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Chapter 3: Fixed-Income Securities
Chapter 4: Common Stocks
Chapter 5: Forwards and Futures
Chapter 6: Options
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Lecture Lecture 33: Fixed-Income Securities: Fixed-Income Securities
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Intermediaries:
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Investors:
1. Governments2. Pension Funds3. Insurance Companies4. Commercial Banks5. Mutual Funds6. Hedge Funds7. Foreign Institutions8. Individuals
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Maturity (year) 1/4 1/2 1 2 5 10 30Price 0.991 0.983 0.967 0.927 0.797 0.605 0.187
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t 1 2 3 4 5Bt 0.952 0.898 0.863 0.807 0.757
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B =6
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time to maturity t (years) 1 2spot interest rate rt 0.05 0.07
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Year 0 1 21-yr borrowing 9.524 ¡ 10.000 02-yr lending ¡ 9.524 0 10.904Repatriation 0 10.000 0Net 0 0 10.904
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(1 + rt)t = (1 + rt¡ 1)t¡ 1(1 + ft)Spot and forward rates
-
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year1 2 3 4 5
-r1 = f1-r1 -f2
-r2 -f3-r3 -f4
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t!1(1+ ft )
ft =(1+ rt )
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t 1 2 3 4Bt 0.9524 0.8900 0.8278 0.7629rt 0.05 0.06 0.065 0.07
f4 = 8.51%
f4 = 8.51%
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(20, 000, 000)(0.8278) = $16, 556, 00016, 556, 000/0.7629 = $21, 701, 403
Year 0 1{2 3 4Purchase of 3-year bonds -16.556 0 20.000 0Sale of 4-year bonds 16.556 0 0 -21.701Total 0 0 20.000 -21.701
21, 701, 403
20, 000, 000 ¡ 1 = 8.51%
21,701,40320,000,000
!1 = 8.51%
$20m0.8278 = $16.556m
$16.556m / 0.7629 = $21.701m
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ft = E [r1(t)]ft = E [r1(t)] + LiquidityPremium
ft = E[r1(t)]
ft = E[r1(t)]+ LP
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ft = E [r1(t)]ft = E [r1(t)] + LiquidityPremium
Consider a situation where the short rate is 5%.
Expectations hypothesis:Suppose that the expected short rate for the following year is 6%.What is the price of a 2-year zero?
(1+ r2 )2 = (1+ r1)(1+ E[r2 ]) = 1.05 1.06B2 = $1000 / (1+ r2 )2 = $898.47
Suppose that an investor wants to invest for 1 year. Strategy A: invest into the 1-year zero.Strategy B: invest into the 2-year zero, and sell it after 1 year.
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ft = E [r1(t)]ft = E [r1(t)] + LiquidityPremium
What is the expected return of strategy B?
If the future short rate is 6%, the future bond price is
$1000/1.06=$943.4.
In this case, the return is ($943.4-$898.47)/$898.47=5%.
So, is the investor indifferent between strategies A and B?
What if the investor requires a discount to hold the 2-year bond?Suppose that the bond trades at $890.
B2 = $1000 / (1+ r2 )2 = $890
r2 = 6%
f2 =1.062
1.05= 7% = E[r1]+1%
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ft = E [r1(t)]ft = E [r1(t)] + LiquidityPremium
The liquidity preference hypothesis is based on the idea thatinvestors require a risk-premium to invest in long-term bonds.
Why?
Consider an investor who wants to invest for a 2 year period.
Such an investor bears roll-over risk if she invests in 1 year bonds.To hold 1 year bonds, the investor will require a risk premium...
! ft < E[r1(t)]
Forward rates contain information about 1. expected future short rates and2. risk premia.
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ft = E [r1(t)]ft = E [r1(t)] + LiquidityPremium
Yieldconstant liquidity premium
constant forward rate fyield curve:(1+yt)=(1+r1)(1+f)…(1+f)
constant expected short rate r1
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Bond yields
Prices in the bond market contain information aboutthe time-value of money:
Spot interest ratesForward interest rates
The term structure of interest rates
Next time: managing bond portfolios.
