1 Internal rate of return, bonds, yields

1 Copyrightc 2005 by Karl Sigman1 Internal rate of return , bonds, Internal rate of returnGiven a deterministic cash flow steam, (x0, x1, .. , xn), wherexi(allowed to be positive, 0 ornegative) denotes the flow at time periodi(years say), we already studied thenet present value,N P V=n i=0xi(1 +r) the known (annual say) interest rate available to us all. For comparison purposes,if the investment stream actually came from only withdrawing and depositing money in abank account at interest rater, thenN P V= 0 (for example if you place $100 in a savingsaccount for 1 year at fixed rater, then the cash flow stream is ( 100,100(1 +r)) and NPV=100 + 100(1 +r)(1 +r) 1= 0). But an arbitrary cash flow does not come from such a simplebank account scheme, henceN P Vis typically quite different from 0. (Hopefully positive!) Butthis motivates hunting for a value ofrthat would result inN P V= 0:Definition Internal rate of return (IRR) of the stream is a numberr >0such thatn i=0xi(1 +r)i= is the interest rate implied by the cash flow stream (not the current real interest rate, whateverit may be).

2 Changing variablesa= (1 +r) 1, equivalently we must solve for a positive root0< a <1of a degreenpolynomial:n i=0xiai=x0+ax1+ +anxn= 0;(1)thenr= 1/a general, there is no such solution but there always is under suitable sufficient conditions suchasIRR Sufficient Conditions:Ifx0<0andxi 0,1 i nand ni=0xi>0, then a solutionr >0exists. This conditionmeans that we put money in only at time0(initial investment), and then get back money at allother times, and that the sum over all those other times exceeds our initial investment. (Easyexample: buy a bond ).A proof of the above: Letf(a) =x0+ax1+ +anxn, as from (1). Then under thesufficient conditions we havef(0)<0 andf(1)>0; thus by the continuity of the functionf=f(a) ( , the intermediate value theorem of calculus) there must exist a value ofa (0,1)for whichf(a) = can be used as an alternative to NPV for purposes of comparing two different streamsto decide which is better.

3 The idea is that of the two, you would choose the one having the1largest IRR. (Of course, just as any stream withN P V <0 would be avoided, any stream withIRR< r= current interest rate would be avoided too.) Unfortunately it is possible that thetwo methods yield different conclusions; that is, IRR might rank your first steam higher thanthe second, while NPV might rank your second steam higher than your first!Finally we point out that a solution to IRR must be solved for numerically in general, usingNewton s method, for Bonds, fixed income securitiesA bond is an example of afixed incomesecurity, meaning that the payoff is essentially predeter-mined, deterministic, fixed: You invest a fixed amount of money now and are guaranteed fixed,known payoffs in the future. (Stock on the other hand is a so-calledrisky security, differentsince the payoff is random and potentially highly volatile.) Here we introduce some basics ofbonds, their payoff structure, yields , and with no couponsBuying a bond means you are lending money now (timet= 0) to some institution (government,business, etc.)

4 That needs to raise capital, and are promised back the money back at a pre-specified future time with a profit. Along the way, there may be so-calledcouponpayments,meaning, for example, that every 6 months until maturity, you receive a fixed amount simplest case, however, is when there are no coupons, azero coupon bond . For example,suppose you buy a 5-year $1000 bond , which means that 5 years from now you will receive aface-valueof $1000, but nothing in between. The pricePthat you pay now is the present value,P= 1000/(1 +r)5, whereris the interest rate. For example ifr= , thenP= 822. Theinterest rates used for such bonds depend on the length of maturity; you would receive a higherrate for a longer time period. For example, if instead of 5 years, you bought the bond for 10years then you might receive a rate of instead of , and thenP= 1000/( )10= various interest rates are referred to asspot the lifetime of your bond before maturity, interest rates might change causing theprice of new bonds to be different than what you paid.

