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Assess how a bond's price is affected when its time to maturity changes.

FRM® Part I · Valuation and Risk Models · Interest Rates · FRM1.VR.10.f

The idea

A bond's price is the present value of its remaining cash flows, so as time passes with the yield curve unchanged the price moves in a predictable way. A zero-coupon bond accretes toward par, because its price is Price = par ÷ (1 + y)ⁿ, where n is the remaining years. Each year closer to maturity raises that price by a factor of 1 + y, so by roughly the yield. A coupon bond priced at a premium declines toward par and one priced at a discount rises toward par. At maturity every bond equals its face value. At a constant yield, the total return over a period equals the yield when coupons are reinvested at that yield: the price change plus the coupon income. Do not confuse this pull to par with the effect of yield changes. The total price change is the sum of a time effect and a yield effect.

Formulas

Price of a zero-coupon bond with n years left

P = F ÷ (1 + y)ⁿ

F
the par (face) value
y
the annually compounded yield
n
the remaining years to maturity

As n falls by one at a constant y, the price rises by a factor of 1 + y. Separate this time effect from any change in y.

A practice question

From the question bank, with the reason each wrong option is wrong.

A bond trades at a premium to par and its yield stays constant as time passes. Its price will:

  1. rise further above par, because the bond becomes safer over time.Why not: Confuses credit risk with the pull to par; at a constant yield a premium bond's price falls.
  2. stay constant, because the yield has not changed.Why not: Ignores that fewer coupon payments remain, so the premium amortizes.
  3. decline toward par, reaching par at maturity. - correct
  4. fall below par, reaching a discount at maturity.Why not: Overshoots: at maturity every bond is worth its face value, not less.

A premium bond's coupons exceed what the yield requires, and the premium shrinks as fewer coupons remain. At constant yield the price converges to par at maturity.

A harder one

On an upward-sloping curve the annually compounded spot rate is 3% for 2 years and 4% for 3 years. An investor buys a 3-year zero-coupon bond and holds it for one year, and the spot curve is unchanged at the end of the year. The one-year holding return is closest to:

  1. 4.00%Why not: Assumes the return equals the 3-year spot rate; the bond also rolls down to a lower-yield point.
  2. 6.03% - correct
  3. 3.00%Why not: Uses the 2-year spot rate as the return, ignoring the price gain from yield at purchase.
  4. 5.69%Why not: Divides the price gain by the ending price instead of the starting price.

Price now = 100/1.04³ = 88.90; in one year it is a 2-year zero priced at 100/1.03² = 94.26. Return = 94.26/88.90 - 1 = 6.03%. It exceeds 4% because the bond rolls down the upward curve.

The hardest tier

A 5-year bond with face value 100 pays an 8% annual coupon and is priced to yield 4%. An investor holds it for 2 years, the yield stays at 4%, and coupons are reinvested at 4%. The total 2-year return, including the price change, coupons and reinvestment income, is closest to:

  1. -5.69%Why not: Counts only the price change and ignores coupon income.
  2. 7.89%Why not: Adds the two coupons but not their reinvestment income.
  3. 13.58%Why not: Counts only the coupon income, ignoring the price decline of the premium bond.
  4. 8.16% - correct

Price now 117.81; after 2 years 111.10. Future value = 111.10 + 8 x 1.04 + 8 = 127.42, a return of 8.16% = 1.04² - 1. At a constant yield, price decline plus coupons earns exactly the yield.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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