thm. The price of a bond is the following:

where…

  • is the current market price of the bond
  • is the periods left to maturity (where is the years left to maturity)
    • not periods passed! periods left
  • is the Principal amount
    • Normally use 1000
  • is the per-period (=semi-annual) coupon payment
    • normally, coupon is quoted at annual coupon rate (in percent) so
  • (or ) is the yield to maturity (discount rate)
    • is the current yield.
  • is the number of coupon payments (semi-annual for Treasury bonds)

The Three Rates of Bonds

  1. : Yield to Maturity—“What’s the return rate for the coupons if I hold until maturity?”
  2. : Current Yield—“Coupon rate, but at the current bond price”
  3. : Coupon Rate—“Coupon rate (at principal price).”
  • Price and rates:
    • Trades at premium
    • Trades at discount

Observe…

  • .

  • Causality: Exogenous factors → yield → price. 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.|260

  • In a portfolio of bonds…

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:

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

  • 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:= = “ market value for 1% yield change” =
  • DV01 has no relationship with volatility

Example calculation

Time to MaturityDV01Yield Vol.e.g. change in yieldchange in price
2 year$1.810%p20bp$0.36
10 year$7.25%p10bp$0.72

Yield(–Maturity) Curve (empirical)

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Yield 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

Risk & Volatility

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

def. Yield volatility = (in %p)

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

  • Yield Vol > RoR Vol ← mathematical relationship (price formula)
  • Maturity Price Vol.