5 If the rates go up then the bond pricegoes down, whereas if the rates go down, then the bond price goes up. Bonds are typicallytraded on the open market, so you could, for example, sell your 5 year $1000 face-value bond (bought at timet= 0) one year later (at timet= 1) as a 4 year $1000 face-value bond at aprice of the formP= 1000/(1 +r)4, whereris the appropriate spot rate for 4-year bonds attimet= with couponsMost bonds give you fixed amount payments at regular intervals up to the face value termination(maturity) date; these payments are referred to ascouponpayments since some time ago whenbuying a bond you would be given paper coupons to turn in for your payments. For example,USA Treasury Notes and Bondsare bonds that pay you a fixed amount every 6 months up totermination at which time you get both the last fixed amount plus the face value. The price ofsuch a bond can be computed by using present values with current spot rates ( , the currentzero coupon rates ).

6 2-year$1000bond exampleFor example, consider a 2-year $1000 bond , that has coupons every 6 months in the amount of$25, for a total of four times untilt= 2 years at which time you receive $1025. To price this2bond we need to know the current , r1, ; the spot rates for bonds havingmaturity from 6 months to 2 years. The first payment of 25 has PV of 25(1 + ) 1, thesecond has PV of 25(1 +r1) 1, the third has PV of 25(1 + ( )) 1and the last has PV of1025(1 + (2)) 1; thus the pricePis given byP= 25(1 + ( ) ) 1+ 25(1 + (1)r1) 1+ 25(1 + ( ) ) 1+ 1025(1 + (2)r2) the above formula we used simple linear scaling for the interest rate computations (sincewe did not specify the type of compounding). Let s instead assume continuous compoundingin which case the price becomesP= 25e ( ) + 25e (1)r1+ 25e ( ) + 1025e (2)r2.(2)For example, suppose , r1= , andr2= ThenP yieldsGiven a bond , we can solve for the implied interest rate, that is, the IRR of the cash flow streaminduced by the bond (IRR is defined in Section ); (x0, x1.)

7 , xn) wherex0= P <0, andxi= the payment at theithperiod. This rate, denoted by , is called theyieldof the bond ,and it always exists because the IRR sufficient conditions given in Section us suppose for example, that a 2-year $1000 bond is issued with pricePas in (2). Thenthe yield is the solution to25e ( ) + 25e (1) + 25e ( ) + 1025e (2) =P= 999.(3)Lettinga=e ( ) we equivalently must solve for a zero of a 4th degree polynomial,25a+ 25a2+ 25a3+ 1025a4 999 = solution (solved numerically) isa .976; = 2 ln (a) = yield formulaHere we offer a general formula for finding the yield of a given bond that has priceP. Let usassume that the face value is denoted byF, the coupon payments are givenm 2 times peryear (every 1/myears). Let us assume further thatKdenotes the coupon amount per period,and that there are 1 n mperiods remaining. We wish to compute the IRR, that is, theimplied annual interest rate that, when compounded every 1/munits of time for computingthe PV of the bond payoffs, would produce the priceP: Solve for >0 such thatP=F[1 + ( /m)]n+n i=1K[1 + ( /m)]i(4)=F[1 + ( /m)]n+K( /m){1 1[1 + ( /m)]n}.

8 (5)The formula is derived as follows: There aren 1 periods remaining, so the PV of theface valueFis given byF[1+( /m)]n. Meanwhile, theithremaining coupon payment has a PV ofK[1+( /m)]i, and these payments are summed over allnremaining periods. The final simplifiedform of the answer is due to the formulas for geometric series discussed in Lecture Notes 1 (in3the Appendix, for example). In our 2-year $1000 bond example,m= 2,n= 4,F= 1000 andK= compounding is continuous, then the formula is given byP=F e (nm)+n i=1Ke (im)(6)=F e ( m)n+Ke ( m){1 e ( m)n1 e ( m)}.(7)By changing variables,a= /m, e mrespectively, in either case, finding reduces to theproblem of finding the zero of a polynomial (ina) of degreen+ of Deposit (CD)ACertificate of Deposit (CD)is a fixed income security offered by your bank. It is just likea bond except there are no coupons and it can t be re-sold on the open market: You mustwait until maturity to get the agreed upon face valueF.

9 Typically maturity for a CD is from6 months to several years, unlike bonds which might have maturities as long as 30 years. Inessence, a CD is like a savings account that offers a higher interest rate since you are agreeingnot to take any withdraws until a fixed date in the future (at which time you withdrawF).Just as for a zero coupon bond , the price of such a CD with maturity (say)Tis given byF e rTwhereris the spot rate for such a