# Review of Time Value of Money. FUTURE VALUE Fv = P V ( 1 + r) t FUTURE VALUE OF A SUM F v INVESTED TODAY AT A RATE r FOR A PERIOD t :

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Review of Time Value of Money

FUTURE VALUE Fv = P V ( 1 + r) t FUTURE VALUE OF A SUM F v INVESTED TODAY AT A RATE r FOR A PERIOD t :

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 year for 6 years. \$16,956,500 Suppose a pension fund manager invests \$10 million in a financial instrument that promises to pay 9.2% per year for 6 years with interest paid twice a year \$17,154,600

FUTURE VALUE OF AN ANNUITY [(1+r) t – 1] FV annuity = C ------------------- r Annuity : when the same amount of money is invested periodically C: amount of the annuity R : risk free rate t : period of the annuity

Suppose a portfolio manager purchases \$20 million par value of a 15-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\$

PRESENT VALUE OF AN ANNUITY 1 - [1/(1+r) t ] PV annuity = C ------------------- r

_____C_____C_____C_____C_____C_____C_____C_____C+P0 4 C = COUPON P = PRINCIPAL 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 SOLVE FOR P V F v Σ P V = 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 EQUAL TO THE PRESENT VALUE OF ITS EXPECTED CASH FLOWS. P v =  P n / (1+r) t 1.IDENTIFY THE EXPECTED CASH FLOWS 2.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 = COUPON P = PRINCIPAL 4 YEAR BOND CASH FLOWS 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 1 st Coupon Principal+ last coupon CF of 2 nd Coupon Sum of all Cash 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 YIELD GOES 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 ] PV annuity = 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 PRICE AS 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 SELLER OF THE BOND WITH THE COUPON INTEREST EARNED FROM THE 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… \$855 DISCOUNT IF A BOND QUOTES 102, IT MEANS IT IS TRADING AT… \$1020 PREMIUM 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 \$905 Payments are made each March1 and Sept. 1 Coupon rate = 8%=\$80 You buy the bond on July 3… Treasury bond : act/act basis 03/0109/0103/0109/01 July 3 Last coupon date

# days since last coupon date -------------------------------------------------------------------- x coupon amount \$ 365 124 accrued interest = --------x 80 = \$27.178 365 So, you'll pay the bond 905.0 + 27.1 = \$932.1 Clean priceDirty price

Consider a corporate bond trading at 105 or maturing March 30th 2015 Coupon 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 75 accrued interest = --------x 60 = \$12.50 360 So, you'll pay the bond 1050 + 12.50 = \$1062.50 Clean priceDirty price

HAVE A GOOD WEEK !

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