**In this article we will discuss about the trade-off between risk and return of investment. **

Let us suppose that a person wants to invest his savings in two assets—Treasury bills which are almost risk-free, and a representative group of stocks. He would have to decide how much to invest in each asset. He might, for instance, invest only in Treasury bills, only in stocks, or in some combination of the two.

Let us denote the risk-free return on the Treasury (T.) bill by R_{f}. Since the return is risk-free, the expected and actual returns are the same. In addition, let the expected return from investing in the stock market be R_{m} and the actual return be r_{m}. The actual return is risky.

At the time of the investment decision, we know the set of possible outcomes and the probability of each, but we do not know what particular outcome will occur. The risky asset will have a higher expected return than the risk-free asset (R_{m} > R_{f}). Otherwise, risk-averse investors would buy only Treasury (T) bills and no stocks.

**The Investment Portfolio:**

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To determine how much money the investor should put in each asset, let us set the fraction of his savings placed in the stock market equal to b and the fraction used to purchase Treasury bills equal to (1 – b).

**The expected return on his total portfolio, R _{p}, is a weighted average of the expected returns on the two assets: **

R_{p} = bR_{m} + (1-b)R_{f} (7.6)

Let us suppose, for example, that R_{f} = 0.4 (or 4%), R_{m} = .12 (or 12%), and b =1/2. Then

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R_{p} = 8%. How risky is the portfolio? One measure of its riskiness is the SD of its return, i.e.,

**The Investor’s Choice Problem: **

We have now to determine how the investor should choose the fraction b. To do so, we have to obtain the equation that would give his risk-return trade-off.

**This equation, again, is obtained by rewriting equation (7.1): **

**Risk and the Budget Line:**

Equation (7.9) is a budget line because it describes the trade-off between risk (σ_{Rp}) and expected return (R_{p}). Let us note that it is the equation of a straight line. Since R_{m}, R_{f} and σR_{m} are_{ }positive constants, the slope of the line (R_{m} – R_{f})/ σR_{m}, is also a positive constant as is the intercept R_{f}.

The equation says that the expected return of the portfolio, R_{p}, increases as the SD of that return, σR_{p}, increases. We call the slope of this budget line, (R_{m} – R_{f})/ σR_{m}, the price of risk, because it tells us the rate of change of R_{p} w.r.t. σR_{p}, i.e., how much extra return he requires if he is to accept an additional unit of risk.

For example, he could invest all his funds in stocks (b = 1), earning an expected return R_{m}, but incurring an SD of σ_{Rp} = σ_{Rm} (from 7.8) or, he might invest some fraction of his funds in each type of asset, earning an expected return somewhere between R_{f} and R_{m}, and facing an SD (σ_{Rp}) less than σ_{Rm} but greater than zero.

**Risk and ICs: **

Figure 7.6 also shows the solution to the investor’s problem. Three ICs of the investor between risk and return are drawn in the figure. Each curve describes combinations of risk and return that leave the investor equally satisfied. The curves are upward sloping because risk is undesirable. Thus, with a greater amount of risk, it takes a greater expected return to make the investor equally well-off.

Of the three given ICs, IC_{3} yields the highest level of satisfaction and IC_{1} the lowest level. For a given amount of risk, the investor earns a higher expected return on IC_{3} than on IC_{2}, and a higher expected return on IC_{2} than on IC_{1}.

Therefore, the investor would prefer to be on IC_{3}. This position, however, is not feasible, because IC_{3} does not take him on to the budget line. The curve IC_{1} is feasible because the point T_{1} on IC_{1} takes him on to his budget line, but the investor can do better.

The investor does the best by choosing a combination of risk and return at the point T_{2} where an IC, viz., IC_{2}, is tangent to the budget line. The point of tangency between the investor’s budget line and an IC takes him on to the highest possible IC, i.e., the highest possible level of satisfaction, subject to his budget constraint. At that point (here T_{2}) the investor’s return has an expected value R* and at an SD of σ*.