Relationships

Interest rates up, bond prices down
Interest rates down, bond prices up

the longer the maturity, the higher the volatility
sensitivities

Volatility
When people talk about bond volatility they are referring to how to measure the changes in price

Coupon Rate

The relationship between price and coupon

Notice that the present value of the coupon payments decreases the further into the future you go.
The further away the payment the less this is worth today.
Distant cash flows are more sensitive to interest rates.
the higher the coupon, the lower the volatility
A high-coupon bond will be exposed more to short and intermediate interest rates than will a low coupon bond with the same maturity.
A zero coupon bond will be exposed only to the interest rate associated with its maturity.

The smaller the coupon, the greater the price movement
A bond with a higher coupon rate will return a higher percentage of its current market value

Maturity Date

The relationship between price and maturity price
The price of longer-term bonds are generally affected more by changes in interest rates.
However for coupon bonds, maturity is a somewhat crude indicator of interest rate sensitivity.
price approaches par as bond approaches maturity
The longer the maturity, the greater the price movement

Yield

The relationship between price and yield is that they are inverse
When one goes up the other goes down.
All option free bonds have the same relationship
For any change in yield, the price movement is different for different bonds
For a small change in yield, the price movement of a bond is approximately symmetrical (increase or decrease)
For a large change in yield, the price movement of a bond is not symmetrical
For a large increase in yield the price movement is larger than an equivalent decrease in yield.
The lower the yield the greater the price movement

If the interest rate falls below the coupon rate, the bond trades at a premium and the bond price is greater than its nominal
If the interest rate rises above the coupon rate, the bond trades at a discount and the bond price is less than its nominal.
The present value of a bond can be viewed as a collection of independent cash flows
Each one of these cashflows could be treated like a zero coupon instrument
For example a 3 year treasury bond paying an 7% coupon with a par value of $100,000 is equivalent to:
5 zero coupon instruments maturing every 6 months, each one with a par value of (7%/2 * 100,000)
1 zero coupon instrument maturing in 3 years with a par-value of (semi-annual coupon + 100,000)

We know the treasury does not issue zero-coupon bonds/instruments so how can we calculate the present value of these zero-coupon instruments
The answer if that we can derive/create an artificial zero coupon yield curve using coupon paying treasury instruments
It is this yield curve that is used to obtain the different rates for the different cash flows.

SS - Equation

The following always holds:
A bond is said to be selling at a:
Par Value - Yield to Maturity = Coupon Rate
Discount - (below par) Yield to Maturity > Coupon Rate
Premium - (above par) Yield to Maturity < Coupon Rate

GRAPH

This relationship is convex
Price Up, Yield Down
Price Down, Yield Up
We talk about the sensitivity between the two

Pull to Par

As the maturity date approaches the price approaches par
Premium priced bonds will get cheaper
Discount priced bonds will get more expensive

Yield is a Price

Yield is as much a quote as a price