Financial Mathematics Fundamentals: Time Value of Money, Interest Rates, and Discount Factors

The word “discount” has the rare ability to cause peoples’ hearts to race for joy.

I certainly get excited when items are on sale, though I am wary of situations where the original price might be high to begin with.

The mathematics of discounts can be confusing for many people. I once had a student who was very fast at answering simple discount problems posed to her verbally: a dress that ordinarily sells for $200 is on sale for 40% off. Quick! How much does it sell for?

She could say the answer of $120 in about a second!

However, if I gave her a problem with “uglier” numbers in written form, she had no idea what to do. She had no idea how to translate her intuitive knowledge in simple situations to a rule that would work in general. Perhaps another part of the difficulty had to do with her processing skills for reading. Dyslexia seems to be more common than was once thought, perhaps 10-15% of the population.

I’m not exactly sure what her main difficulty was. It was also frustrating for both of us that even though I taught her how to think about the problem and what to do, and she seemed to understand after we talked it through and worked examples, she forgot what to do when it came to doing her written homework from the textbook the next day.

What is the general logic of how to solve such a problem? First, you must realize what the word “percent” means. You can essentially translate it as “per hundred” or, “for every hundred”. The root word “cent” is Latin for “one hundred”. Therefore, when an item is, for example, 27% off, that means the dollar amount taken off is $\frac{27}{100}$ (twenty-seven one-hundredths) of the original price. In decimal form, this number is 0.27.

What this means is that, to find the amount taken off the original price, multiply by this number. If the original price is $189.99, then the amount taken off is $\$189.99\times 0.27\approx \$51.30$ , making the sale price be $189.99 – $51.30 = $138.69 (taxes not included).

The method used in the previous paragraph is a two-step process: first multiply, then subtract. Perhaps that was another reason it was confusing to my student. However, I did teach her that you can convert this into a one-step process.

The two-step process can be written in one equation as $\$189.99-\$189.99\times 0.27=\$138.69.$ Since $\$189.99=\$189.99\times 1,$ this is the same as $\$189.99\times 1-\$189.99\times 0.27=\$138.69.$ By the algebraic distributive property, the common factor of $189.99 can be factored out to write this as $\$189.99\times (1-0.27)=\$138.69$ , or $\$189.99\times 0.73=\$138.69.$

The number 0.73 is called a “discount factor” because 1) it can be used to find a discounted price and 2) it is a factor of the last equation.

Think about this with simpler numbers! If you are someone who can very quickly calculate that when an item that normally sells for $200 is 40% off, the sales price is $120, then you essentially did the same algebra very quickly in your head! Why? if the item is 40% off, the discount factor would be 1 – 0.40 = 0.60 = 60%. Then you quickly found 60% of $200 in your head by multiplication: $\$200\times 0.60=\$120!$

In fact, maybe you even used the distributive property quickly in your head in this last step without even realizing it! Maybe you computed $\$200\times 0.50=\$100$ and $\$200\times 0.10=\$20$ and just added the results. This is the distributive property! Why? Because you intuitively knew that

$\$200\times 0.60=\$200\times (0.50+0.10)$ $=\$200\times 0.50+\$200\times 0.10=\$100+\$20=\$120.$

Alternatively, maybe you used the associative property in your head without even realizing it! Maybe you computed $\$200\times 0.60=(2\times \$100)\times 0.60=2\times (\$100\times 0.60)=2\times \$60=\$120.$

If you did it quickly either way, congratulations! Now go back and think through the harder example again. Make sure it makes complete sense before continuing.

The Time Value of Money, Interest Rates, and Future Values

You may have heard from someone, somewhere, sometime, that “money has a time value“. Of course, hearing does not equal understanding.

Essentially the meaning of this fact can be boiled down to the statement: “given a choice, a person will prefer to have $1 today over $1 one year from now.”

Why is this true? There are many reasons. Some reasons could be: 1) lack of self-control, 2) inflation (money loses value over time, so things generally cost more over time), 3) opportunity cost (you could invest the money you receive right now and have even more one year from now), 4) you might die before the year is over, and, even just, 5) “why not?”.

Interest is essentially the monetary compensation a person demands to wait one year for the money instead of taking it now. If a person demands (at a minimum) an effective annual interest rate of 10%, that person is saying they are willing to not take the $1 now in exchange for receiving 10% extra (or more) on top of the dollar in one year. This 10% is called the effective annual interest rate, and the product $\$1\times 10\%=\$1\times 0.10=\$0.10$ is called the amount of interest, or just “interest”, that accrues. Therefore, the person is willing to not take the $1 now in exchange for receiving $1.10 in one year.

