06_Present_Value1

06_Present_Value1 - Economics 251a John Geanakoplos Fall...

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Economics 251a John Geanakoplos Fall 2006 Present Value with Constant Interest Rates Suppose now that that we imagine a world with multiple periods and constant annualized interest rates in equilibrium. This is a good place to start because it is a world which is easy to understand, so that we can learn valuable lessons quickly. We might also think of it as a crude approximation to the real world. Perhaps most importantly, many real-world instruments have payo f sde f ned as if they existed in a world with constant interest rates. So suppose we are in equilibrium, and that at time 0 we have two sets of present value prices. For simplicity we assume that there is only one perishable good in the economy, called apples. Let π t = price at time 0 of $1 at time t p t = price at time 0 o foneapp leatt ime t WesawinFisher’stwo-periodmodelhowitmight make sense to suppose that these prices actually correspond to (implicitly) traded (portfolios of actual) securities. In a later chapter we shall show how at least the π t are calculated every morning in the real world by nearly every large business, bank, and hedge fund. For now let us assume they are known to us. We shall suppose that π t takes a very simple form given by constant nominal interest rates, and then later that p t also takes a simple form given by constant real interest rates. These considerations will give us some idea of the power of compound interest, of how coupon bonds, mortgages, perpetuities, and annuities work. Then we shall see what a good idea Irving Fisher had in describing lifetime or multi-period sequential budget constraints in terms of one present value budget constraint. 1 Present Value of Cash Flows with Constant Nominal Interest Rates Suppose that π t = 1 (1 + i ) t for all t =1 ,...,T Given any future cash F ows, we can always compute their present value as the price of "buying" them today. Theorem : If a stream of future cash F ows ( m 1 ,m 2 ,...,m T ) canbebough t(o racqu i redv iaaninve s tmen t )a tt ime0 ,andi fthe rei snoa rb i t rage ,theni t s price must be PV ( m, π )= π 1 m 1 + π 2 m 2 + ··· + π T m T ( m, i 1 1+ i m 1 + 1 i 1 i m 2 + + 1 i 1 i ... 1 i m T 1
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Cash Flow: ( m 0 ,m 1 ,...,m T ) 01 2 3T m 1 m 2 m 3 m 0 = 0 time m T PV ( m, i ) T X t =1 m t / (1 + i ) t 0123 T time FV ( m, i ) T X t =0 m t (1 + i ) T t =(1+ i ) T ( m, i ) T time 1.1 Doubling Time Doubling Time Theorem If i is near 7%, the number of years it takes to double your money at interest rate i is near . 72 /i . Proof $1 for n years at interest rate i gives (1 + i ) n =2 n log(1 + i )=log2 . 69 n = . 69 log(1 + i ) 2
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Now, recall the Taylor expansion of any di f erentiable function f around a : f ( x ) f ( a )+ f 0 ( a )( x a 1 2 f 00 ( a )( x a ) 2 + ··· If f ( x )=log x ,and a =1 , f 0 ( a )=1 /a , f 00 ( a )= 1 /a 2 = 1 . So around a , log(1 + i ) 0 + 1((1 + i ) 1) + 1 2 ( 1)(1 + i 1) 2 0+ i i 2 2 For i very small, i i 2 / 2 i ,so log(1 + i ) i n . 69 / log(1 + i ) . 69 /i =69 / 100 i .F o r i . 07 , log(1 + i ) i i 2 / 2= . 07 . 0049 / 2 . 0675 . 69 i i 2 2 . 69 . 0675 10 . 2 . 72 . 07 . 72 i .
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06_Present_Value1 - Economics 251a John Geanakoplos Fall...

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