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Transcript of Introduction to Fixed Income – part 2 Finance 30233 - Fall 2003 Advanced Investments Associate...
Introduction to Fixed Income – part 2
Finance 30233 - Fall 2003Advanced InvestmentsAssociate Professor Steven C. MannThe Neeley School of Business at TCU
Forward Interest ratesyield curves
spotparforward
Introduction to Term Structure
Term structure
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
7.0
6.5
6.0
5.5
5.0
yield
Maturity (years)
Typical interest rateterm structure
“Term structure” may refer to various yields:
“spot zero curve”: yield-to-maturity for zero-coupon bonds source: current market bond prices (spot prices)
“forward curve”: forward short-term interest rates: “short rates” source: zero curve, current market forward rates
“par bond curve”: yield to maturity for bonds selling at par source: current market bond prices
Forward rates
Introductory example (annual compounding) :
one-year zero yield : 0y1 =5.85% ; B(0,1) = 1/(1.0585) = 0.944733
two-year zero yield: 0y2 =6.03% ; B(0,2) = 1/(1.0603)2 = 0.889493
$1 investment in two-year bond produces $1(1+0.0603)2 = $1.1242 at year 2.
$1 invested in one-year zero produces $1(1+0.0585) = $1.0585 at year 1.
What “breakeven” rate at year 1 equates two outcomes?
(1 + 0.0603)2 = (1 + 0.0585) [ 1 + f (1,2) ]
breakeven rate = forward interest rate from year 1 to year 2 = f (1,2) (one year forward, one-year rate)
1 + f (1,2) = (1.0603)2/(1.0585) = 1.062103 f (1,2) = 1.0621 - 1 = 6.21%
and $1.0585 (1.0621) = $1.1242.
Forward and spot rate relationships : annualized rates
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Example: Using forward rates to find spot rates
n spot rate
(year) 0yn+1
0 f (0,1) = 8.0% B(0,1) = 0.92593 8.000%1 f (1,2) = 10.0% B(0,2) = 0.84175 8.995%2 f (2,3) = 11.0% B(0,3) = 0.75833 9.660%3 f (3,4) = 11.0% B(0,4) = 0.68318 9.993%
f (n,n+1)
forward rate
B(0,n+1)
bill price
6.0%7.0%8.0%9.0%
10.0%11.0%12.0%
0 1 2 3
Forward rates Spot rates
Given forward rates, find zero-coupon bond prices, and zero curve
Bond paying $1,000:maturity Price yield-to-maturityyear 1 $1,000/(1.08) = $925.93 0y1=[1.08] (1/1) -1 =8%
year 2 $1,000/[(1.08)(1.10)] = $841.75 0y2 = [(1.08)(1.10)](1/2)- 1 =8.995%
year 3 $1,000/[(1.08)(1.10)(1.11)] = $758.33 0y3 =[(1.08)(1.10)(1.11)] (1/3) = 9.660%
year 4 $1,000/[(1.08)(1.10)(1.11)(1.11)] = $683.18 0y4 =[(1.08)(1.10)(1.11)(1.11)] (1/4) = 9.993%
Yield curves
maturity
maturity
rate
rate
Forward ratezero-coupon yieldcoupon bond yield
Coupon bond yieldzero-coupon yieldforward rate
Typical upward slopingyield curve
Typical downward slopingyield curve
Coupon bond yield is “average” of zero-coupon yields
Facey
Cy
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Coupon bond yield-to maturity, y, is solution to:
10%
T B(0,T) 0y T B(0,t)Ct B(0,3)$100
1 0.92593 8.00% 9.262 0.84175 9.00% 8.423 0.75833 9.66% 7.58 75.83 Bond Value
Bond yield = 9.56% 25.26 75.83 101.09
bond: 3-year $100 face; annual coupon =
Bonds with same maturity but different coupons will have different yields.
15%
T B(0,T) 0y T B(0,t)Ct B(0,3)$100
1 0.92593 8.00% 13.892 0.84175 9.00% 12.633 0.75833 9.66% 11.37 75.83 Bond Value
Bond yield = 9.52% 37.89 75.83 113.72
bond: 3-year $100 face; annual coupon =
5%
T B(0,T) 0y T B(0,t)Ct B(0,3)$100
1 0.92593 8.00% 4.632 0.84175 9.00% 4.213 0.75833 9.66% 3.79 75.83 Bond Value
Bond yield = 9.61% 12.63 75.83 88.46
bond: 3-year $100 face; annual coupon =
Determination of the zero curve
B(0,t) is discount factor: price of $1 received at t; B(0,t) = (1+ 0yt)-t .