Here is a way to describe this situation: “the time value of $1 in the present is $1.10 one year from now, at an effective annual interest rate of 10%.”

In financial mathematics, it is customary to use the letter $i$ or the letter $j$ for effective periodic (e.g. annual) interest rates, though this is actually not so typical in mathematics textbooks whose main topic is not financial mathematics, such as calculus texts. If a person demands an interest rate of $i$ in exchange for waiting one period (e.g. one year) instead of receiving a monetary amount $A$ in the present, then the amount the person demands after one period is $A+Ai=A(1+i)$ (I have now switched to juxtaposition of symbols for multiplication rather than using the times symbol $\times$ ).

The amount $A(1+i)$ is called the future value at time $t=1$ of the amount $A$ at time $t=0$ . In the specific situation from above, we can now also say: “$1.10 is the future value, one year from now, of $1 in the present, at an effective annual interest rate of 10%.”

The Time Value of Money, Present Values, and Present Value Discount Factors

We can turn the idea of the time value of money around from the previous section. That is, we can ask a person: what amount of money would you be willing to receive now (in the present) instead of waiting to receive $1 one year from now? Alternatively: what amount of money would you be willing to forgo now (in the present) to receive $1 one year from now?

Again, because of the time value of money, the person would rather not wait for one year to receive money unless they are compensated with interest. If they could have some money now instead of $1 one year from now, they would be willing to take less than $1. Perhaps they are willing to receive $0.90 now instead of having to wait for one year to receive $1. Alternatively, perhaps instead of receiving $0.90 now, they are demanding to receive $1 if they must wait one year to get the money.

At the “boundary value” of the ideas of “being willing” and “demanding”, is the “negotiated” interest rate $i$ for this situation. As described at the end of the previous section, the effective annual interest rate $i$ will satisfy $\$0.90(1+i)=\$1$ (note that $A=\$0.90$ here). We can solve for $i$ now to get $1+i=\frac{\$1}{\$0.90}\approx 1.111$ and so $i\approx 0.111=11.1\%.$

The equation $\$0.90(1+i)=\$1$ can also be written as $\$0.90=\frac{\$1}{1+i}=\$1\cdot \frac{1}{1+i}=\$1\cdot (1+i)^{-1}$ (now I am using a dot $\cdot$ for multiplication). If we let $v=\frac{1}{1+i}=(1+i)^{-1},$ then this equation can be written as $\$0.90=\$1\cdot v.$ Here’s a link to a description of negative exponents if you forget what they mean.

The quantity $v$ is called the present value discount factor. It is often just called a “discount factor” for short, though that can cause confusion with “ordinary” discounts like the ones at the beginning of this post, where the time value of money is not being considered.

In this situation, the present value discount factor is $v= \frac{1}{1.1111\ldots}=0.90$ , which is also obvious from the equation $\$0.90=\$1\cdot v.$

More Advanced Topics: Functional and Graphical Relationships Between i and v

Both the effective periodic interest rate $i$ and the corresponding (effective periodic) present value discount factor $v$ are useful for solving financial mathematics problems. The value of $v$ is useful, for example, in the problem I solved in the video below.

Actuarial Exam 2/FM Prep: Deposit (Present Value) for Future Trust Fund Payments. Video #3, Exercise 1.2.2S in “Mathematics of Investment and Credit”, Samuel A. Broverman, 6th Edition.

But $i$ and $v$ are also variables that are related to each other. We already know that $v=\frac{1}{1+i}=(1+i)^{-1}.$ This equation defines $v$ as a function of $i,$ making $i$ an independent variable (input) and $v$ a dependent variable (output). If we want, we can use function notation and write $v=f(i)=\frac{1}{1+i}.$ For any given effective periodic interest rate $i$ , such as $i=8\%=0.08$ , we can find the corresponding (effective periodic) present value discount factor $v=\frac{1}{1+i}=\frac{1}{1.08}=0.925925\ldots\approx 92.6\%.$

We can also graph the function $v=f(i)=\frac{1}{1+i}.$

The graph of $v=f(i)=\frac{1}{1+i}$ for $-0.2\leq i\leq 2.0$ (so $-20\%\leq i\leq 200\%$ ).

Don’t just make graphs for their own sake (though they can be pretty), make sure you learn how to “read” them! This is not easy, but it is worth learning in the long run.

