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VALUATION OF FIXED INCOME SECURITIES
Bond: A debt instrument with periodic payments of interest and repayment of principal at maturity
rM rM rM rM rM rM rM+M|___|____|____|____|____|...…..|___ |0 1 2 3 4 5 n-1 n
r: coupon interest rateM: maturity (par value)n: term to maturity
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Bond Valuation
V= rM(PVIF)i,1+rM(PVIF)i,2 +………rM(PVIF)i,n + M(PVIF)i,n
i: market rate of interest
Coupon payments (rM) can be regarded asan annuity,
V= rM(PVIFA)i,n + M(PVIF)i,n
or
(1+i)n -1 1V = rM ------------- + M ------------
(1+i)n (1+i)n
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Bond Valuation examplen=10 years, coupon rate: 8%M= $1,000 Market rate : 10%
$80 $80 $80 $80 $80 $80 $180|___|____|____|____|____|...…..|___ |0 1 2 3 4 5 9 10
V= $80x(PVIFA)10%,10 + $1,000x(PVIF)10%,10
= $877.11
If i > r V < M (discount)i < r V > M (premium)i = r V = M (par)
Yield-to-maturity: the rate of return on a bondIn the example, the YTM is 10%.
A bond’s YTM is the market rate of interest forthat risk group and maturity.
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Valuation Between Interest Payment Dates
1
11/ )1()1()1(
1 n
tntgc i
M
i
rMrM
iV
V: invoice price of the bondc: days until first paymentg: number of days between two payment periods
P= quoted price = V - accrued interestAccrued Interest = rM (g-c)/g
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Valuation Example
Eg. N=5 years,semiannual coupon r=8%, i=10%, first payment 2 months from today.
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196/2 )05.01(
1000
)05.01(
4040
)05.01(
1
tt
V
V= Invoice Price = $953.29Accrued Interest = 40 x (4/6)
= $26.67
Quoted price = $926.62
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Risks Faced by a Bond Investor
• Default risk
• Interest rate risk (price risk)
• Reinvestment risk
• Call risk
• Inflation risk
• Foreign exchange risk
• Liquidity risk
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Rating
Category Moody’s S&P------------------------------------------High Grade Aaa AAA
Aa AA-------------------------------------------Investment A AGrade Baa BBB-------------------------------------------Speculative Ba BB
B B-------------------------------------------Default Caa CCC
Ca CC C C
D
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Interest Rate Risk
Bond ValueMarket Rate of
InterestFirst Issue:N = 1 yr
Second Issue:N = 10 yrs
5% 100.00 100.006% 99.06 92.647% 98.13 85.958% 97.22 79.87
Example: Two bond issues of ABC Co.N1=1 yr N2= 10 yrs r = 5%
As term to maturity increases, value of the bond becomes more sensitive to movementsin market interest rate.
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Bond Value and Coupon RatesExample:Two issues of ABC Co.
n=20 yrs, r1=10%, r2=6%
MarketInterest Rate
Bond 1R=10%
Percentchange
Bond 2R=6%
Percentchange
8% 119.64 80.369% 109.13 -8.78% 72.61 -9.64%10% 100.00 -8.36% 65.95 -9.17%11% 92.04 -7.96% 60.18 -8.75%12% 85.06 -7.58% 55.18 -8.31%
• Low coupon bonds are more sensitive to changes in market interest rates
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Value of a Bond in Time
Example: Market rate stays at 10%, values oftwo bonds with coupon rates of 8% and 12%as the term to maturity approaches:
Maturity Bond 1R=8%
Bond 2R=12%
5 92.42 107.584 93.66 106.343 95.03 104.972 96.53 103.471 98.18 101.820 100.00 100.00
Assuming that interest rates remain the same,bond value approaches to par over time asterm to maturity shortens.
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Term Structure of Interest Rates
Relationship between yield and time to maturity.
Example: n=1 i=6%n=5 i=8%n=20 i=9%
Maturity
i
Yield Curve
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Possible Explanations of the Term Structure
1. Expectations Hypothesis
1 + in =[(1+ i1)(1+ 1i2)…….(1+n-1 in)]1/n
Example: i2=8% i1=6% 1i2=?
1 + 0.08 = [(1+ 0.06)(1+ 1i2)]1/2
1i2 = 0.1004 or 10%
2. Liquidity Preference Hypothesis
Slope of the yield curve is higher than specified in expectations hypothesis
3. Segmented Markets Hypothesis
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Duration
Volatility in bond price is directly proportionalto term to maturity but inversely proportionalto coupon payments. Duration of a bond is a measure that incorporates both factors thataffect volatility.
n
ttt Vi
CtD
1 )1(
)(
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Duration Examplen=5 yrs, r=8%, i=10%
(1)Year
(2)PMT
(3)PVIF
(4)(2)x(3)
(5)(4)/V
(6)(1)x(5)
1 8 0.9091 7.27 0.0787 0.0787
2 8 0.8264 6.61 0.0715 0.1430
3 8 0.7513 6.01 0.0650 0.1950
4 8 0.6830 5.46 0.0591 0.2364
5 108 0.6209 67.06 72.57 3.6284
Total 92.41 4.28
Bond Value = $92.41Macaulay Duration = 4.28 years
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Hedging Interest Rate Risk
$12 $12 $12 $12 $12 $12 $112|___|____|____|____|____|...…..|___ |0 1 2 3 4 5 9 10
V0=$84.94 when i=15%
After i declines to 12%, V = $100V when term to maturity is 4 years:V6 = $100
Future value of the first 6 coupon paymentsreinvested at 12%: 12 x PVIFA 12%,6 = $97.38Total savings = $100 + $97.38 = $197.38
$84.94 in 6 years grows to $197.38Annual growth of 15%.