Example: find 2-year zero yielduse 1-year zero-coupon bond price and 2-year coupon bond price:bond price per $100: yield1-year zero-coupon bond 94.7867 5.500%2-year 6% annual coupon bond 100.0000 6.000%
B(0,1) = 0.9479. Solve for B(0,2): 6% coupon bond value = B(0,1)($6) + B(0,2)($106)
$100 = 0.9479($6) + B(0,2)($106) 100 = 5.6872 + B(0,2)($106) 94.3128 = B(0,2)(106)
B(0,2) = 94.3128/106 = 0.8897
so that 0y2 = (1/B(0,2))(1/2) -1 = (1/0.8897)(1/2) -1 = 6.0151%
“Bootstrapping” the zero curve from Treasury prices
Example:six-month T-bill price B(0,6) = 0.974812-month T-bill price B(0,12) = 0.9493
18-month T-note with 8% coupon paid semi-annually price = 103.77
find “implied” B(0,18):
103.77 = 4 B(0,6) + 4 B(0,12) + (104)B(0,18)= 4 (0.9748+0.9493) + 104 B(0,18)= 7.6964 + 104 B(0,18)
96.0736 = 104 B(0,18)B(0,18) = 96.0736/104 = 0.9238
24-month T-note with 7% semi-annual coupon: Price = 101.25
101.25 = 3.5B(0,6) + 3.5B(0,12) + 3.5B(0,18) + 103.5B(0,24)= 3.5(0.9748+0.9493+0.9238) + 103.5B(0,24)
B(0,24) = (101.25 - 9.9677)/103.5 = 0.9016
n spot rate
(year) 0yn+1
0 f (0,1) = 8.0% B(0,1) = 0.92593 8.000%1 f (1,2) = 10.0% B(0,2) = 0.84175 8.995%2 f (2,3) = 11.0% B(0,3) = 0.75833 9.660%3 f (3,4) = 11.0% B(0,4) = 0.68318 9.993%
f (n,n+1)
forward rate
B(0,n+1)
bill price
Holding period returns under certainty (forward rates are future short rates)
One year later:f (0,1) = 0y1 = 10%f (1,2) = 11%f (2,3) = 11%
One-year holding period returns of zero-coupons:invest $100:one-year zero: $100 investment buys $100/92.92593 = $108.00 Face value.
At end of 1 year, value = $108.00 ; return = (108/100)-1 = 8.0%
two-year zero: $100 investment buys $100/84.175 = $118.80 Face value.at end of 1 year, Value = $118.80/1.10 = $108.00 ;
return = (108/100) -1 = 8.0%three-year zero: $100 investment buys $100/75.833 = $131.87 face value
at end of 1 year, value = $131.87/[(1.10)(1.11)] = $108.00 ;return = (108/100) -1 = 8.0%
If future short rates are certain, all bonds have same holding period return
n spot rate
(year) 0yn+1 now
0 f (0,1) = 8.0% B(0,1) = 0.92593 8.000%1 f (1,2) = 10.0% B(0,2) = 0.84175 8.995% 11.00%2 f (2,3) = 11.0% B(0,3) = 0.75833 9.660% 8.00%3 f (3,4) = 11.0% B(0,4) = 0.68318 9.993% 9.00%
one year later
possible short rate (0y1) evolution:
f (n,n+1)
forward rate
B(0,n+1)
bill price
Holding period returns when future short rates are uncertain
One year holding period returns of $100 investment in zero-coupons:one-year zero: $100 investment buys $100/92.92593 = $108.00 Face value.
1 year later, value = $108.00 ; return = (108/100)-1 = 8.0% (no risk)
two-year zero: $100 investment buys $118.80 face value. 1 year later: short rate = 11%, value = 118.80/1.11 = 107.03 7.03% return
short rate = 9%, value = 118.80/1.09 = 108.99 8.99% return
Risk-averse investor with one-year horizon holds two-year zero only if expected holding period return is greater than 8%:only if forward rate is higher than expected future short rate.
Liquidity preference: investor demands risk premium for longer maturity
Term Structure Theories
1) Expectations: forward rates = expected future short rates2) Market segmentation: supply and demand at different maturities3) Liquidity preference: short-term investors demand risk premium
maturity
rate
Expected short rate is constant
Forward rate = expected short rate + constant
Par Bond yield curve is upward sloping
Yield Curve: constant expected short ratesconstant risk premium
Yield curves with liquidity preference
rate
Expected short rate is declining
Forward rate
Par bond yield curve
Liquidity premiumincreasing with maturity
maturity
maturity
rate
Expected short rate is declining
Forward rateHumped par bond yield curve
Constant Liquidity premium