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yield (%)
6
Price (in log)
5.0
100 r
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D =PV[CFt ]
Bt = 1
Bt=1
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tt=1
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=D1+ y
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t CF PV(CF ) t¢PV(CF )1 3.5 3.40 3.402 3.5 3.30 6.603 3.5 3.20 9.604 3.5 3.11 12.445 3.5 3.02 15.106 3.5 2.93 17.597 3.5 2.85 19.928 103.5 81.70 653.63
103.50 738.28
D = (738.28)/103.50 = 7.13MD = D/(1 + y) = 7.13/1.03 = 6.92
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t CF PV(CF ) t¢PV(CF )1 3.5 3.40 3.402 3.5 3.30 6.603 3.5 3.20 9.604 3.5 3.11 12.445 3.5 3.02 15.106 3.5 2.93 17.597 3.5 2.85 19.928 103.5 81.70 653.63
103.50 738.28
D = (738.28)/103.50 = 7.13MD = D/(1 + y) = 7.13/1.03 = 6.92
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Duration
How does duration depend on bond maturity?What happens if the maturity increases?
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t CF PV(CF ) t¢PV(CF )1 3.5 3.40 3.402 3.5 3.30 6.603 3.5 3.20 9.604 3.5 3.11 12.445 3.5 3.02 15.106 3.5 2.93 17.597 3.5 2.85 19.928 103.5 81.70 653.63
103.50 738.28
D = (738.28)/103.50 = 7.13MD = D/(1 + y) = 7.13/1.03 = 6.92
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Par and premium bonds: duration increases.Discount bonds: duration can decrease!
How does the duration depend on the bond yield?What happens if the yield increases?
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t CF PV(CF ) t¢PV(CF )1 3.5 3.40 3.402 3.5 3.30 6.603 3.5 3.20 9.604 3.5 3.11 12.445 3.5 3.02 15.106 3.5 2.93 17.597 3.5 2.85 19.928 103.5 81.70 653.63
103.50 738.28
D = (738.28)/103.50 = 7.13MD = D/(1 + y) = 7.13/1.03 = 6.92
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Negative relation between the duration and the bond yield
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Y ield Price Using MD Di®erence0.040 96.63 96.35 0.290.035 100.00 99.93 0.070.031 102.79 102.79 0.000.030 103.50 - -0.029 104.23 104.23 0.000.025 107.17 107.09 0.080.020 110.98 110.67 0.32
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0.1%
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-
yield (%)
6
Price (in log)
3.0
103.5 r
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Duration gives the slope
? Convexity gives curvature
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(¢ B ) = ¢ B¢ y (¢ y) + 1
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Year 0 value In° ation rate (%) Year 1 nom. payo® Year 1 real payo®1000 0.10 1100 10001000 0.08 1100 10191000 0.06 1100 1038
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Description Moody's S&PG ilt-edge Aaa AAAVery high grade Aa AAUpper medium grade A ALower medium grade Baa BBBLow grade Ba BB
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Promised YTM =≥1000:00
321:97´1=10
¡ 1= 12%
Expected YTM =≥762:22321:97
´1=10¡ 1= 9%
Default Premium = Promised YTM ¡ Expected YTM= 12% ¡ 9% = 3%
Risk Premium = Expected YTM ¡ Default-free YTM= 9% ¡ 8% = 1%
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Promised YTM = (1000 / 321.97)110 !1 = 12%
Expected YTM = (762.22 / 321.97)110 !1 = 9%
Default premium = 12% - 9% = 3%Risk premium = 9% - 8% = 1%
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Y ield-to-maturity for a risky bond
8% - Default-free YTM9% - Expected YTM
12% - Promised YTM
Default-freerate
R iskpremium
DefaultpremiumY ield
spread
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