Money Market

Sukrit Mittal Franklin Templeton Investments

Outline

1. Motivation: why money markets matter
2. Risk-free assets and modeling assumptions
3. Zero-coupon bonds: definition and pricing
4. Discount factors and term structure
5. Coupon bonds: cash-flow decomposition
6. Yield measures and their limitations
7. Money market account
8. No-arbitrage relationships
9. Worked examples
10. Exercises

1. Motivation: Why Study the Money Market?

The money market is where time itself is priced.

Before equities, before derivatives, before fancy models, finance had one problem:

How much is a sure rupee tomorrow worth today?

Everything else in mathematical finance is built on the answer to that question.

If you misunderstand the money market, all later pricing formulas are numerology.

What the Money Market Is Not

• It is not about speculation
• It is not about beating benchmarks
• It is not about taking risk

Its job is boring by design:

• Preserve capital
• Provide liquidity
• Anchor the risk-free rate

Boring markets are the most important ones.

2. Risk-Free Assets and Modeling Assumptions

In theory, we assume the existence of a risk-free asset.

Meaning:

• Known payoff
• No default
• No uncertainty

In reality, nothing is perfectly risk-free.

But theory is about controlled lies.

What Makes an Asset “Risk-Free”?

Three conditions must hold:

1. Deterministic payoff: We know exactly what we’ll receive
2. Zero credit risk: The counterparty cannot default
3. Known timing: Payments occur at specified dates

In practice:

• US Treasury bills ≈ risk-free for USD calculations
• German bunds ≈ risk-free for EUR calculations
• Government securities of stable economies

The approximation is good enough for modeling. Perfect is the enemy of useful.

Common Abstractions

We collapse many instruments into a few idealized objects:

• Treasury bills
• Bank deposits
• Interbank lending

All are modeled as deterministic cash flows.

This abstraction is old. It works. And it scales.

The Modeling Philosophy

Real markets feature:

• Credit spreads
• Liquidity premia
• Operational risk

Our model ignores these. Why?

Because we’re isolating pure time value.

Once we understand discounting in a frictionless world, we can add back complexity piece by piece.

Start simple. Complexify only when forced.

3. Zero-Coupon Bonds

A zero-coupon bond is the cleanest financial instrument imaginable.

Definition:

• Pays exactly 1 unit of currency at time $T$
• Pays nothing before maturity

Notation:

• $P(0,T)$: price at time 0

Think of it as a time machine for money.

Why Zero-Coupon Bonds Matter

They isolate time value from everything else.

No coupons. No reinvestment assumptions. No ambiguity.

If you understand zero-coupon bonds, you understand:

• Discounting
• Interest rates
• Present value

They are the atoms of fixed income.

Pricing with Continuous Compounding

Assume a constant continuously compounded rate $r$.

Then:

Interpretation:

• Future certainty discounted exponentially
• Time erodes value smoothly

Exponential discounting is not a choice. It is forced by consistency.

Why exponential?

Consider dividing time $T$ into $n$ intervals. In each interval $\Delta t = T/n$, the discount factor is $(1+r\Delta t)^{-1} \approx (1-r\Delta t)$ for small $\Delta t$.

Compounding:

Standard limit:

Setting $x = -rT$ gives $P(0,T) = e^{-rT}$.

This isn’t arbitrary. It’s the only way to make time-consistent discounting work.

Pricing with Discrete Compounding

If interest is compounded once per period:

If compounded $n$ times per year:

Different conventions. Same economics.

Numerical Example:

For $r = 6\%$ and $T = 5$ years:

• Annual: $P(0,5) = (1.06)^{-5} \approx 0.7473$
• Semi-annual ($n=2$): $P(0,5) = (1.03)^{-10} \approx 0.7441$
• Quarterly ($n=4$): $P(0,5) = (1.015)^{-20} \approx 0.7425$
• Continuous: $P(0,5) = e^{-0.3} \approx 0.7408$

Notice: more frequent compounding → lower present value.

The differences are small but matter at scale.

4. Discount Factors and Term Structure

Each maturity $T$ has its own discount factor $P(0,T)$.

The collection:

is called the term structure of interest rates.

This curve encodes the market’s view of time and liquidity.

Properties of Discount Factors

Well-behaved term structures satisfy:

1. Boundary condition: $P(0,0) = 1$ (money today equals money today)
2. Monotonicity: $P(0,T_1) > P(0,T_2)$ if $T_1 < T_2$ (longer waits mean deeper discounts)
3. Positivity: $P(0,T) > 0$ for all $T$ (money doesn’t become worthless)
4. Smoothness: $P(0,T)$ varies continuously in $T$

These aren’t assumptions. They’re arbitrage-free requirements.

