4 ways to calculate the discount factors PDF

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4 ways to calculate the discount factors t

1+ y 1 ( t ) ¿ Method 1) 1 d(t )= ¿

BD 6.6

Where y 1 (t) is the zero-coupon rate for a t year zero coupon bond. It is safe to assume that the zero-coupon bonds are risk-free. This implies that y 1 (t) is also the t year risk-free rate, r f . The zero-coupon rate, y 1 (t) , is the yield to maturity (YTM) for a zero-coupon bond. (EAR = YTM). YTM can also be called effective yield. This method is used when Zero-coupon rates are given. Method 2) Bootstrapping (idiot task) Identification: The prices on the bonds are given. Installment ¿ MMULT PA d( 1) Bond A Y 1 , A Y 1 , A Y 1 , A Bond B Y 1 , B Y 1 , B Y 1 , B ¿ · d( 2) = PB Bond C Y 1 ,C Y 1 ,C Y 1 ,C d (3) PC ⏟

[

][ ] [ ]

⏟ ¿

d ( t ) vector

⏟

Price ventor

Naming: Y 1 , name Y = Installment In the problem all prices on different bonds will be given. text.

P A , PB

and PC

d (t) ¿ (matrix form), where  Y is the installment matrix,  d (t) Y¿ ¿ and  P , is a column-vector with bond prices. d (t) ¿ ¿ ¿

will be given in the

is the discount factor vector,

(Y ) Y −1=Minverse ⏟ Ctr + Shift+ Enter

Method 3: Price(s) of zero-coupon bond has (have) to be given. d ( t) =

Price of a t year zero−coupon bond Face value of at year zero−coupon bond

Zero-coupon bonds d(t) are Market d(t)

Example: The price of a 4-year zero-coupon bond with a face value of 100 DKK is 94,25. d ( 4 )=

94,25 =0,9425 , which tells us that the price today of 1 DKK in t=4 is worth 0,9425 DKK 100

today.

Method 4: When we have YTM (EAR=Effective yield=YTM) given in the text. d ( t) =

1 1 = t ( 1+YTM ) (1+ r )t

So r is the discount rate in general, while the YTM (yield to maturity) is a specific discount rate that makes the NPV of a bond or loan = 0. YTM =IRR of a bond=EAR=Effective yield=Effective an

Conversion from d(t) to discount rate: −1 ( t ) r=d ( t ) −1

F .23 : Lower simple arbitrage bound :C 0 ≥0∧P0 ≥0

USA

EU

USA

EU

F .24 : EU VSUSA :C 0 ≥ C 0 ∧P 0 ≥ P0

F .25 : USA : American pull options can not have a negative time vale USA CUSA ≥ S t − K ∧ Pt ≥ K −St t

F .27 Call optionupper arbitrage bound :C 0 ≤ S0...