# Positive bond returns and positive yield change in the same month

Can a bond return over a month be positive while the bond also has a positive yield change for the month? How does this occur?

• In general, when the yield curve is positive, this indicates that investors require a higher rate of return for taking the added risk of lending money for a longer period of time. – user16651 Aug 18 '16 at 9:12
• As written, the question is simple. If a bond with 2pct coupon is priced at 100 and is still at 100 one month later, then the annualized return is 2pct for the month, obviously. – dm63 Aug 18 '16 at 10:05
• I guess you mean positive yield change? Yes, while the yield change decreases the value of the bond, the passage of time increases it thanks to the yield at the starting point. It is in the balance of yield change and "carry". – Mats Lind Aug 19 '16 at 5:12
• OP: Can you confirm @MatsLind edit? – Bob Jansen Aug 20 '16 at 10:29
• "positive yield change" is correct, yes. – Gregmf90 Aug 21 '16 at 13:17

## 1 Answer

The price change of a bond over a time period $dt$ is composed of:

1) An increase in value due to all coupons and principal repayment getting closer in time and so increasing their PV (assuming no yield change). This price change is roughly equal to $P .y . dt$ where $y$ is the bond yield and $P$ is the bond price.

2) An increase/decrease in value due to a decrease/increase in bond yields. This may be due to market moves. It will also include a component due to "roll-down" i.e. the tendency of the yield curve to be upward sloping such that as time passes, the corresponding yield declines. The size of the price change is $D . P . dy$ where $D$ is the modified duration of the bond.

3) Coupon payments when an amount $c/f$ is paid (coupon $c$ paid with frequency $f$). The bond price will fall by exactly $c/f$ across the coupon payment date. The coupon is a received payment that forms part of the period return and may be reinvested.

A single period total return will take into account all of these.

It is possible that the positive contribution of (1) can exceed the negative contribution of (2) if the yield increase is small and the bond duration is low, thereby resulting in a positive total return.