Deriving the Self-Financing Portfolio Value Identity
Summary
The document explains the discrete-time identity for a self-financing investment strategy. It starts with portfolio value as holdings multiplied by asset prices, then compares value at consecutive times. The change separates into a price-movement component and a component caused by changing holdings at the previous prices.
Self-financing means reallocations do not add or remove portfolio value, so the holdings-change component is zero. The portfolio’s value change over a period is then the new holdings multiplied by the asset price change. Summing this one-step identity across periods gives the cumulative portfolio value relation, with the timing of holdings made explicit. A two-asset example shows how selling five shares priced at 80 can fund the purchase of four shares priced at 100 without adding capital. The explanation assumes the stated discrete-time portfolio setup and does not address transaction costs or market frictions.
Key ideas
- Portfolio value at each time is the inner product of asset holdings and prices.
- A one-period value change separates into price changes and changes in holdings valued at prior prices.
- A strategy is self-financing when the holdings-change component has zero value.
- The cumulative value identity uses holdings from the start of each period to weight that period’s price change.
- The example illustrates a value-neutral exchange between two assets.
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Full text
# Discrete self financing strategy
# Discrete self financing strategy
> Let $H$ be an investment strategy in a discrete price model. Proof $H$ is self financing if and only if the following holds for the portfolio process $P_t$: $$P_t = P_0 + \sum_{s=1}^tH_{s-1}(X_s-X_{s-1}) \quad \forall t=1, \dots,T$$
$\textbf{Definition:}$ $H_t$ self financing strategy $\iff (\Delta H_t)^TX_{t-1}=0\ \forall t=1, \dots,T$.
We did not define what a portfolio process is so I guess the portfolio value process is meant here: $V=V(H)=H^TX$ with prices $X$.
I tried $$(\Delta H_t)^TX_{t-1}=0 \iff H_t^TX_{t-1}=H_{t-1}^TX_{t-1} \iff \Delta(H_t^T)_t=\Delta(H\circ X)_t \\ \iff H_t^TX_t=H_0^TX_0+(H\circ X)_t$$
$\forall t=1,...,T$. With $P_t:=H_t^TX_t$ I get $$P_t = P_0 + \sum_{s=1}^tH_s(X_s-X_{s-1})\ \forall t=1,...,T$$ but I need $H_{s-1}$ instead of $H_s$. I also tried integration by parts and got the same result... How do I proof the claim?
Thank you in advance!
## Answer by Kurt G. (score 3)
https://quant.stackexchange.com/a/75692
It is enough to consider one time step. The portfolio value changes by \begin{align} P_t-P_{t-1}&=H_t^\top X_t-H^\top _{t-1}X_{t-1}\\ &=\underbrace{H_t^\top(X_t-X_{t-1})}_{\textstyle(A)}+\underbrace{(H_t^\top-H^\top_{t-1})X_{t-1}}_{\textstyle (B)}\,. \end{align} The term $(B)$ reflects changes in the portfolio value that are due to reallocations and/or withdrawals resp. additions of assets to the portfolio with newly added funds. The first term is the value change due to the assets having new values at the end of the period.
When the strategy $H$ is self-financing only reallocations are allowed, that is, $B$ must be zero. Nothing is allowed that adds or withdraws value from the porfolio. An example with two assets having prices $$ X^1_{t-1}=100\,,\quad X^2_{t-1}=80 $$ and current allocations $$ H^1_{t-1}=8\,,\quad H^2_{t-1}=10\,. $$ Then you are allowed to sell five shares of the second asset to buy four of the first: $$ H^1_t-H^1_{t-1}=4\,,\quad H^2_t-H^2_{t-1}=-5\,. $$ Conclusion: $H$ is self-financing if and only if one of the two equivalent conditions hold
- $P_t-P_{t-1}=H_t^\top(X_t-X_{t-1})\,,$
- $(H_t^\top-H^\top_{t-1})X_{t-1}=0\,.$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.