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• ### 2.1 principal and interest

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• #### effective interest rate

r' --> 1+r'=(1+(r/m))^k
the equivalent yearly interest rate that would produce the same result after 1 year without 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)^n-1 + ... + xn

Example:
(-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)^n

continuous compounding:
PV = xk / (1 + r/m )^k

Example:
(-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 equal

x = (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 xk 0 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^n

Furthermore, if xk > 0, then the corresponding internal rate of return r=(1/c)-1 is positive

• ### 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 payment
-adjustable-rate 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 forever

P= A/(1+r)^k = A/(1+r) + P/(1+r) = A/r
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