# 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

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