Lecture 1, Part III: Bond “Mathematics”

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).

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.