# Calculating expected price of the stock, risk premium and expected return

Assume that there are two stocks that have the following payouts next year depending on the development of the economy Weak Economy Strong Economy Stock A EUR 478.00 EUR 1,196.00 Stock B EUR 966.00 EUR 248.00 The share price for stock A is EUR 1, 007.40. Furthermore there is a risk-free zero-coupon bond that pays EUR 1, 444.00, with a coupon of 5.00%. What are the price, expected return and risk premium for stock B?

Select one: • a. 367.84; 65.02%; 60.02% True.

b. 1, 743.08; —22.29%; —27.29%

c. 1, 743.08; —65.18%; —70.18%

d. 367.84; 81.31%; 86.31%

e.367.84; 81.31%;76.31%

The ans is (a) but I’ve no idea how they arrived at those numbers. Can anyone help me figure it out?

my guess here is that you need to use the binomial pricing method.
i could be wrong, and if that is the case, i would appreciate a CFA correcting the flaw in my argument, thanks.

you know the price of stock A at times 0 and the two possible price paths at time 1. this gives you u, the rate at which the stock price increases when the economy is strong. you also know the zero coupon bond, which gives you the information on the risk-free rate, r.
you can use u, r and maturity T (1 year) to find the risk-neutral probability that the stock price increases in price, p*. this is the risk-neutral probability that the economy will be strong (alternatively, you could do the same calculations using the price when the economy is weak. you would find and use d (= 1- percentage decrease) rather than u, but the computations should be similar).
because the stock B is trade at the same timeline as stock A, the risk neutral probability for A should also be the same as the risk-neutral probability for B (if the economy is strong for stock A, the economy is also strong for stock B). this means you can use p* to find the price of stock B at time 0, as well as compute the expected value of the stock
risk premium = expected return - risk free rate

edit: i now noticed that the price of stock B is lower when the economy is strong. given that is the case, p* would be the risk-neutral probability that stock B goes down, rather than up. you could use 1 - p* as the risk-neutral probability of stock B going up instead.

late update, and you probably already figured it out, but.
first, set the one-period binomial tree for stock A.
you have the root node Sa(t = 0) = 1,007.40
the up node Sa(t = 1, up) = 1196.00
the down node Sa(t= 1, down) = 478.00
you also have an effective risk -free rate rf = 5%
you have u = 1,196 / 1,007.40
d = 478.00 / 1,007.40
finally, the risk neutral probability in this tree is
p neutral = ( (1 + rf) - d ) / ((u - d) / Sa)
substituting the values give
p neutral = .807479109
(this is the risk-neutral probability, not the actual probability)

now, construct the binomial tree for stock B. you dont know the root node, but you know that the stock goes up to 966.00 with a risk-neutral probability of 1 - p neutral, and goes down to 248.00 with risk neutral probability p. the price of the stock is the risk-neutral expected value of stock B at time 1, discounted at the risk free rate.
that is,
price at time 0 = (966.00 x (1 - p neutral) + 248.00 x p neutral) / 1.05
this gives a value of 367.838095,
or 367.84 rounded to two decimal digits.

to find the expected return, recall that
return = (final price - initial price ) / initial price
E(return) = (E(final price) - initial price ) / initial price
this time, you need to use the actual probability rather than the risk-neutral probability to find the expected price.
it turns out that you need to use 50% as the probability of stock increasing or decreasing in price. maybe a CFA can elaborate why this (using 50% as the actual probabilities) works or makes sense.
the problem doesnt say that both scenarios (strong economy and weak economy) are equally likely.

expected return = ((.5 x 966 + .5 x 248) - 367.84 ) / 367.84 = .65017…
or 65.02% rounded to two digits.

risk premium = expected return - risk-free rate
= 65.02% - 5% = 60.02%

edit: why is my edit bolded?
edit2 fix.ed

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