What it means
A bond is a promise: pay the holder a fixed amount of interest at set dates and return the principal at the end. The value of that promise depends on how much those future payments are worth today, which depends on the rate of return an investor can get elsewhere on similar promises.
If a bond pays 5% and new bonds of the same quality pay 6%, no one will pay full face value for the 5% bond; its price falls until its yield to a buyer matches 6%. If new bonds pay 4%, the 5% bond is attractive and its price rises above face value.
The mechanics are discounted cash flow. Each coupon and the final principal are discounted at the required yield for the number of periods until they arrive, and the sum is the price.
A bond priced above face value trades at a premium; below, at a discount; at face value, at par. The yield that makes the discounted cash flows equal the market price is the yield to maturity, the standard measure of a bond's return.
Two factors drive the required yield. The risk-free rate for the bond's maturity, set by government bond yields, reflects the time value of money.
The credit spread above that rate reflects the risk that the issuer will not pay, and it widens as the issuer's creditworthiness falls or markets become nervous. A bond's price can therefore fall because interest rates in general have risen, because the issuer's credit has deteriorated, or both.
Sensitivity to yield changes depends on the bond's maturity and coupon. Longer bonds and lower-coupon bonds are more sensitive, because more of their value lies in distant payments.
Duration measures that sensitivity: a bond with a duration of 7 will fall about 7% in price for a 1 point rise in yield. Investors use duration to manage interest rate risk, and issuers use the same mathematics in reverse to decide when and at what maturity to borrow.
In practice
Real-world examples.
Example
An insurer holding a portfolio of ten-year bonds sees their market value fall 15% when central bank rates rise 2 points, though it will still receive every coupon and the full principal if it holds to maturity.
Example
A company issuing a five-year bond sets the coupon at 5.5%, the yield investors require on the day, so that the bond prices at par.
Example
A fund manager buys a bond at $92 when its issuer is downgraded, judging that the credit spread has widened more than the issuer's real risk justifies.
Think of it
“Bond valuation is like calculating what future rent payments are worth today. You're finding the present value of a series of cash flows.
Formula
Calculation
Bond Price = Sum over each period t of [ Coupon / (1 + y) to the power t ] + Face Value / (1 + y) to the power n
where y is the required yield per period and n the number of periods to maturity.
For a bond with level coupons: Price = Coupon x [ (1 minus (1 + y) to the power minus n) / y ] + Face Value / (1 + y) to the power n
Current Yield = Annual Coupon / Price
Worked example. A corporate bond has a face value of $1,000, an annual coupon of 5% ($50 a year), and four years to maturity.
If investors require a yield of 6%:
- Present value of coupons = $50 x [ (1 minus 1.06 to the power minus 4) / 0.06 ] = $50 x 3.465 = $173.26
- Present value of face value = $1,000 / 1.06 to the power 4 = $1,000 / 1.2625 = $792.09
- Price = $965.35 (a discount to par, because the coupon is below the required yield)
If investors require 4%:
- Present value of coupons = $50 x 3.630 = $181.49
- Present value of face value = $1,000 / 1.1699 = $854.80
- Price = $1,036.30 (a premium)
If investors require exactly 5%, the price is $1,000 (par).
Sensitivity: a 2-point rise in yield from 4% to 6% cut the price by $70.95, or 6.8%. For a similar bond with 15 years to maturity, the same yield change would cut the price by about 19%, which is why long bonds are riskier for holders when rates rise.
Current yield at the $965.35 price = $50 / $965.35 = 5.18%, lower than the 6% yield to maturity because the current yield ignores the $34.65 gain the holder will make when the bond repays $1,000 at maturity.Case study
Seen in the real world.
A regional bank's treasury held $400 million of long-dated government bonds bought when yields were 1.5%, classified as available for sale and carried at fair value. When yields rose to 4% over eighteen months, the bonds' market value fell by $85 million, the loss was recognised in equity, and the bank's regulatory capital ratio fell towards its minimum. The bonds themselves were as safe as ever; the government would pay every coupon and repay every dollar.
But the valuation was correct: an investor could now buy the same cash flows for $85 million less. The bank had funded long fixed-rate assets with short-term deposits whose cost rose with rates, and the mismatch, not the credit, was the risk.
The bank raised equity, shortened the duration of its bond portfolio, and introduced a limit on the interest rate sensitivity of its balance sheet. The treasurer's summary was that the bonds had never been risk-free; they had been default-free, which is a different thing.
Watch out
Common mistakes.
- Assuming a bond's value is its face value. Between issue and maturity the price moves with yields and credit.
- Comparing bonds on coupon rate rather than yield to maturity. The coupon is fixed at issue; the yield reflects the price actually paid.
- Ignoring duration. Two bonds with the same yield can have very different price sensitivity to rate changes.
Questions
People also ask.
Why do bond prices fall when interest rates rise?
Because the bond's fixed coupons become less attractive than new bonds paying the higher rate, so the price must fall until the bond's yield matches the market.
What is the difference between yield to maturity and current yield?
Current yield is the annual coupon divided by the price. Yield to maturity also accounts for the gain or loss between the price paid and the face value repaid, and is the true return for a holder to maturity.
Does a fall in a bond's price matter if I hold to maturity?
Not for the cash you receive, which is unchanged if the issuer pays. It matters for the accounts if the bond is carried at fair value, for the opportunity cost of holding a below-market yield, and if you need to sell before maturity.
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