The portfolio whose return is the stochastic discount factor

Question

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 1

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 2

$\alpha$ 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 3

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.