FM for Actuaries

236 CHAPTER 7
hypothesis. Note that under the pure expectations hypothesis equation (7.25) can
be generalized to
E[ t i S τ ]=iF t,τ , for 0 <t,τ, (7.26)
so that the prevailing forward rates have important implications for the expected
future values of the spot rates at any time t over any horizon τ.
We have seen that if the term structure is upward sloping, the forward rate of
interest is higher than the spot rate of interest. Thus, under the pure expectations
hypothesis, an upward sloping yield curve implies that the future spot rate is expected
to be higher than the current spot rate. 9
On the other hand, if long-term bonds command a risk premium, an upward
sloping yield curve may not imply that the spot rate is expected to rise. For example,
assuming the constant-premium model and taking expectations of equation (7.24),
we have
E[ 1 i S 1 ]=iS 1 + iF 2 − E[i(2) H ]=iS 1 + iF 2 − (iS 1 + ϱ) =iF 2 − ϱ.
Now as an upward sloping yield curve implies i F 2 >iS 1 , we conclude E[ 1i S 1 ] >
i S 1 − ϱ. However, this does not imply E[ 1i S 1 ] >iS 1 . Thus, after taking account of
the risk premium, the expected future spot rate may be lower than the current spot
rate even if the term structure is upward sloping.
7.8 Summary
1. A bond’s yield to maturity is the internal rate of return an investor would
achieve if she purchases the bond at its current market price and holds it until
maturity, assuming all coupon and principal payments are received as scheduled.
There are other bond yield measures such as current yield, nominal
yield and yield to call.
2. The par yield is the coupon rate of interest of a bond such that the bond
price is at par based on the prevailing term structure. The par yield may be
regarded as a summary measure of the existing term structure.
3. The holding-period yield is the average rate of return of a bond over the
holding (multiple) period. It takes account of the actual interest-on-interest
earnedonanex post basis, and is also called the realized compound yield
or total return. On an ex ante basis, the holding-period yield may be computed
under different assumed scenarios and is a useful tool in active bond
management.
9 For example, since (1 + i S 2 ) 2 =(1+i S 1 )(1 + i F 2 ), i S 2 >i S 1 (upward sloping term structure)
implies E( 1i S 1 )=i F 2 >i S 2 >i S 1 .