For example, note that the graph above is continuous and smooth. Also note from the graph that $v<1$ when $i>0$ (the typical case) and that $v>1$ when $i<0$ (and, of course, $f(0)=1$ so that $v=1$ when $i=0$ ). Furthermore, if you look very carefully at the graph, you’ll note that when $i$ is close to zero, that a given small increase in $i$ causes approximately the same small decrease in $v$ .

Use the formula to confirm this: for example, if $i$ increases from 0.01 to 0.02 (an increase of 0.01), then $v$ decreases from $\frac{1}{1.01}\approx 0.9901$ to $\frac{1}{1.02}\approx 0.9804$ (a decrease of about 0.01).

Also note that the graph is decreasing the whole way: $v$ goes down in value as $i$ goes up in value. More subtly, the graph is concave up (it looks like part of a smile): the rate at which $v$ goes down in value gets closer to zero (the graph gets flatter) as $i$ goes up in value. This means that a given small increase in $i$ will produce a smaller decrease in $v$ when $i$ is large compared to when $i$ is small. For example, when $i$ increases from 1.5 to 1.51, then $v$ decreases from $\frac{1}{1.5}\approx 0.66666667$ to $\frac{1}{1.51}\approx 0.66225166$ . This is a total decrease of about 0.00441501, which is smaller than the decrease of approximately 0.01 from the previous paragraph (when $i$ went up by the same amount).

If you know calculus, all these things can be confirmed with derivatives. Note that $\frac{dv}{di}=f'(i)=-(1+i)^{-2}=-\frac{1}{(1+i)^{2}}<0$ for all $i\not=-1$ and that $\frac{d^{2}v}{di^{2}}=f''(i)=2(1+i)^{-3}=\frac{2}{(1+i)^{3}}>0$ for all $i>-1$ .

On the flip side, the equation $v=\frac{1}{1+i}$ also defines $i$ as a function of $v$ , making $v$ an independent variable (input) and $i$ a dependent variable (output). To find the formula for this function, we need to solve the preceding equation for $i$ as follows: $v=\frac{1}{1+i}\Rightarrow 1+i=\frac{1}{v}\Rightarrow i=\frac{1}{v}-1=\frac{1-v}{v}.$

Using inverse function notation, we would write $i=f^{-1}(v)=\frac{1}{v}-1=\frac{1-v}{v}$ (either of these last two expressions is okay for the formula — the first one is actually a bit quicker to use).

You may have had a teacher that said, at this point, to “swap the variables”. That’s okay to do if your variables don’t have any real-life meaning. However, that is NOT the situation here. The meanings of $i$ and $v$ are specific and NOT interchangeable. Do NOT swap them. Leave the formula for the inverse function as it is.

For any given (effective periodic) present value discount factor, such as $v=92\%=0.92$ , this formula gives you the corresponding effective periodic interest rate, in this case $i=\frac{1}{0.92}-1\approx 0.08696=8.696\%.$

We can also graph this function. There are two ways this can be done. We could actually say the graph in the figure above is good enough, though we would have to change our usual perspective and think of the vertical axis as being the independent variable (or else look at the graph sideways in a mirror). Alternatively, we could relabel the axes and make the $v$ axis horizontal.

In doing this, the graph would be, in a purely-geometric way, the reflection of the original graph about the $45^{\circ}$ diagonal line through the origin. However, we should not graph it in the same picture because we have relabeled the axes!

Instead, we should graph it in a new picture with our relabeled axes as so:

*The graph of $i=f^{-1}(v)=\frac{1}{v}-1$ for $0.4\leq v\leq 1.5$ .*

*The graph of $i=f^{-1}(v)=\frac{1}{v}-1$ for $0.4\leq v\leq 1.5$ .*

As with the graph of $v=f(i)$ , you should work at gaining skill in reading/describing the graph of $i=f^{-1}(v)$ , perhaps using calculus to confirm your descriptions. Give it a try!

We end by noting that, as with all functions and their inverses, there are a couple key identities that arise under function composition because these functions “undo” each other.

In the present example, we note that: $(f\circ f^{-1})(v)=f(f^{-1}(v))=f\left(\frac{1}{v}-1\right)=\frac{1}{1+(\frac{1}{v}-1)}=\frac{1}{1/v}=v$ for all $v\not=0$ and $(f^{-1}\circ f)(i)=f^{-1}(f(i))=f^{-1}\left(\frac{1}{1+i}\right)=\frac{1}{\frac{1}{1+i}}-1=(1+i)-1=i$ for all $i\not=-1.$