Replicating a Stochastic Discount Factor with a Portfolio

# The portfolio whose return is the stochastic discount factor


# The portfolio whose return is the stochastic discount factor












I am trying to construct a portfolio whose return is $a + bm_{t+1}$ where $a$ and $b$ are some constants for a certain investor. $m_{t+1}$ is the stochastic discount factor at time $t+1$.

I am guessing that to have $a$ as return, you would have to buy $\frac{a}{1+r_f}$ units of the risk-free asset, whose existence we take for granted. Here $r_f$ is the risk-free interest rate.

Then for the portfolio that returns $m_{t+1}$, I proceed as follows. I have been given the hint that I should consider the regression $m_{t+1} = \sum_j^J \beta_jR_{j,t+1} +\varepsilon_{t+1}$ with $E[\varepsilon_{t+1}R_{i,t+1}] = 0$ for all $i=1,\ldots,J$. Here $R_{i,t+1}$ is the return on asset $i$ and $J$ is the total number of assets.

My guess was to find a $(\beta_j)_{j=1,\ldots,J}$ such that $m_{t+1} = \sum_j^J \beta_jR_{j,t+1}$ almost surely. Then if I show that $$E[\lvert m_{t+1}-\sum_j^J \beta_jR_{j,t+1}\rvert^2] = 0$$ I will have what I want.

This gives me something like $E[m_{t+1}^2] = \sum_j^J \beta_j$. I am not sure where I am going with this to be honest. I am fairly new to this stuff. I would appreciate it if someone could put me on the right path in solving this problem. Thanks.

## Answer by fni (score 1)

https://quant.stackexchange.com/a/20711

There is more than one way to construct the portfolio that mimics the evolution of the Stochastic Discount Factor (SDF). In these regards, one of the best references is Balduzzi&Robotti, Journal of Business & Economic Statistics, 2008 .

## Answer by Kiwiakos (score 0)

https://quant.stackexchange.com/a/25326

$\alpha$ units of cash and $\beta$ bonds? Presumably you mean 'value' rather than 'return', since the SDF is not a percentage return but a 'discount factor'.

## Answer by user9403 (score 0)

https://quant.stackexchange.com/a/30009

I'm assuming that you have a one period discrete model; that is you are currently at time $t$.

Let $S_t=\frac{1}{m_t}$ be the asset or portfolio of assets that is used as the discount factor. I assume that there are only two possible outcomes for this asset at time $t+1$: $S_u$ and $S_d$. Assuming the existence of a risk free asset, I can formulate the problem using matrices as follows:

$$\begin{bmatrix} S_u & 1+r_f \\ S_d & 1+r_f \end{bmatrix}\begin{bmatrix} \Delta \\ \Gamma \end{bmatrix} = \begin{bmatrix} a+b \frac{1}{S_u} \\a+b\frac{1}{S_d}\end{bmatrix}$$

Where $\Delta$ and $\Gamma$ are the units of each asset that I hold to construct the portfolio. Solving the matrix for $\Delta$ and $\Gamma$ should give you the answer.