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FixedIncomeUnderstanding.rmd
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---
title: "Fixed Income Understanding"
author: "Mark Bergh"
date: "2023-03-02"
output: pdf_document
---
```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```
## What are Bonds?
Bonds are financial instruments that represent a collection of future cashflows. These cashflows will be $N \geq 0$ "coupon payments" and one payment of the principle or "par" value. For example, should an investor purchase a 5 year \$100 5% Treasury bond "at par value", and suppose the bond pays coupons semi-annually, she would be entitled to receive 10 coupon payments of \$2.5 every six months and one payment of \$100 at the end of 5 years.
## Bond Valuation
Assuming no credit risk, the price of a bond is the sum of the present values of its future cashflows. The present value of a cashflow is the value of that cashflow, discounted at a given rate. In such fashion we can calculate the price $P$ of a bond with coupon payments $c$ and principle $M$ as:
$$
P = \sum_{i=1}^n\frac{c}{(1+r_i)^i}+\frac{M}{(1+r_n)^n}
$$
where $r_i$ is the periodic discount rate for the $i$'th cashflow.
### Payment Frequency
Care should be taken to ensure that analysts understand the implications when coupons are paid multiple times per annum.
Some things to keep in mind
* If a bond with par value $P$ has a coupon of $X\%$ with quarterly payments, the payments will be $\$\frac{X \times P}{4 \times 100}$
*
## Yield
The yield to maturity of a bond is the annual discount rate that you would need to use in order for the present value of the bond's cashflows to equal the market price of the bond.
Solve for $r$, given $P, M, c$:
$$
P = \sum_{i=1}^n\frac{c}{(1+r)^i}+\frac{M}{(1+r)^n}
$$
The annual yield will then be either $(1+r)^m$ (effective yield) or $m\times r$ (bond equivalent yield // This is not totally clear to me.) where $m$ is the number of periods in a year.
From the equation for bond valuation, we can see that there is an inverse relationship between the price of a bond and its yield, all else being equal.
### Yield Curve Spot Rate Curve
Plotting the yields for bonds with varying maturities but similar credit risk results in the so-call yield curve. The shape of the yield curve contains information about market expectations over different time-frames. For example, in the context of a Treasury yield curve, the shape of the short-term (0-5 year) yield curve reflects expectations about monetary policy, while the long-term section contains information about expectations of fiscal policy.
## Rates: Spot and Forward
Of interest to us are the spot (or zero) rates offered by a bond issuer, particularly a national Treasury. The spot rate is the yield on a bond paying all interest at maturity. In general, issuers do not offer bonds of this type, so the spot rate must be inferred from the yields of bonds using a bootstrapping procedure.
One reason we are interested in spot rates is that it gives us an estimate for an appropriate discount rate for cashflows at a given time point (the $r_i$s in the above equation).
Using the Spot Rate curve, we can derive the Forward Rates implied by a set of bond yields. The forward rate is the interest rate at some future point in time that would need to be earned to equal the return from investing a given bond. This should not be confused with the market's expected interest rate in the future, but more like a way to hedge against future interest rates.
## Volatility
// This is likely inaccurate.
As discussed earlier, there is an inverse relationship between the price of a bond and its yield. This same property applies to a portfolio of bonds, for some measure of the yield of a portfolio. Of interest is the degree of volatility in price implied by changes in yields and vice versa. We have developed a number of measures to approximate this volatility.
```{r}
yield_seq = seq(0,0.5, length=100)
get_price = function(yield){
c = 5
M=100
n=20
price = 0
for (i in 1:n){
price = price + c/(1+yield)^i
}
price = price + M/(1+yield)^n
return(price)
}
prices = sapply(yield_seq, get_price)
plot(prices~yield_seq, type = "l", ylab = "Price", xlab = "Yield")
```
### Duration
One measure of volatility of the price of a portfolio of bonds as yields change is duration.
### Convexity
## Questions
1. The whole Yield-Price thing still confuses me. I would understand more if yield was something we could calculate independently of price. For example, can we ever say "We expect yields to increase by 1%, which will *cause* the price to drop X%"? How can we expect yields to do anything without first expecting price to do something? My current thinking is that this has more to do with volatility; Something like "High-Yield bond prices are less volatile in the presence of changing yields compared to Low-Yield", but even there I am confused at how we can measure something like "yield volatility". It all feels a bit circular and I don't know which factors are driving the others.
2. What is the relationship between Central Bank Interest Rates and Bond Yield? If interest rates are very high, I would assume that institutions are more likely to issue bonds as those would be an alternative way to raise capital. On the other hand, I don't actually know the effects of raising interest rates on investment opportunities. Do banks offer a higher return to institutions who lend them money?