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Immunization Example
$1,000 $2,000 $2,500 $2,000 $1650|_____|______|______|______|______|0 1 2 3 4 5
Total Premiums = Assets = $6,830.82Market rate = 10% Flat yield curve
Strategy 1: Invest in 1-yr bills with 10% interest
6830.82 -> 7513.90 (1000.00) 6513.90 --> 7165.29
(2000.00) 5165.29 --> 5681.82
(2500.00) 3181.82 ->3500
(2000) 1500 ->1650
(1650)
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Immunization Example (Cont’d)
However, if interest rates fall, assets will be short of liabilities
Strategy 2: Invest in 3-yr zero coupon bondsyielding 10%
Duration of Liabilities:
1 1000 909.09 0.133 0.1332 2000 1652.89 0.242 0.4843 2500 1878.29 0.275 0.8254 2000 1366.03 0.200 0.8005 1650 1024.52 0.150 0.750
2.990
Duration = 2.99 years
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Immunization Example (Cont’d)
Market rate 10%, V = $6,830.82M = $9,091.82 Duration = 3 years
If interest rates fall from 10% to 8%,V= $9,091.82 x PVIF 8%,3 = $7,217.38
7217.38 ->7794.77 (1000.00) 6794.77 ->7338.35
(2000.00) 5338.35->5765.42 (2500.00)
3265.42->3526.66 (2000.00) 1526.66->1650
(1650)
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Modified Duration
DMD = -----------
(1 + i)
In the example above, MD = 4.28/1.10 = 3.89
Approximate Change in V = -MD x Change inyield
Example:If the yield decreases from 10% to 8%
% Change in V= -4.28 x (-2) = 8.56%
In fact when i=10% V = $92.41 i=8% V = $100 increase 8.21%
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Convexity
Price-Yield Relationship
V
Yield
The shape of the curve depends on the coupon rate and term to maturity
High coupon + Short term -----> LinearLow coupon + Long term ------> Convex
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Convexity (Cont’d)
Higher convexity means that when interestrates go up, bond value declines slowly; but when rates decline, increase in bond price is large
Therefore high convexity is a desirablefeature.
Factors that increase convexity:
* Low coupon* Long term to maturity* Low yield
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Convexity (Cont’d)
n
tt
t tti
C
idi
Vd
VdiVd
1
222
2
2
2
)()1()1(
1
Convexity
(1) (2) (3) (4) (5)Year Ct PVIF(8%,n) (1) x (2) t2 + t (3) x (4)1 8 0.9091 7.27 2 14.552 8 0.8264 6.61 6 39.673 8 0.7513 6.01 12 72.134 8 0.6830 5.46 20 109.285 108 0.6209 67.06 30 2011.79
92.42 2247.41
Convexity = [1/(1.10)2][2247.41][1/92.42] = 20.10Appox. Change in V = -MD x i + K x (i)2
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Alternative Measures of Yield
• Current Yield = rM / V
• Yield-to-maturity– Bond is held until maturity– All coupon and principal
repayments are made on time– Bond is not called before maturity– Coupon payments are reinvested
at yield-to-maturity
• Yield-to-call
• Holding period yield Vt+1 - Vt + rMHPY = -------------------- Vt
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Approximate yield-to-maturity
2MVnVM
rMi
Example V= $877.11 n=3 yrs r=8% M=$1000
0983.0
2100011.877
1011.8771000
80
i
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Bond Investment Strategies
I. Passive Strategies
Investing $100 in 1925T-billDepositsStock MarketAAA Corporate BondsGoldInflation
Passive Strategies are better when:Interest rate risk is low, andInflation is low and stable
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II. Active Strategies
• Strategies based on maturity structure– Maturity matching - duration– Spreading the maturity– Investing only in short term bills and
long term bonds
• Strategies based on forecasting interest rate movements– Interest rate fluctuations
• Buy when rates are high, sell when low• Increase duration if higher rates are
forecast, reduce duration otherwise
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- Riding the yield curve
• Investing in bonds assuming that the yield curve will not shift
i
Maturity
BA
Eg. 1 year bill i=6% V1 = $943.40 B 2 year zero coupon i=8% V2 = $857.34 A
Buy the 2-year bond at $857.34, sell it next yearat $943.40
HPY = (943.40 - 857.34) / 857.34 = 10.04%
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Strategies based on lack of market efficiency
• Junk bonds
• Bond swaps– Yield swap : same coupon, rating,
maturity and industry, different yield
– Exchange swap: same rating, maturity, industry, yield, different coupon. Exchange current yield for capital gains
– Tax swap: Selling a bond to realize a loss, and replacing it with a similar bond
– Swapping bonds with different tax status: eg. AAA corporate bond vs. municipal bond