Question #70426

• A coupon bond with a coupon rate of 8% and a face value of $1,000. Coupons
are paid out annually and the bond has 1 year to maturity. The current coupon
has just been paid out. The current price of the bond is $1018.772.
• A zero coupon bond with a face value of $1,000 and 2 years to maturity. The
bond trades at $907.029.
• An annuity that pays $50 every year for the next 3 years. The next payment will
be a year from now and the last payment will be 3 years from now. The annuity
is currently worth $136.967.
All these securities are risk-free. Note that there is no direct borrowing and lending
here, so if you want to borrow (lend) you need to sell (buy) an appropriate bond.
(a) Find the term structure of spot interest rates (i.e., r1, r2 and r3: rates for the
next 1, 2, and 3 years).
(b) Find the term structure of forward interest rates (i.e., rates between years 0 and
1, between years 1 and 2, and between years 2 and 3).

Expert's answer

a) The one year spot rate r1: 1,08/(1+r1)= $1018.772/$1000=> r1=6,01%

The two year spot rate r2: 1/(1+r2)2=$907,209/$1000=> r2=3,11%

The two year spot rate r3: $50*(1+r3)3= $136.967=> r3=40%.

b) The one-two year forward rate r0,1: 1,08/(1+r1)= $1018.772/$1000=> r1=6,01%.

The one-two year forward rate r1,2: 0/(1+r1)+1/(1+r1)(1+r1,2)=0,907209=>r1,2≈4%.

The one-two year forward rate r2,3: 0,4/(1+r1)+0,4/(1+r1)(1+r1,2)+1,4//(1+r1)(1+r1,2)(1+r2,3)= 0,136967=>r2,3=10%.

The two year spot rate r2: 1/(1+r2)2=$907,209/$1000=> r2=3,11%

The two year spot rate r3: $50*(1+r3)3= $136.967=> r3=40%.

b) The one-two year forward rate r0,1: 1,08/(1+r1)= $1018.772/$1000=> r1=6,01%.

The one-two year forward rate r1,2: 0/(1+r1)+1/(1+r1)(1+r1,2)=0,907209=>r1,2≈4%.

The one-two year forward rate r2,3: 0,4/(1+r1)+0,4/(1+r1)(1+r1,2)+1,4//(1+r1)(1+r1,2)(1+r2,3)= 0,136967=>r2,3=10%.

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