Bond Price

thm. The price of a bond is the following:

B (n) = \frac{1 - ( 1 + r _{y} /2 ) ^{- n}}{r _{y} /2} C + \frac{M}{( 1 + \frac{r _{y}}{2} ) ^{n}}

where…

$B (n)$ is the current market price of the bond
$n = 2 τ$ is the periods left to maturity (where $τ$ is the years left to maturity)
- not periods passed! periods left
$M$ is the Principal amount
- Normally use $100 or$ 1000
$C$ is the per-period (=semi-annual) coupon payment
- normally, coupon is quoted at annual coupon rate $r_{C}$ (in percent) so $C = \frac{r _{C}}{2} M$
$r_{y}$ (or $y$ ) is the yield to maturity ( $\approx$ discount rate)
- $r$ is the current yield. $r = \frac{2 C}{B ( n )}$
$n$ is the number of coupon payments (semi-annual for Treasury bonds)

The Three Rates of Bonds

$r_{y}$ : Yield to Maturity—“What’s the return rate for the coupons if I hold until maturity?”
$r$ : Current Yield—“Coupon rate, but at the current bond price”
$r_{C}$ : Coupon Rate—“Coupon rate (at principal price).”

Price and rates:
- Trades at premium $B (n) > M$ ⇔ $r_{y} < r < r_{C}$
- Trades at discount $B (n) < M$ ⇔ $r_{y} > r > r_{C}$

Observe…

Market Value = \frac{Price of par}{100} \times # of pars

The causality of the variables of a bond. The market’s expected rate of return of a bond is the yield. With this yield we can calculate its price.

Price—Yield Curve (mathematical)

Price and Yield are inversly correlated → Price-yield curve looks like this:

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Duration is the gradient of this price-yield function:

Duration := \frac{d P}{d y}

Dollar Duration (DV01) is the change in price due to 1% change in yield [=$1 change in yield per par]

DV01 := \frac{Δ P}{\pm 1% \cdot y}

Dollar Duration can be only used to approximate price changes for small changes in yield (~50bp)
Duration is used to estimate risk (Price sensitivity against yield = DV01 risk
Dollar Duration is quoted as an absolute value. (We all know that it’s mathematically negative)
DV01 Risk:= $\frac{Δ Market Value}{Δ1% yield}$ = “ $Δ$ market value for 1% yield change” = $\frac{# par}{100} \cdot DV01$
DV01 has no relationship with volatility

Example calculation

Time to Maturity	DV01	Yield Vol.	e.g. change in yield	change in price
2 year	$1.8	10%p	20bp	$0.36
10 year	$7.2	5%p	10bp	$0.72

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⇒ Yield $\propto$ Maturity, because the farther away the cash flows are, the more risky it is [ $∵$ uncertainty]

Inverted Yield Curve: Bond investors think a recession is looming → Treasury will increase rates
Left intercept is the fed rates & short-mat bonds → high movement right side is long-mat bonds → low movement

def. Price[rate of return] volatility = $SD [Price]$ (in $)

def. Yield volatility = $SD [Yield]$ (in %p)

def. DV01 Risk = Change in MV per change in 1% yield