Sukrit Mittal Franklin Templeton Investments
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.
Its job is boring by design:
Boring markets are the most important ones.
In theory, we assume the existence of a risk-free asset.
Meaning:
In reality, nothing is perfectly risk-free.
But theory is about controlled lies.
Three conditions must hold:
In practice:
The approximation is good enough for modeling. Perfect is the enemy of useful.
A zero-coupon bond is the cleanest financial instrument imaginable.
Definition:
Notation:
Think of it as a time machine for money.
They isolate time value from everything else.
No coupons. No reinvestment assumptions. No ambiguity.
If you understand zero-coupon bonds, you understand:
They are the atoms of fixed income.
Assume a constant continuously compounded rate \(r\).
Then:
\[ P(0,T) = e^{-rT} \]
Interpretation:
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:
\[ P(0,T) = \lim_{n \to \infty} (1-r\cdot T/n)^{-n} \]
Standard limit:
\[ \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x \]
Setting \(x = -rT\) gives \(P(0,T) = e^{-rT}\).
This isnât arbitrary. Itâs the only way to make time-consistent discounting work.
Each maturity \(T\) has its own discount factor \(P(0,T)\).
The collection:
\[ \{P(0,T) : T > 0\} \]
is called the term structure of interest rates.
This curve encodes the marketâs view of time and liquidity.
Well-behaved term structures satisfy:
These arenât assumptions. Theyâre 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.
Define the spot rate \(r(0,T)\) by:
\[ P(0,T) = e^{-r(0,T)T} \]
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:
\[ r(0,T) = -\frac{1}{T} \ln P(0,T) \]
Example: If \(P(0,3) = 0.85\), then:
\[ r(0,3) = -\frac{1}{3} \ln(0.85) = -\frac{1}{3}(-0.1625) \approx 0.0542 = 5.42\% \]
This is the annualized continuously compounded rate for 3-year money.
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:
\[ P(0,T_2) = P(0,T_1) \cdot e^{-f(0,T_1,T_2)(T_2-T_1)} \]
Solving:
\[ f(0,T_1,T_2) = \frac{\ln P(0,T_1) - \ln P(0,T_2)}{T_2-T_1} \]
Forward rates let you lock in future borrowing costs today.
A coupon bond pays periodic interest plus principal.
Parameters:
Cash flows:
\[ c, c, \dots, c, 1 + c \]
This is a bundle, not a primitive object.
A coupon bond is a portfolio of zero-coupon bonds.
Price at time 0:
\[ P = \sum_{t=1}^{T} c \cdot P(0,t) + P(0,T) \]
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:
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.
Because it tells you:
If you price bonds any other way, you are guessing.
Market convention compresses all cash flows into a single number.
\(y\) solves:
\[ P = \sum_{t=1}^{T} \frac{c}{(1+y)^t} + \frac{1}{(1+y)^T} \]
This number is convenient.
It is also dangerous.
What YTM represents:
Itâs the internal rate of return (IRR) of the bondâs cash flows.
If you: * Buy the bond at price \(P\) * Hold to maturity * Reinvest all coupons at rate \(y\)
Then your realized return equals \(y\).
Thatâs a lot of âifsâ.
YTM is a quoting convention, not a pricing principle.
Why these limitations matter:
Flat term structure: YTM treats all maturities as having the same discount rate. Reality: 1-year rates â 10-year rates.
Reinvestment assumption: If you canât reinvest coupons at \(y\), your realized return differs from YTM.
Non-additivity: If Bond A has YTM = 5% and Bond B has YTM = 6%, their portfolio doesnât have YTM = 5.5%.
When YTM is useful:
When YTM fails:
The money market account models continuous reinvestment.
Let \(A(t)\) denote its value.
With constant rate \(r\):
\[ A(t) = A(0)e^{rt} \]
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 ârisk-free growth process.â
Why it matters:
Starting with \(A(0) = 1\), we get \(A(t) = e^{rt}\).
The reciprocal \(e^{-rt}\) is exactly our discount factor.
In discrete time:
\[ A(t+1) = (1+r)A(t) \]
Starting from \(A(0)=1\):
\[ A(t) = (1+r)^t \]
This will later become the numeraire.
