(17B/19B) GPM: initial payment

[Albert Chan]

(05-27-2024 02:23 PM)Albert Chan Wrote:  We can remove loops, and get direct formula for pv/pmt

Let q = s*d12/c12 = s / (1+i)^12
...
--> pv/pmt = -q^5 / c300 - (q^5-1)/(q-1) / c12

There is a simpler way to show this, without too much math.
Let t = (1+i)^-12      --> q = s*t

Year 6 to 30: c300/t^5 * pv6' + s^5 * pmt = 0      --> pv6' / pmt = -q^5/c300
Year 5:            c12/t^4 * pv5' + s^4 * pmt = 0      --> pv5' / pmt = -q^4/c12

--> pv4' / pmt = -q^3/c12
--> pv3' / pmt = -q^2/c12
--> pv2' / pmt = -q/c12
--> pv1' / pmt = -1/c12

All the pv(k)' is "pulled" to the beginning of loan --> pv = Σ pv(k)'

pv/pmt = -1/c12 - q/c12 - q^2/c12 - q^3/c12 - q^4/c12 - q^5/c300

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This setup break the connection between years, making analysis simple.
We could do Year 6 to Year 30 individually, combine them, and get same result.

Year 6: c12/t^5 * pv6' + s^5 * pmt = 0      --> pv6'/pmt = -q^5/c12
Year 7: c12/t^6 * pv7' + s^5 * pmt = 0      --> pv7'/pmt = -q^5/c12 * t
Year 8: c12/t^7 * pv8' + s^5 * pmt = 0      --> pv8'/pmt = -q^5/c12 * t^2
...

c12 = i / (1 - (1+i)^-12) = i / (1 - t)

Σ(pv(k)', k=6 .. 30) / pmt
= -q^5 * (1 - t) / i * (1 + t + t^2 + ... + t^(30-6))
= -q^5 * (1 - t^25) / i
= -q^5 / c300