Time Value of Money

Time Value of Money

If given a choice, would you rather receive $1,000 today or $1,000 a year later? With simple intuition, we know to receive it today.

After all, if we are going to receive the same amount regardless, why should we not receive it today? It could be used to business or investment that could have a positive return. The $1,000 received today might become $1,100 the next year. Even if holding it as cash, there is no reason to delay the cashflow.

There is an opportunity cost associated with delaying the $1,000. That is to say that present value of money is higher than the future value.

Next, we will look into how to discount the future value to its present value.

Discounting to Present Value

Risk-free rate is the theoretical return of an investment with zero risk. Its return is guaranteed.

Of course, such a thing does not exist in the real world. However, most investors take the US Treasuries as the risk-free rate. US is seen as the least likely to default on their bonds.

US Treasury Yield
US Treasury Yield

Usually, T-Bills (Treasuries with maturity less than a year) are used as a proxy for the risk-free rate. However, Treasuries with othertime to maturity could also be used as risk-free rate.

With this risk-free rate, we are able to find the present value of the $1,000 that would be received a year later. Note that other sources might useĀ discount rate instead of risk-free rate, but they are interchangeable, the words used are just based on the context.

\[\text{Let r = risk-free rate; pv = present value; fv = future value (\$1,000)} \\pv(1+r)=fv \\ pv=\frac{fv}{1+r} \]

For discounting of future value n-years, below is the general formula.

\[ pv=\frac{fv}{(1+r)^n} \]

In the real-world, there are rarely cases where there is only 1 future cashflow, there are usually multiple future cashflow that comes at different time in the future. A good example would be bonds.

In the next article, discounted future cashflow, we will go more in depth into bonds and how the risk-free rate affects their pricing.

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