If $P(0,2) > P(0,1)$, you could:

• Borrow at 2 years
• Lend at 1 year
• Relend for another year
• Lock in risk-free profit

Markets eliminate such opportunities quickly.

Spot Rates

Define the spot rate $r(0,T)$ by:

Each maturity has its own rate.

Flat curves are special cases, not defaults.

Markets are rarely that polite.

Solving for spot rates:

Given discount factors, we can extract:

Example: If $P(0,3) = 0.85$, then:

This is the annualized continuously compounded rate for 3-year money.

Forward Rates

We can also define forward rates $f(0,T_1,T_2)$: the rate agreed today for borrowing between times $T_1$ and $T_2$.

No-arbitrage requires:

Solving:

Forward rates let you lock in future borrowing costs today.

5. Coupon Bonds

A coupon bond pays periodic interest plus principal.

Parameters:

• Face value: 1
• Coupon: $c$
• Maturity: $T$

Cash flows:

This is a bundle, not a primitive object.

Coupon Bonds as Portfolios

A coupon bond is a portfolio of zero-coupon bonds.

Price at time 0:

No assumptions. No probabilities. No equilibrium.

Just accounting and no-arbitrage.

Decomposition logic:

Think of buying:

• $c$ units of 1-year zeros
• $c$ units of 2-year zeros
• …
• $c$ units of $T$-year zeros
• 1 unit of $T$-year zero (for principal)

Total cost must equal bond price, otherwise arbitrage exists.

Example with numbers:

Suppose we have a 3-year bond with $c=0.05$ (5% coupon) and discount factors:

• $P(0,1) = 0.96$
• $P(0,2) = 0.92$
• $P(0,3) = 0.88$

Price:

\(P = 0.05(0.96) + 0.05(0.92) + 1.05(0.88)\) \(= 0.048 + 0.046 + 0.924 = 1.018\)

This bond trades at a premium (above par) because coupons exceed the market rate.

Why This Decomposition Matters

Because it tells you:

• What drives bond prices
• Why yield measures can mislead
• How to hedge interest-rate risk

If you price bonds any other way, you are guessing.

6. Yield to Maturity (YTM)

Market convention compresses all cash flows into a single number.

$y$ solves:

This number is convenient.

It is also dangerous.

What YTM represents:

It’s the internal rate of return (IRR) of the bond’s cash flows.

If you:

• Buy the bond at price $P$
• Hold to maturity
• Reinvest all coupons at rate $y$

Then your realized return equals $y$.

That’s a lot of “ifs”.

Limitations of YTM

• Assumes flat term structure
• Assumes coupons reinvested at the same rate
• Not additive across portfolios

YTM is a quoting convention, not a pricing principle.

Respect it. Don’t worship it.

Why these limitations matter:

1. Flat term structure: YTM treats all maturities as having the same discount rate. Reality: 1-year rates ≠ 10-year rates.

2. Reinvestment assumption: If you can’t reinvest coupons at $y$, your realized return differs from YTM.

3. Non-additivity: If Bond A has YTM = 5% and Bond B has YTM = 6%, their portfolio doesn’t have YTM = 5.5%.

When YTM is useful:

• Comparing similar bonds
• Quick market quotes
• Rough intuition

When YTM fails:

• Pricing exotic structures
• Risk management
• Hedging calculations

For serious work, always go back to discount factors.

7. Money Market Account

The money market account models continuous reinvestment.

Let $B(t)$ denote its value.

With constant rate $r$:

This asset defines the baseline growth of wealth.

Interpretation:

Imagine a bank account where:

• Interest is credited continuously
• You never withdraw
• The rate stays constant

This is the “risk-free growth process.”

Why it matters:

• It’s the benchmark for all other returns
• In derivatives pricing, we discount at this rate
• It defines what “risk-neutral” means

Starting with $B(0) = 1$, we get $B(t) = e^{rt}$.

The reciprocal $e^{-rt}$ is exactly our discount factor.

Discrete-Time Version

In discrete time:

Starting from $B(0)=1$:

This will later become the numeraire.

Relationship to discounting:

If $B(t) = (1+r)^t$ grows wealth forward, then $P(0,t) = (1+r)^{-t}$ brings it back.

These are inverse operations:

Preview of risk-neutral pricing:

Later, we’ll price derivatives by:

1. Computing expected payoffs
2. Discounting at the risk-free rate

The money market account is the denominator in that calculation.

For now: it’s just a modeling device for time value.

8. No-Arbitrage Relationships

No-arbitrage principle:

Identical cash flows must have identical prices.

