Expected Spot Price Computation

[teetaz]

Hi David

May you please assist in the computation of the Expected future spot price. I thought it was computed using the following formular but im not getting it right.

EST=Soexp(Rf -K-Q)t

where
T=Time
Rf- Risk free rate
K- Rf+Beta( Market Risk Premium)

167.1. Assume the current (spot) price of the S&P 500 Index is 1,300 and the dividend yield is 2.0% per annum. The overall market return is 7.0% and the riskfree rate is 4.0% per annum; i.e., the market risk premium (a.k.a., equity risk premium, ERP) is 3.0%. Assume all yields/rates are continuously compounded and that we can use the capital asset pricing model (CAPM) where the index has a beta of 1.0 to predict the expected return of the index. What are, respectively, the expected future spot price in one year, E[S(1.0)], and the one-year forward price, F(0,1.0)?
a) E[S(1.0)] = 1,367 and F(0, 1.0) = 1,326
b) E[S(1.0)] = 1,394 and F(0, 1.0) = 1,326
c) E[S(1.0)] = 1,394 and F(0, 1.0) = 1,367
d) E[S(1.0)] = 1,394 and F(0, 1.0) = 1,394

Thank you.

 

[David Harper CFA FRM]

Hi teetaz,

(please note that unless i am mistaken, yours would simplify to:
E(ST)=S0exp(Rf - K-Q)t = S0(Rf - (Rf + beta*MRP) - Q))t = S0(- beta*MRP - Q)t, such that for an market uncorrelated commdodity, beta = 0, reduces to S0(- Qt), and this implies a lower future spot price or stock with a TSR of zero)

If we start with McDonald:
E[S(t)] = S(0)*exp[gT], where growth (g) = discount rate (k) - dividend yield (q).
E[S(t)] = S(0)*exp[(k-q)T]; please note (k) already has the riskfree rate, so we do not need it again.
As per CAPM, discount rate (k) = Rf + beta*MRP, could further expand to:
E[S(t)] = S(0)*exp[((Rf + beta*MRP) - q)*T].

Note this is consistent with Hull, where he generally assumes commodity has no systematic risk (i.e., beta = 0), such that:
E[S(t)] = S(0)*exp[((Rf + 0*MRP) - q)*T] = S(0)*exp[(r - q)*T]
... i think it is easy to miss that that beta is really meant to be here but just drops out under the assumption it equals zero!

So, here is: E[S(t)] = S(0)*exp[((Rf + beta*MRP) - q)*T] = 1300 *exp(4% + 1*3% - 2%) = 1300*exp(5%) = 1,367

And connecting it back to the fact that the forward price is a function of expected future spot and the risk premium,
where risk premium = riskless rate - discount rate:
F(0) = E[S(t)]*exp(r-k) --> 1326 = 1367*exp(4% - 4%); ie., the forward price does not use the discount rate, rather F(0) = exp(r - q)

If it helps, note the discount rate is 7% but the growth rate is 5%; if you owned the stock, your discount rate would be 7% because your total return (appreciation + dividends) is 7%. But 2% of that is dividends and does not contribute to the future expected spot price. Put another way, for the same discount rate, if dividends were 7%, you would expect no growth in the stock price.

Hope that helps, David