WHAT DOES « PRESENT VALUE (PV) OF AN INVESTMENT »
MEAN ?
WHAT NEEDS TO BE INVESTED TODAY IN ORDER TO REALIZE A SPECIFIC FUTURE VALUE
Suppose a pension fund manager invests $10 million in a financial instrument that promises to pay 9.2% per yearfor 6 years.
$16,956,500
Suppose a pension fund manager invests $10 million in a financial instrument that promises to pay 9.2% per yearfor 6 years with interest paid twice a year
$17,154,600
FUTURE VALUE OF AN ANNUITY
[(1+r)t – 1]FVannuity= C -------------------
r
Annuity : when the same amount of money is invested periodically
C: amount of the annuityR : risk free ratet : period of the annuity
Suppose a portfolio manager purchases $20 million par value of a15-year bond that promises to pay 10% interest per year. The issuer makes a payment once a year with the first payment a year from now. Annual interest payments are reinvested at 8% annually
What will the portfolio manager have at the end of the 15-year period ?
•$20 million when the bond matures•Interest earned by investing the anual interest payments at 8%
$20M + 48,429,840 = 68,429,840$
_____C_____C_____C_____C_____C_____C_____C_____C+P0 4
C = COUPONP = PRINCIPAL
6 month
6 month
6 month
6 month
6 month
6 month
6 month
6 month
THE PRICING OF A 4-YEAR BOND
PV OF ITS EXPECTED CASH FLOWS
THE PRICING OF A BOND
SUM PV OF ITS EXPECTED CASH FLOWS
SOL
VE
FO
R P
V
Fv
Σ PV = Σ ------------- ( 1 + r)t
EXAMPLE
SUPPOSE AN INVESTOR EXPECTS TO RECEIVE $1000 SEVEN YEARS FROM NOW. SUPPOSE THE INVESTOR CAN EARN 5% ANNUALLY COMPOUNDED ON ANY SUM INVESTED TODAY.
WHAT IS THE PV OF THAT SUM ?
$710.68
THE PRICE OF ANY FINANCIAL INSTRUMENT IS EQUALTO THE PRESENT VALUE OF ITS EXPECTED CASH FLOWS.
Pv = Pn/ (1+r)t
1. IDENTIFY THE EXPECTED CASH FLOWS2. ESTIMATE THE APPROPRIATE YIELD
IDENTIFY THE EXPECTED CASH FLOW:
1. COUPONS PAID EVERY 6 MONTHS
2. COUPON RATE IS FIXED
3. THE NEXT COUPON PAYMENT IS PAID EXACTLY SIX MONTHS FROM NOW.
_____C_____C_____C_____C_____C_____C_____C_____C+P0 4
C = COUPONP = PRINCIPAL
4 YEAR BOND CASH FLOWS
6 month
6 month
6 month
6 month
6 month
6 month
6 month
6 month
P= C/(1+r)t + C/(1+r)t + C/(1+r)t …+(C+P) /(1+r)t
P= C/(1+r)t + (C+P) /(1+r)t
CF of 1st CouponPrincipal+ last coupon
CF of 2nd Coupon
Sum of allCash flows
Sum of last cash flow + principal
EXERCISE
WHAT IS THE PRICE (using both methods) OF A 4-YEAR BOND (FACE VALUE $1000) WITH A 5% COUPON PAID ONCE A YEAR WHEN THE YIELD IS AT 6%?
WHAT HAPPENS TO THE BOND PRICE IF THE YIELDGOES UP TO 8%
Approx. $965
Approx. 89.90
Multi yearly payments.……
With annual coupon payments, the price of our bond would be computed as the presente value of an annuity:
+
PV of the par maturity value 1000/(1+R)t
$173
$792
$965
1 - [1/(1+r)t ]PVannuity= C -------------------
r
WHAT IS THE PRICE OF A ZERO COUPON BOND EXPIRING IN 30 YEARS WITH A YIELD OF 9.4%
IT IS THE PRESENT VALUE OF ITS MATURITY VALUE
? 1000------------- = $67.52(1+0.094)30
P= C/(1+r)t + C/(1+r)t + C/(1+r)t …+(C+P) /(1+r)t
WHAT ARE THE FACTORS THAT WILL AFFECT THE PRICE OF A BOND ?
•CHANGE IN RATING
•TIME LEFT TO MATURITY
•CHANGE IN INTEREST RATES
•WHETHER THE BOND TRADES AT A DISCOUNT OR AT A PREMIUM
•CREDIT RISK
PRICE QUOTE AND
ACCRUED INTEREST
•PREMIUM BOND >100•DISCOUNT BOND < 100•PAR BOND=100
WHEN QUOTING BONDS, TRADERS QUOTE THE PRICEAS A PERCENTAGE OF PAR VALUE
Cash price = Quoted price +Accrued Interest
WHEN AN INVESTOR PURCHASES A BOND BETWEEN COUPON PAYMENTS, THE INVESTOR MUST COMPENSATE THE SELLEROF THE BOND WITH THE COUPON INTEREST EARNED FROMTHE TIME OF THE LAST COUPON PAYMENT TO THE SETTLEMENT DATE OF THE BOND .
IF A BOND QUOTES 95, IT MEANS IT IS TRADING AT $950
IF A BOND QUOTES 85.5, IT MEANS IT IS TRADING AT…
$855DISCOUNT
IF A BOND QUOTES 102, IT MEANS IT IS TRADING AT…
$1020PREMIUM
GOT IT ?????
Coroporate bonds quoting system
IF A BOND QUOTES 100, IT MEANS IT IS TRADING AT…
$1000 Par
ACCRUED INTEREST
The acrrued coupon is the coupon which the seller of bond has « earned » so far by holding the bond since the last coupon date.
Consider a Treasury bond trading at 90.50 or $905Payments are made each March1 and Sept. 1Coupon rate = 8%=$80
You buy the bond on July 3…
Treasury bond : act/act basis
03/01 09/01 03/01 09/0109/01
July 3Lastcoupon date
# days since last coupon date-------------------------------------------------------------------- x coupon amount $ 365
124accrued interest = --------x 80 = $27.178
365
So, you'll pay the bond 905.0 + 27.1 = $932.1
Clean price Dirty price
Consider a corporate bond trading at 105 or maturing March 30th 2015Coupon rate = 6%=$60
You buy the bond on June 15th.
Calculate the “dirty price”.
Corp. 30/360 basis
$1050
# days since last coupon date-------------------------------------------------------------------- x annual coupon rate $ 360
75accrued interest = --------x 60 = $12.50
360
So, you'll pay the bond 1050 + 12.50 = $1062.50
Clean price Dirty price