10/4/2023 0 Comments Drawdown formula excel![]() The relationships are now v(i+1) = (v(i)+b)*(1+r) - pĪpplying the same sort of analysis as set out above about leads to the analogy of Equation m' as v(i) is now the amount of loan outstanding at the start period i, immediately before the amount b is added to the loan. Here, assume that at the start of each period (including the first), an additional amount b is borrowed. Which is, effectively, the formula found previously. Or, dividing the numerator and denominator on right hand side by (1+r)^n Which, with a bit of re-arranging can be written as Now, set the general value m to the number of periods n, set i to 1 and note the boundary conditions thatĭoing this provides a version of Equation m' as Or, simplifying the final denominator term In Equation m, the geometric series is written backwards and x = 1+r, so the equation can be simplified to A geometric series (Google it) is a sum in which each successive term is the previous term multiplied by a constant amount.įor a general finite geometric series written The factor that p multiplies is a finite geometric series written backwards. There is an emergent pattern (*) in Equations 1, 2 and 3 which can be used to write a general Equation m as So, substituting for v(i+2) using Equation 2 yields Which, with a little re-arranging can be written as Now using Equation 1 to substitute for v(i+1) in this second equation yields Since Equation 1 applies to all periods, it follows that This starts with the relationship between the loan amount outstanding in successive periods ![]() ![]() The analysis below, which determines p in terms of L, r and n, assumes payments are made at the end of each period. Note that if the payments p are made at the start of each period (rather than the end) the first equation would change to:Īnd v(i) would be the amount outstanding at the start of period i immediately before the payment p is made. The other two equations simply state the start and end conditions of the loan. Simply says, in words, that the amount outstanding at the beginning of period i is increased by adding interest accrued during period i before being reduced by the payment amount made at the end of period i to get the amount outstanding at the beginning of the following period (period i+1). v(i) is the amount of the loan outstanding at the beginning of the i'th period. The basic mathematical relationships involved are: v(i+1) = v(i)*(1+r) - pĪn amount L is borrowed for n periods at an interest rate of r per period and payment amounts of p are made at the end of each period. To get to the type of expression quoted in it is necessary to understand those mathematics and then adapt them to deal with the scenario outlined. Functions such as PMT shield users from having to understand the mathematics underlying Excel's financial calculation capabilities.
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