoP Ijarah Sukuk	Government	3, 5, 10 years	SBP auction
TFCs	Corporates	3–7 years	Privately placed / PSX
Bank certificates	Commercial banks	Short-term	Bank-to-bank

SBP Monetary Policy Committee: Sets the policy rate (currently 12% as of early 2026, down from 22% peak in 2023). The policy rate anchors the entire yield curve.

SBP OMOs: Open Market Operations — SBP uses repo and reverse repo to manage overnight rates. Repos inject liquidity; reverse repos absorb.

Pakistan institutional buyers:

Banks: largest buyer — SLR compliance, excess liquidity parking
Insurance companies: duration matching
Pension funds: ALM-driven buying
Mutual funds: fixed income money market funds

Gulf Fixed Income

UAE:

CBUAE manages liquidity; Dirham pegged to USD → UAE rates shadow US Fed
ADIB, Emirates NBD, ENBD regularly issue international bonds
ADNOC, Mubadala: benchmark-level USD issuers

Saudi Arabia:

Saudi Arabia Debt Management Office (DMO): world-class bond program since 2016
SAR domestic market + USD international program
Aramco bonds: among the most liquid EM credits globally

Qatar:

Qatar Investment Authority (QIA): significant secondary market investor
QNB: regional benchmark bond issuer
Qatar sovereign: regular USD and EUR issuance

Gulf rate linkage: All GCC pegged currencies (SAR, AED, QAR, BHD, OMR) track US Fed Funds — Gulf bond markets move almost in lockstep with US Treasuries. This is very different from Pakistan where PKR rates are set independently.

Self-Assessment

FERROQUANT Capital holds a PKR 500 million position in a 10-year PIB with:
- Annual coupon: 14%, paid semi-annually
- YTM: 12% annualized
- Modified Duration: 6.8 years
- Convexity: 61
(a) Is the bond trading at a premium or discount? Why? (b) Calculate the DV01 of the position (use approximate price ≈ PKR 113 per PKR 100 face value). (c) If the SBP unexpectedly raises the policy rate by 50bps and the yield rises by 50bps, estimate the price change using both duration alone and duration + convexity. Which is more accurate and why? (d) FERROQUANT wants to hedge this interest rate exposure by shorting PIB futures. If each futures contract has a DV01 of PKR 5,000, how many contracts should FERROQUANT short?
You are bootstrapping the PKR spot curve from the following instruments:
- 3-month T-bill: YTM = 11.50%
- 6-month T-bill: YTM = 12.00%
- 1-year PIB (annual coupon 12.5%, trading at par): YTM = 12.50%
- 2-year PIB (annual coupon 13%, trading at par): YTM = 13.00%
(a) What is the 3-month spot rate? 6-month spot rate? (b) Using bootstrapping, derive the 1-year spot rate from the 1-year PIB price. (c) Using the 1-year spot rate from (b), derive the 2-year spot rate from the 2-year PIB price. (d) Calculate the 1-year forward rate one year from now. If actual 1-year rates one year from now are 14%, was the market's forward rate forecast accurate? What might explain the miss?
FERROQUANT is analyzing a PKR 5B TFC issued by a Pakistani cement company:
- Maturity: 5 years
- Coupon: KIBOR + 175bps, paid quarterly
- Trading at 98.50 (flat price, % of par)
- Current KIBOR: 12.00%
- 5-year PIB yield: 13.50%
- Assumed flat term structure for simplicity
(a) Calculate the approximate Z-spread for this TFC (hint: treat as a floating-rate bond — Z-spread ≈ yield − benchmark rate + spread above KIBOR). (b) How would you calculate the OAS if the TFC had a call option at par in year 3? Would OAS be higher or lower than Z-spread? (c) FERROQUANT's credit analyst estimates the company has a 3% annual probability of default with 40% recovery. Is the Z-spread adequate compensation for the credit risk? Show your calculation. (d) The TFC is rated A- by PACRA. A comparable listed bond in Turkey (B+ Moody's) trades at 450bps over local government. Does Pakistan's credit spread look cheap or expensive on a cross-market basis? What structural factors might explain the difference?