# Lecture 1, Part III: Bond “Mathematics”

Source: https://www.youtube.com/watch?v=NZ3Mva95UsQ
Recap page: https://rapidrecap.app/video/NZ3Mva95UsQ
Generated: 2025-12-03T16:10:37.484+00:00

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## Quick Overview

The lecture, titled "Bond 'Mathematics'" from MIT Course 18.642, focuses on discounting and bond mathematics, specifically deriving the present value of a future cash flow using the concept of compounding interest ($1 invested grows to $(1+r)^n$ in $n$ years) and extending this to continuous compounding, leading to the formula $\lim_{m \to \infty} (1 + r/m)^{m \cdot n} = e^{rn}$. The instructor then sets up a scenario to determine the present value ($X$) one should borrow today to receive $1 in one year, solving for $X = 1/(1+\tau)$ where $\tau$ is the prevailing annual interest rate, which is the definition of the discount factor.

**Key Points:**
- The lecture introduced discounting by first establishing compounding interest: $1 grows to $(1+r)^n$ in $n$ years, and continuously compounded growth leads to $e^{rn}$.
- The concept of discounting was illustrated by asking how much ($X$) one should borrow now to receive $1 in one year, yielding the present value $X = 1/(1+\tau)$, where $\tau$ is the annual interest rate.
- The instructor showed the formula for the present value of $N$ dollars received in $n$ years under annual compounding: $P = N / (1+r)^n$, and under continuous compounding: $P = N \cdot e^{-rn}$.
- The discussion extended to coupon bonds, where the price $P$ is the sum of discounted coupons ($C$) and the discounted notional payment ($N$), leading to the formula $P = C \frac{(1+r)^n - 1}{r(1+r)^n} + \frac{N}{(1+r)^n}$.
- The concept of yield was defined as the discount rate needed to obtain the current price of a bond, and the relationship between yield and price was shown graphically: price decreases as yield increases (07:14, 11:52).
- The concept of the yield curve (term structure of yields) was introduced using U.S. Treasury yield data from various dates (e.g., 9/11/92, 9/1/24), showing different yields for different maturities (14:35).

![Screenshot at 00:16: 07:The instructor begins deriving the relationship between compounding frequency \($m$\) and continuous compounding, showing the limit expression $\\lim\_{m \\to \\infty} \(1 + r/m\)^{m \\cdot n} = e^{rn}$ on the blackboard.](https://ss.rapidrecap.app/screens/NZ3Mva95UsQ/00-00-16.png)

**Context:** This segment is part III of Lecture 1 from MIT Course 18.642, "Topics in Mathematics with Applications in Finance," presented by Vasily Strela. The lecture focuses on the fundamental mathematical concepts underpinning bond pricing, specifically introducing discounting, the time value of money, and the relationship between interest rates, compounding frequency, and bond valuation. The instructor uses blackboard derivations to explain discounting fundamentals before transitioning to the yield curve for fixed-income securities.

## Detailed Analysis

The lecture segment focuses on the mathematics of bond valuation, starting with the time value of money. The instructor first establishes compounding interest, noting that a $1 investment grows to $(1+r)^n$ in $n$ years with annual compounding, and if reinvested $m$ times per year, it grows to $(1+r/m)^{mn}$. For continuous compounding, this limit as $m \to \infty$ yields $e^{rn}$ (00:16). The core concept of discounting is then introduced by asking the equivalent present value ($X$) of receiving $1 one year from now, given an annual interest rate $\tau$. This leads to the equation $1 - X(1+\tau) = 0$, solving for the discount factor $X = 1/(1+\tau)$ (07:04). This is explicitly defined as discounting. The principle is then generalized to an $N$ dollar future payment, resulting in the present value formula $P = N / (1+r)^n$ for annual compounding and $P = N \cdot e^{-rn}$ for continuous compounding (08:01). Subsequently, the instructor moves to coupon bonds, where the price $P$ is the sum of discounted coupons ($C$) and the discounted notional payment ($N$), providing the formula derived from a geometric series sum (09:31). The lecture then transitions to the concept of bond yield, defined as the discount rate needed to obtain the current price. A key takeaway is the inverse relationship between bond price and yield, illustrated by a graph showing multiple yield curves for different coupon rates (C=1%, 5%, 10%) across various maturities (11:52). The instructor points out that the yield curve reflects the term structure of yields, noting historical examples like the 1992 and 2007 curves, and emphasizes that longer-term bonds are generally more sensitive to yield changes (duration).

### Compounding and Discounting Fundamentals

- $1 investment grows to $(1+r)^n$ in $n$ years
- Continuous compounding leads to $e^{rn}$
- Present value $X$ of $1 received in one year at rate $\tau$ is $X = 1/(1+\tau)$ (the discount factor)

### Bond Pricing Formulas

- Present value of $N$ in $n$ years is $P = N/(1+r)^n$ (annual) or $P = N \cdot e^{-rn}$ (continuous)
- Coupon bond price $P$ is the sum of discounted coupons and notional payment, summarized by $P = C \frac{(1+r)^n - 1}{r(1+r)^n} + \frac{N}{(1+r)^n}$ (09:31)

### Yield and Yield Curve

- Yield is the discount rate needed to obtain current price
- Price decreases as yield increases (demonstrated graphically)
- Yield curves show term structure of yields, with historical data used for comparison (e.g., 1992 vs 2024 curves) (13:35)

### Bond Sensitivities

- Duration ($D$) measures price sensitivity to yield changes (first derivative)
- Convexity ($D^2 P / dy^2$) measures the curvature of the price-yield relationship (19:29)

![Screenshot at 00:00: 00:Initial visual setup showing a mass $m$ attached to a spring, representing simple harmonic motion, before the lecture begins.](https://ss.rapidrecap.app/screens/NZ3Mva95UsQ/00-00-00.png)
![Screenshot at 00:16: 00:The instructor writing the formula for continuous compounding, $\\lim\_{m \\to \\infty} \(1 + r/m\)^{m \\cdot n} = e^{rn}$, on the blackboard \(00:16\).](https://ss.rapidrecap.app/screens/NZ3Mva95UsQ/00-00-16.png)
![Screenshot at 07:04: The derivation showing the present value $X$ of $1 received in one year is $X = 1/\(1+\\tau\)$, labeled as the discount factor \(07:14\).](https://ss.rapidrecap.app/screens/NZ3Mva95UsQ/00-07-04.png)
![Screenshot at 11:52: A slide displaying multiple yield curves for US Treasury bonds across different dates, illustrating the term structure of yields \(11:52\).](https://ss.rapidrecap.app/screens/NZ3Mva95UsQ/00-11-52.png)
![Screenshot at 19:29: A slide summarizing key bond price sensitivities, including definitions for Duration \($D$\), Duration of a zero-coupon bond \($D\_Z$\), Duration of a continuous par bond \($D\_P$\), and Convexity \($D^2 P / dy^2$\) \(19:29\).](https://ss.rapidrecap.app/screens/NZ3Mva95UsQ/00-19-29.png)