BOND VALUATON
EXAMPLE
F = FACE OR “PAR” VALUE
IF TELETRON ELECTRONICS HAS A BOND
OUTSTANDING THAT PAYS A $75 COUPON SEMIANNUALLY, WHAT IS ITS PRICE? R = 10%; 15 YEARS TO
MATURITY.
Ct = COUPON AMOUNT AT DATE t
SOMETIMES C1 = C2 = … = Cn = C
r = 10/2 = 5%
C = COUPON “RATE” AS A % OF PAR VALUE; THEN: C =
cF
P = 75 (PVIFA 5%, 30) + 1000(PVIF 5%, 30)
= 75 (15.3725) + 1000(0.2314)
= $1384.34
n = NUMBER OF PERIODS TO MATURITY
r = DISCOUNT RATE, DETERMINED BY FINANCIAL
MARKET AND BOND CHARACTERISTICS
(CHANGES DAY-TO-DAY)
P=
n = 15*2 = 30 PERIODS
C1
C2
C  Fn
+
+....+ n
2
(1 + r)
(1 + r)
(1 + r) n
USUALLY, COUPONS ARE PAID SEMI-ANNUALLY; THEN
r AND n SHOULD BE SEMI-ANNUAL.
1
SOME RULES OF BOND PRICING
(i)
(ii)
(iii)
(iv)
(v)
SUMMARY OF BOND PRICING RULES
WHEN c = r , THE BOND SELLS AT PAR
WHEN c > r, THE BOND SELLS AT A PREMIUM
(FOR n > 0);
WHEN c < r, THE BOND SELLS AT A DISCOUNT
(FOR n > 0);
WHEN r  , PRICE FALLS; AND WHEN r  ,
PRICE INCREASES.
THE LONGER THE MATURITY, THE MORE
SENSITIVE THE BOND PRICE TO INTEREST
RATE CHANGES.
AS n  0 (THE BOND APPROACHES
MATURITY), P  F (FACE VALUE); AT n = 0,
P = F.
Bond Price, p
r<c
r=c
F
r>c
2
0
Time to Maturity, n

PASSAGE OF TIME
YIELD TO MATURITY
REAL INTEREST RATE ( rc )
THIS IS THE RATE OF RETURN WE WOULD EARN IF WE
BOUGHT A BOND AT A GIVEN PRICE P, AND HELD IT TO
MATURITY: IN OTHER WORDS, IT IS THAT VALUE OF “r”
THAT SOLVES THE BOND PRICING EQUATION FOR A
GIVEN P.
IS DEFINED AS THE INTEREST RATE THAT WOULD
PREVAIL IN A WORLD WITHOUT INFLATION. DEPENDS
ON:
(1)
WHAT DETERMINES THE YIELD OF BONDS?
1.
2.
3.
(2)
REAL INTEREST RATE
INFLATION RATE
TIME TO MATURITY
INVESTOR PREFERENCES (CURRENT VS.
FUTURE CONSUMPTION)
PRODUCTION OPPORTUNITIES
INFLATION RATE
INFLACTION EXISTS IN MOST ECONOMIES, MEASURED
AS:
(1 + p) =
PRICE OF A BASKET OF GOODS NEXT YEAR
PRICE OF SAME BASKET THIS YEAR
p = (Ratio of PRICES) – 1
E.G. CONSUMER PRICE INDEX
3
RELATIONSHIP BETWEEN REAL AND NOMINAL
(MONEY) INTEREST RATES
(FISHER’S THEORY)
SPOT RATES
THESE ARE THE YTM OF BONDS THAT HAVE ONLY ONE
CASH FLOW TO THE INVESTOR (“PURE DISCOUNT
BONDS”)
(1 + r) = (1 + rc) (1 + p)
F
; r1 IS THE ONE YEAR SPOT RATE
(1 + r1 )
F
P02 =
; r1 IS THE TWO YEAR SPOT RATE,
(1 + r1 ) 2
P01 =
THE TERM STRUCTURE OF INTEREST RATES OF THE
YIELD CURVE
DEFINED AS THE RELATIONSHIP BETWEEN THE YTM
AND TIME TO MATURITY.
