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What's the rate of return on a mortgage?

If $T = \infty$, $V_t$ simplifies to $$V_t = S_t - S_0 (1 - \alpha) $$ and we have $$ \frac{\mathrm{d} V_t}{V_t} = \frac{\mathrm{d} S_t}{S_t} \frac{1}{1 - \frac{S_0}{S_t}(1-\alpha)}.$$ Letting $S_t = ...
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