Implications:

• Discount factors are unique
• Coupon bonds are pinned down
• Risk-free growth is dominant

Break consistency, and arbitrage strategies appear.

What is arbitrage?

A trading strategy that:

• Costs nothing (or makes money) today
• Never loses money in the future
• Makes money with positive probability

Example of arbitrage:

Suppose:

• A 1-year zero-coupon bond costs $P_1 = 0.95$
• A 2-year zero-coupon bond costs $P_2 = 0.91$
• A 1-year forward contract starting in 1 year costs $F = 0.94$

No-arbitrage requires: $P_2 = P_1 \cdot F$, so $F = P_2/P_1 = 0.91/0.95 \approx 0.9579$.

If market quotes $F = 0.94 < 0.9579$:

Arbitrage strategy:

• Buy the 2-year zero for $0.91$
• Sell the 1-year zero for $0.95$ (borrow)
• Sell forward at $F=0.94$

Cash flows:

• Today: $-0.91 + 0.95 = +0.04$ (profit now)
• Year 1: Pay back 1 unit (from short), receive $1/F = 1/0.94$ forward
• Year 2: Receive 1 from bond, pay $1/F$ forward = net $1 - 1.0638 < 0$…

Actually, let me recalculate. The correct arbitrage:

• Sell 2-year zero for $P_2 = 0.91$
• Buy 1-year zero for $P_1 = 0.95$
• Enter forward to lock in year-2 rate

If $F < P_2/P_1$, you can arbitrage. Markets prevent this.

Preview: What Comes Next

Soon, interest rates will:

• Vary over time
• Become random
• Interact with risky assets

But all of that rests on today’s foundation.

Weak foundations collapse silently.

9. Worked Example

Suppose:

• $P(0,1)=0.95$
• $P(0,2)=0.90$
• $P(0,3)=0.85$

Coupon bond:

• $c=0.04$
• $T=3$

Price:

Nothing mystical happened.

Let’s extend this example:

Part (a): Extract spot rates

\(r(0,1) = -\ln(0.95) \approx 0.0513 = 5.13\%\) \(r(0,2) = -\frac{1}{2}\ln(0.90) \approx 0.0527 = 5.27\%\) \(r(0,3) = -\frac{1}{3}\ln(0.85) \approx 0.0542 = 5.42\%\)

The term structure is upward sloping: longer maturities command higher rates.

Part (b): Compute forward rate from year 1 to year 2

Part (c): What if the bond traded at $P = 0.95$?

Then it’s mispriced relative to the discount curve.

Arbitrage:

• Sell the bond for $0.95$
• Buy the replicating portfolio for $0.958$
• Lock in $0.95 - 0.958 = -0.008$… wait, that’s a loss.

Actually: sell the replicating portfolio, buy the bond.

• Buy bond for $0.95$
• Short $0.04$ units of 1-year zero (receive $0.04 \times 0.95 = 0.038$)
• Short $0.04$ units of 2-year zero (receive $0.04 \times 0.90 = 0.036$)
• Short $1.04$ units of 3-year zero (receive $1.04 \times 0.85 = 0.884$)

Total received: $0.038 + 0.036 + 0.884 = 0.958$

Net today: $0.958 - 0.95 = 0.008$ risk-free profit.

At maturity, bond pays exactly what shorts require. Free money.

10. Exercises

Exercise 1

Compute $P(0,5)$ for a zero-coupon bond under:

1. Simple interest $r=6\%$
2. Annual compounding $r=6\%$
3. Continuous compounding $r=6\%$

Compare results.

Solution hints:

1. Simple: $P = 1/(1+rT)$
2. Annual: $P = 1/(1+r)^T$
3. Continuous: $P = e^{-rT}$

Observe how the differences grow with maturity.

Exercise 2

Given discount factors:

Price a 3-year coupon bond with $c=0.05$.

Extended parts:

(a) Compute the spot rates $r(0,1)$, $r(0,2)$, $r(0,3)$.

(b) Compute forward rates: $f(0,1,2)$ and $f(0,2,3)$.

(c) If this bond trades at $P=1.02$, describe the arbitrage strategy.

(d) What coupon rate $c^*$ would make the bond trade at par ($P=1$)?

Exercise 3

A bond trades at price $0.92$ with:

• $T=2$
• $c=0.06$

Compute its yield to maturity.

Then explain why two bonds with the same YTM may have different prices.

Solution approach for YTM:

Solve: \(0.92 = \frac{0.06}{1+y} + \frac{1.06}{(1+y)^2}\)

This requires numerical methods (Newton-Raphson) or financial calculator.

Answer: $y \approx 9.23\%$

Explanation part:

Two bonds with same YTM but different prices could have:

• Different coupon rates
• Different maturities
• Different credit quality (if not truly risk-free)
• Different liquidity

YTM is not a complete descriptor of a bond.