CALCULATED FROM THE PRICE OF A TWO YEAR PURE
DISCOUNT BOND.
TO STUDY THE YIELD CURVE, IT IS USEFUL TO THINK
OF “SPOT” RATES.
P03 =
F
; r3 IS THE THREE YEAR SPOT RATE,
(1 + r1 )3
CALCULATED FROM THE PRICE OF A THREE YEAR
PURE DISCOUNT BOND.
4
ONCE WE HAVE ALL THE SPOT RATES, WE CAN WRITE:
P=
EXAMPLE
C1
C2
C  Fn
+
+....+ n
2
(1 + r1 )
(1 + r2 )
(1 + rn ) n
COMPUTE THE SPOT RATES GIVEN THE FOLLOWING
ZERO COUPON BOND PRICES: P01 = $900; P02 = $820;
P03 = $725 (ASSUME $1000 FACE VALUES).
THE EARLIER FORMULA FOR PRICING BONDS APPLIES
ONLY WHEN THE YIELD CURVE IS FLAT.
P01 = 900 =
1000
1000
 r=
-1
(1 + r1 )
900
P02 = 820 =
1000
1000
 r2 =
-1
2
(1 + r2 )
820
P03 = 725 =
1000
1000
 r3 = 3
-1
3
(1 + r3 )
725
YTM
TYPICAL YIELD CURVES
0
Time to Maturity, n
5
DECOMPOSING A ZERO COUPON BOND INTO A
PORTFOLIO OF ZERO COUPON BONDS
ARBITRAGE IN THE BOND MARKET
1.
IF THE YIELD CURVE IS UPWARD SLOPING, THE
YIELD ON A COUPON BOND WILL BE LOWER
THAN ON A ZERO COUPON BOND (WHY?)
LAW OF ONE PRICE  SECURITIES, (OR PORTFOLIOS
OF SECURITIES) WHICH HAVE THE SAME RISKINESS
AND WHICH GIVE THE HOLDER THE SAME CASH
FLOW STREAM MUST SELL FOR THE IDENTICAL
(SAME) PRICE TODAY.
2.
IF THE YIELD CURVE IS DOWNWARD SLOPING,
THE YIELD ON A COUPON BOND WILL BE
HIGHER THAN ON A ZERO-COUPON BOND.
EXAMPLE: ARBITRAGE EXISTS?
BOND
 THE KEY IS TO REALIZE THAT THE YIELD ON
COUPON BONDS ARE WEIGHTED AVERAGES OF
THE CORRESPONDING SPOT RATES.
6
CASH FLOW AT DATE:
1
2
PRICE
A
80
1080
982
B
1100
-
880
C
120
1120
1010
LET US TRY TO REPLICATE BOND C WITH BONDS A
AND B:
nA (80) + nB (1100) = 120
(1)
nA (1080) + nB (0) = 1120
(2)
VALUING PERPETUAL BONDS
P=

FROM (2),
nA =
CONSOL BOND
COUPONS RECEIVED IN PERPETUITY; NO FACE
VALUE PAYMENT.
EXAMPLE
1120
= 1.037
1080
C = $50/yr
r = 12% = 0.12
P = 50/0.12 = $416.67
SUBSTITUTING IN (1),
nB =
C
r
120 - 1.037(80)
= 0.03367
1100
BY THE LAW OF ONE PRICE,
PC = nA (PA) + nB (PB)
= 1.037 (982) + 0.03367 (880)
= 1047.96  1010
THEREFORE, ARBITRAGE OPPORTUNITY EXISTS. BUY
BOND C AND SELL AN EQUIVALENT PORTFOLIO OF
BONDS A AND B.
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