VALUATION OF FIXED INCOME SECURITIES

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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
rMM ___________________.....___ 0 1
2 3 4 5 n-1 n r
coupon interest rate M maturity (par value) n
term to maturity
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Bond Valuation
V rM(PVIF)i,1rM(PVIF)i,2 rM(PVIF)i,n
M(PVIF)i,n i market rate of interest Coupon
payments (rM) can be regarded as an annuity, V
rM(PVIFA)i,n M(PVIF)i,n or (1i)n
-1 1 V rM ------------- M ------------
(1i)n (1i)n
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Bond Valuation example
n10 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 gt r V lt M (discount) i lt r V
gt M (premium) i r V M (par) Yield-to-matu
rity the rate of return on a bond In the
example, the YTM is 10. A bonds YTM is the
market rate of interest for that risk group and
maturity.
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Valuation Between Interest Payment Dates
V invoice price of the bond c days until first
payment g number of days between two payment
periods P quoted price V - accrued
interest Accrued Interest rM (g-c)/g
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Valuation Example
Eg. N5 years,semiannual coupon r8,
i10, first payment 2 months from today.
V Invoice Price 953.29 Accrued 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 Moodys SP -----------------------------
------------- High Grade Aaa AAA Aa
AA ------------------------------------------- Inv
estment A A Grade Baa BBB -----------
-------------------------------- Speculative
Ba BB B B ----------------------------
--------------- Default Caa CCC Ca
CC C C D
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Interest Rate Risk
Example Two bond issues of ABC Co. N11 yr N2
10 yrs r 5
As term to maturity increases, value of the bond
becomes more sensitive to movements in market
interest rate.
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Bond Value and Coupon RatesExampleTwo issues of
ABC Co. n20 yrs, r110, r26

• 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 of two
bonds with coupon rates of 8 and 12 as the term
to maturity approaches
Assuming that interest rates remain the
same, bond value approaches to par over time
as term to maturity shortens.
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Term Structure of Interest Rates
Relationship between yield and time to
maturity. Example n1 i6 n5 i8 n20
i9
i
Yield Curve
Maturity
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Possible Explanations of the Term Structure
1. Expectations Hypothesis 1 in (1
i1)(1 1i2).(1n-1 in)1/n Example
i28 i16 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
proportional to term to maturity but inversely
proportional to coupon payments. Duration of a
bond is a measure that incorporates both factors
that affect volatility.
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Duration Examplen5 yrs, r8, i10
Bond Value 92.41 Macaulay 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 V084.94 when i15 After i declines to
12, V 100 V when term to maturity is 4
years V6 100 Future value of the first 6
coupon payments reinvested at 12 12 x PVIFA
12,6 97.38 Total savings 100 97.38
197.38 84.94 in 6 years grows to
197.38 Annual 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 6,830.82 Market
rate 10 Flat yield curve Strategy 1 Invest
in 1-yr bills with 10 interest 6830.82 -gt
7513.90 (1000.00) 6513.90 --gt
7165.29 (2000.00)
5165.29 --gt 5681.82 (2500.00)
3181.82 -gt3500 (2000) 1500
-gt1650 (1650)
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Immunization Example (Contd)
However, if interest rates fall, assets will be
short of liabilities Strategy 2 Invest in 3-yr
zero coupon bonds yielding 10 Duration of
Liabilities 1 1000 909.09 0.133 0.133 2 2000 1
652.89 0.242 0.484 3 2500 1878.29 0.275 0.825 4 20
00 1366.03 0.200 0.800 5 1650 1024.52 0.150 0.750
2.990 Duration 2.99 years
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Immunization Example (Contd)
Market rate 10, V 6,830.82 M 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 -gt7794.77 (1000.00)
6794.77 -gt7338.35 (2000.00)
5338.35-gt5765.42 (2500.00)
3265.42-gt3526.66 (2000.00)
1526.66-gt1650 (1650)
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Modified Duration
D MD ----------- (1 i) In
the example above, MD 4.28/1.10
3.89 Approximate Change in V -MD x Change
in yield Example If the yield decreases
from 10 to 8 Change in V -4.28 x (-2)
8.56 In fact when i10 V 92.41
i8 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 -----gt Linear Low coupon Long term ------gt
Convex
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Convexity (Contd)
Higher convexity means that when interest rates
go up, bond value declines slowly but when
rates decline, increase in bond price is
large Therefore high convexity is a
desirable feature. Factors that increase
convexity Low coupon Long term to
maturity Low yield
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Convexity (Contd)
Convexity 1/(1.10)22247.411/92.42
20.10 Appox. 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

Vt1 - Vt rM HPY
-------------------- Vt
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Approximate yield-to-maturity
Example V 877.11 n3 yrs r8 M1000
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Bond Investment Strategies
I. Passive Strategies Investing 100 in
1925 T-bill Deposits Stock Market AAA
Corporate Bonds Gold Inflation Passive
Strategies are better when Interest rate risk
is low, and Inflation 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
A
B
Maturity
Eg. 1 year bill i6 V1 943.40 B 2 year
zero coupon i8 V2 857.34 A Buy the 2-year
bond at 857.34, sell it next year at
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

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Strategies based on lack of market efficiency
(contd)

• Possible shortcomings of bond swaps
• time to execute the swap
• taxes
• transaction costs
• risk level of bonds
• Portfolio rebalancing adjusting the bond
  portfolio for the changes in market conditions