Time is money.pdf - Time is money The theory of interest rates Everybody needs money Good Bad Stock Lots of upside Lots of downside Bond Safety No

Time is money.pdf - Time is money The theory of interest...

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Unformatted text preview: Time is money The theory of interest rates Everybody needs money Good Bad Stock Lots of upside Lots of downside Bond Safety No upside Convertible Bond Lots of upside Limited downside The issuing entity Coupons are paid every 6 months. The amount represents the total for one year On this date, the issuer pays $100 to the bond holder (plus the last coupon payment) Bond prices Unlike stocks bond prices do not trade in an exchange prices are determine by independent brokers Clean vs. dirty prices Bonds, Yields With annual compounding Dirty price = X pi (1 + r) ti i = Accrued interest + Clean price Consider a bond that, with a payment of P (t, T ) at time t, p time T (and has no other intermediate payments). If the interest assumed constant, then Bonds, we would Yields have With annual compounding Number of days X P (t, T ) = e r(T t) . Dirty pi (1 + r) ti since theprice last = Hence, coupon payment i log P (t, T ) = Accrued price r = interest + Clean . (T t) This is useful since it is P , n not r that is observed (not quite: Accrued interest = ⇥ Annual Coupon Rate section). When r is not constant, 365 we simply define the yield rate Consider a bond that, with a payment of (t,TT)) at time t, pay log P P (t, r(t, T ) = payments). . If the interest r time T (and has no other intermediate (T t) assumed constant, then we would have Since r determines P , r determines the entire term structure. Canadian Bonds Bonds, Yields With annual compounding Bonds, Yields X With annual compounding Dirty price = pi (1 + r) ti Bonds, Yields X i Dirty price = pi (1 + r) ti With annual compounding = Accrued interest + Clean price i X Dirty price = = Accrued pi (1 interest + r) ti + Clean price log(P/N ) i r(T ) = T ) log(P/N =r(T Accrued interest + Clean price )= n T Accrued interest =on the ⇥ Annual Couponconvention. Rate The definition of interest rates is dependent compounding log(P/N ) 365 r(T=) =n ⇥ Annual Coupon Rate Accrued interest T X 365 r(ti ) ti For annual compounding, a series of cashflows P =Xn pisi ediscounted . r(ti ) ti Coupon Rate Accrued interest i pi⇥ P == 365 e Annual . X i ti X P1 =X pi (1 r(t + ir) . ) ti = i ppi(1e + r) ti.. PP 1 = i If the compounding is n times a year, i X i X Pn =X pi (1 + nr ) ttii n . ti n. 1 = i pp(1 i (1++r r) PP . n = i n) i X i X P1 =X pi er r tit.i n PnP=yields pii (1 In the limit, instantaneous compounding, pi + e nr )ti . . 1 = Compounding • • • • • i i Consider a bond that, with a X payment of P (t, T ) at time t, pays $1 at r a bond that, intermediate with a payment time t, paysrate $1 atr is P1 = pi payments). e oftiP. (t, T ) Ifatthe time Consider T (and has no other interest time T (and has nothen otherweintermediate payments). If the interest rate r is i assumed constant, would have constant, then we with would Instantaneousassumed compounding has many analytical advantages, fromt,now Consider a bond that, a have payment ) at time payson, $1 we at will use it r(Toft)P (t, T and P (t, T ) = e r(T . t) time T (and has no other intermediate If the interest rate r is most of the time. P (t, T ) = e payments). . assumed constant, then we would have Hence, Hence, log P r(T (t, T ) =T )log Pr(t, =P e (t, T ) t). (T t) . r= (T t) Hence, This is useful since it is P , not r that is observed (not quite: see next This is useful since it is P , not r log that is Tobserved (not quite: see next P (t, ) section). When r is not constant, we simply .define the yield rate r = section). When r is not constant, we (T simply t) define the yield rate log P (t, T ) log Pis(t,observed T) . This is useful since it is P , not r that (not quite: see next r(t, T) = r(t, T ) = . Zero coupon bonds Imagine a market where all bonds pay no coupons • Bonds will contain a single cash flow: a single payment of at the time of maturity • A bond will be characterized by three variables: • - The notional, i.e. the payment to occur at maturity. - The price of the bond - The time to maturity Bond price With these concepts in mind, we define the yield Notional yield Time to maturity Bonds, Yields With annual compounding Cashflow formula Dirty price = valuation p (1 + r) X i ti i = Accrued interest + Clean price log(P/N ) The yield curve can then r(T )be =used to calculate the price of T any series of future cashflows: n Accrued interest = ⇥ Annual Coupon Rate 365 X r(ti ) ti P = pi e . i Consider a bond that, with a payment of P (t, T ) at time t, pays $1 ime T (and has no other intermediate payments). If the interest rate r ssumed constant, then we would have i = Accrued interest + Clean price Bootstrapping log(P/N ) T r(T ) = n Accrued interest = ⇥ Annual Coupon Rate 365 If we return to a world where bonds have couponX payments, we can still recover the yield curve from those, avoiding the zero coupon bonds, as follows: P = pi e r(ti ) ti . i zero-coupon bonds, therefore • For maturities less than 6 months, all coupons are X P1 = pi (1 + r) ti . i • Pn = X Valid for 0<T<1 ti n pi (1 + nr ) . For maturities between six months and one year, bonds have a coupon payment within six months, and i another payment between six months and a year. X r ti P = p e . 1 i Known number, Unknown, can be solved i using the previous step as a one-variable equation P = p1 e r(t1 )·t1 + p2 e r(t2 )·t2 , 0 < t1 < Consider a bond that, with a payment of P (t, T ) From the market (Dirty price)T (and has no other intermediate payments). If time 1 2 < t2 < 1. at time t, pays $1 at the interest rate r is process can be extended to infinity, thereforehave allowing us to calculate the yield curve for all maturities, assumed constant, then we would • The assuming coupon bearing bonds for all maturities. P (t, T ) = e Hence, r(T t) . ...
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