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2 The basic theory of interest

2.1 principal and interest
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effective interest rate
r' > 1+r'=(1+(r/m))^kthe equivalent yearly interest rate that would produce the same result after 1 year without compounding 
effective rate of intersest conintuous compounding
1+r'=e^r 
2.3 present and future values of streams
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future value of a stream
FV = x0(1+r)^n + x1(1+r)^n1 + ... + xnExample:(2, 1, 1, 1), 10%FV = 2(1.1)^3 + 1(1.1)^2 + 1.1 + 1 = 0.648 
present value of a stream
PV = x0 + x1/(1+r) + x2/(1+r)^2 + ... + xn/(1+r)^ncontinuous compounding:PV = xk / (1 + r/m )^kExample:(2, 1, 1, 1), 10%PV = 2 + 1/1.1 + 1/(1.1)^2 +1/(1.1)^3 = 0.487 
main theorem on present value
the cash flow streams x and y are equivalent for a constant ideal bank with interest rate r iff the present values of the stream, evaluated at the bank's interest rate, are equalx = (x0, x1, ..., xn)y = (y0, y1, ..., yn) 
2.4 Internal rate of return
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main theorem of internal rate of return
Suppose the cash flow stream x has x0<0 and xk0 for all k with at least one term being striclty positive. Then there is a unique positive root to the equation: 0=x0 + x1c +x2c^2 + ... + xnc^nFurthermore, if xk > 0, then the corresponding internal rate of return r=(1/c)1 is positive 
3 fixedincome security

3.1 the market for future cash
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time deposit account
the interest is guaranteed, where the deposit must be maintained for a given lenght of time 
a banker's acceptance
sell banker's acceptance to someone else at a discount before the time has expired 
variations onthe standard mortage
balloon payment: modestsized periodic payments for several years followed by a final balloon paymentadjustablerate mortgages: mortgage adjust the effective interest rate periodically according to an interest rate index 
3.2 value formulas
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perpetual annuity (perpetuity)
fixed income instrument which pays a fixed sum periodically foreverP= A/(1+r)^k = A/(1+r) + P/(1+r) = A/r
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