Relationship to discounting:
If \(A(t) = (1+r)^t\) grows wealth forward, then \(P(0,t) = (1+r)^{-t}\) brings it back.
These are inverse operations:
\[ A(t) \cdot P(0,t) = (1+r)^t \cdot (1+r)^{-t} = 1 \]
Preview of risk-neutral pricing:
Pricing derivatives by: 1. Computing expected payoffs 2. Discounting at the risk-free rate
The money market account is the denominator in that calculation.
For example:
Risk-neutral probability:
Given \(S(0)=100\), \(S^u(T)=120\), \(S^d(T)=80\), \(r_F=10\%\):
\[q = \frac{e^{r_F T} \cdot S(0) - S^d(T)}{S^u(T) - S^d(T)} = \frac{1.10 \times 100 - 80}{120 - 80} = \frac{30}{40} = 0.75\]
Expected payoff under risk-neutral measure:
\[\mathbb{E}^Q[C(T)] = 0.75 \times 20 + 0.25 \times 0 = 15\]
Discounted value:
\[C(0) = \frac{15}{1.10} \approx 13.64\]
The risk-neutral probability \(q=0.75\) is not the real-world probability. Itâs the probability that makes the discounted expected value equal the market price.
No-arbitrage principle:
Identical cash flows must have identical prices.
Implications:
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 \(F = 0.94 < 0.9579\), the forward is underpriced.
Arbitrage strategy:
Profit analysis:
Present value of year-1 cash flow: \(0.06 \times P_1 = 0.06 \times 0.95 = 0.057\)
Net profit today: \(0.057 - 0.04 = 0.017\) (risk-free profit of 1.7 cents per unit)
Economic intuition: The forward rate \(F = 0.94\) implies you can lock in a 1-year rate starting at year 1 of:
\[\frac{1}{F} - 1 = \frac{1}{0.94} - 1 \approx 6.38\%\]
But the no-arbitrage forward rate should be:
\[\frac{1}{F_{fair}} - 1 = \frac{1}{0.9579} - 1 \approx 4.39\%\]
Youâre getting better borrowing terms (6.38%) than the market implies (4.39%), creating arbitrage.
Suppose:
Coupon bond:
Price:
\[ P = 0.04(0.95+0.90+0.85)+0.85 = 0.04(2.70)+0.85 = 0.958 \]
Nothing mystical happened.
Letâs 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
\[ f(0,1,2) = \frac{\ln P(0,1) - \ln P(0,2)}{2-1} = \ln(0.95) - \ln(0.90) \approx 0.0541 = 5.41\% \]
Part (c): What if the bond traded at \(P = 0.95\)?
Then itâs 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, thatâs a loss.
Actually: sell the replicating portfolio, buy the bond.
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.
Duration formula:
\[ D = \frac{1}{P} \sum_{t=1}^{T} t \cdot C_t \cdot P(0,t) \]
where \(C_t\) is the cash flow at time \(t\).
Duration measures the sensitivity of bond prices to interest rate changes.
Itâs the âcenter of massâ of the bondâs cash flows.
Compute \(P(0,5)\) for a zero-coupon bond under:
Compare results.
Solution hints:
Observe how the differences grow with maturity.
Given discount factors:
| \(T\) | \(P(0,T)\) |
|---|---|
| 1 | 0.97 |
| 2 | 0.94 |
| 3 | 0.90 |
Price a 3-year coupon bond with \(c=0.05\).
Extended parts:
Compute the spot rates \(r(0,1)\), \(r(0,2)\), \(r(0,3)\).
Compute forward rates: \(f(0,1,2)\) and \(f(0,2,3)\).
If this bond trades at \(P=1.02\), describe the arbitrage strategy.
What coupon rate \(c^*\) would make the bond trade at par (\(P=1\))?
A bond trades at price \(0.92\) with:
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.
You observe the following par bonds (bonds trading at face value):
| Maturity | Coupon Rate |
|---|---|
| 1 year | 4% |
| 2 years | 5% |
| 3 years | 5.5% |
Bootstrap the discount factors \(P(0,1)\), \(P(0,2)\), \(P(0,3)\).
Hint: For par bonds, price = 100.
Solve recursively.
This is how market practitioners build the term structure from observable bond prices.
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:
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.
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.