Exercise 4: Duration

Given a bond with:

• Face value = 100
• Coupon rate = 5% (annual)
• Maturity = 4 years
• Current price = 95.50

(a) Compute the Macaulay duration.

(b) If yields increase by 1%, estimate the new price using duration.

Duration formula:

where $C_t$ is the cash flow at time $t$.

Duration measures the sensitivity of bond prices to interest rate changes.

It’s the “center of mass” of the bond’s cash flows.

Exercise 5: Bootstrapping the Yield Curve

You observe the following par bonds (bonds trading at face value):

Bootstrap the discount factors $P(0,1)$, $P(0,2)$, $P(0,3)$.

Hint: For par bonds, price = 100.

• Year 1: $100 = 4 \cdot P(0,1) + 100 \cdot P(0,1)$
• Year 2: $100 = 5 \cdot P(0,1) + 5 \cdot P(0,2) + 100 \cdot P(0,2)$
• And so on…

Solve recursively.

This is how market practitioners build the term structure from observable bond prices.

Exercise 6: Arbitrage Detection

Consider three bonds:

Bond A (1-year zero): Price = 0.96

Bond B (2-year zero): Price = 0.90

Bond C (2-year coupon bond): Coupon = 5%, Price = 0.98

Is there an arbitrage opportunity? If yes, describe the strategy.

Approach:

1. Compute fair price of Bond C using bonds A and B
2. Compare with market price
3. If different, construct arbitrage

Fair price = $0.05 \times 0.96 + 1.05 \times 0.90 = 0.048 + 0.945 = 0.993$

Market price = $0.98 < 0.993$

Arbitrage: Buy Bond C, sell replicating portfolio.

Final Takeaways

• Money markets price time, not risk
• Zero-coupon bonds are the foundation
• Coupon bonds are portfolios
• Yields summarize, discount factors price
• No-arbitrage enforces discipline

If this lecture feels slow, good.

Finance rewards patience before cleverness.

Advanced Topic: Convexity

Duration is a linear approximation. For large rate changes, we need convexity.

Taylor expansion of bond price:

where:

• $D$ = duration
• $C$ = convexity
• $dy$ = change in yield

Convexity formula:

Positive convexity is valuable: bond prices rise more when yields fall than they drop when yields rise.

Advanced Topic: Forward Rate Agreements (FRA)

An FRA is a contract to lock in a borrowing/lending rate for a future period.

Notation:

• FRA$(T_1, T_2)$: agreement to borrow/lend between $T_1$ and $T_2$

No-arbitrage pricing:

The forward rate $f$ must satisfy:

Otherwise, you could:

• Borrow short, lend long
• Or vice versa
• Lock in risk-free profit

Market usage:

• Hedging future borrowing costs
• Speculation on rate movements
• Constructing synthetic positions

FRAs are the building blocks of interest rate swaps.

Connection to Future Lectures

Today’s foundations enable:

1. Portfolio theory: How to combine risky assets with risk-free bonds
2. CAPM: Risk-free rate as the baseline return
3. Options pricing: Discounting expected payoffs
4. Risk management: Term structure models for VaR

Every model we build assumes you understand:

• What a discount factor is
• Why no-arbitrage matters
• How to decompose complex cash flows

If you leave this lecture with nothing else, remember:

Price every cash flow separately, then add them up.

That’s 90% of fixed income in one sentence.

Further Reading

Classics:

• Luenberger, Investment Science (Ch. 3-4)
• Hull, Options, Futures, and Other Derivatives (Ch. 4)
• Tuckman & Serrat, Fixed Income Securities (Ch. 1-3)

For the mathematically inclined:

• Björk, Arbitrage Theory in Continuous Time (Ch. 1-2)
• Shreve, Stochastic Calculus for Finance I (Ch. 1)

Market perspective:

• Fabozzi, Bond Markets, Analysis, and Strategies

The theory is beautiful. The practice is messy. You need both.

Summary: Key Formulas

Zero-coupon bond pricing: \(P(0,T) = e^{-r(0,T) \cdot T}\)

Spot rate extraction: \(r(0,T) = -\frac{1}{T} \ln P(0,T)\)

Forward rate: \(f(0,T_1,T_2) = \frac{\ln P(0,T_1) - \ln P(0,T_2)}{T_2-T_1}\)

Coupon bond pricing: \(P = \sum_{t=1}^{T} c \cdot P(0,t) + P(0,T)\)

Money market account: \(B(t) = e^{rt}\)

These five formulas are your toolkit.

Master them, and the money market becomes transparent.