• 8.3 the term structure of interest rates:
  • Zero coupon bonds, $t>0$;
• 9.2 Duration

8.3 the term structure of interest rates:

interest rate varies with the investment term.

Pure expectation theory:
view a sequence of short term bonds as a perfect substitute for a single longterm bond.

Liquidity preference theory:

• invest money for a sequence of short period so that the money is frequently accessible.

• issuers must offer long-term-bonds with higher rates, called liquidity premium

Preferred habitat theory:

• Investor’s own preferred investment term, but may be induced (感应) to leave their preferred habitat if the compensated yield rates are sufficiently higher, called term premium.

Market segmentation theory:
not one market, divided into separate market for each term.

Yield curve to see one’s annual yield rate.

Zero coupon bonds, $t>0$;

• spot rate / zero-coupon rate $r_t$ is the annual effective interest rate earned by money invested at time 0 for a period of t years.
  $(1+r_t)^t=1+i_{[0,t]}$

• Treasury bills, matures of one year or less, zero coupon bonds, semiannual coupons.

  • if spot rate is more than one year, based on treasury securities, we need to remove coupons. (transfer a coupon bond unto a sequence of zero coupon bonds with different terms, called strips)

• K$ invested at time 0 grows to $K(1+r_t)^t$ at time t.
  if $s>t$, investment grows to $K(1+r_s)^s$ at time s.
  The annual effective interest rate foe the interval $[t,s]$ is $f_{[t,s]}$, such that $(1+f_{[t,s]}^{s-t}) = \frac{(1+r_s)^s}{(1+r_t)^t}$

• $f_{[0,s]} = r_s$

• $f_{[t,s]}$ the theoretical forward rate / implied forward rate.

9.2 Duration

Assumption: compound interest, spot rate are all equal, if one spot rate changes, all change to remain a equal spot rate(parallel shifts).

Present Value $P(i)= \Sigma_{t≥0} C_t (1+i)^{-t}$

$\frac{P(i)-P(i_0)}{P(i_0} = \frac{P^,(i_0)}{P(i_0)}(i-i_0)$

• when $i>-1$, modified duration $D(i,1) = - \frac{P^,(i_0)}{P(i_0)}$

• $D(i,m) = - \frac{\frac{dP}{di^{(m)}}}{P(i_0)}, i^{(m)} = m[(1+i)^{\frac{1}{m}}-1]$

• $D(i,m)(1+\frac{i^{(m)}}{m})= D(i,1)(1+i)$

• $D(i,\infinite)= D(i,1)(1+i)$