# The Time Value of Money

You may have heard the old adage a dollar today is worth more than a dollar tomorrow. But how much more? Would you rather have $100 today or$120 in a month? What about $100 today or$120 in ten years? The study of interest rates (or more precisely risk-free interest rates) tries to answer the question: what is the monetary value of time? How much should you be compensated simply for waiting?

Most discussions of interest rates begin with the assumption that there exists a fictitious bank from which anyone can borrow or deposit any amount of money at a fixed rate $r_f$

Let’s go back to the hypotheticals posed earlier and assume that $r_f$ is 2% (continuously compounded). Should we choose $100 now or$120 in a month? If we really needed money today (to pay bills, buy something immediately) we might be tempted to choose $100 today. However, using the bank we can simply borrow$100 today and pay back the debt with the $120 we get a month from now. Let’s compare the two options:  Receive$100 Today Borrow $100 Today Today $\100$ $\100$ Tomorrow $0$ $\120-\100e^{(.02)(\frac{1}{12})}=\19.83$ Clearly, choosing to receive the$120 in a month is a better option. Now lets consider $100 today or$120 in ten years.

 Receive $100 Today Borrow$100 Today Today $\100$ $\100$ Tomorrow $0$ $\120-\100e^{(.02)(10)}=-\2.14$

In this scenario the $120 isn’t enough to pay back our debt, so we should choose$100 today. If we really didn’t mind waiting the 10 years and would rather have more money in the future, we could simply take the $100 today and put it in the bank. In 10 years, your$100 would turn into $\100e^{(.01)(10)}=\122.14$, which is more than the $120 we would have received. The central idea here is that we can convert money promised in the future into money today, and vice-versa, by using the bank.$100 today is exactly the same as $\100e^{(.02)(1)}=\102.02$ in a year, $\100e^{(.02)(5)}=\110.52$ in five years, and $\100e^{(.01)(10)}=\122.14$ in ten years: We can put $100 in the bank today and withdraw any of those amounts in the future. Similarly,$120 in 10 years is exactly the same as $\120e^{(.02)(-10)} = 98.25$ today, $\120e^{(.02)(-9)}=\100.23$ in one year, and $\120e^{(.02)(-5)}=\108.58$: We can borrow these amounts now, in a year, or in five years and pay back our debt with the $120 in ten years. They are totally and unambiguously interchangeable. Let’s consider another example. Suppose you want to open a restaurant but are deciding between opening a cheap restaurant and a fancy restauran. For each option you know exactly how much money it will cost to open the restaurant, how much money you’ll make in every year for the next ten years.  Year Cheap Fancy 0 -280,000 -720,000 1 120,000 120,000 2 110,400 120,000 3 101,568 120,000 4 93,442 120,000 5 85,967 120,000 6 79,089 120,000 7 72,762 120,000 8 66,941 120,000 9 61,586 120,000 10 56,659 120,000 When comparing these two options, its convenient to convert future cash flows into today-money. This is called discounting future cash flows to their present value or PV. For the Cheap restaurant, this comes out to be: $\displaystyle -\280{,}000 + \120{,}000e^{(.02)(-1)} + \110{,}400e^{(.02)(-2)}+ \cdots + \56{,}659e^{(.02)(-10)}=\655{,}771.12$ Thus, running the cheap restaurant for ten years is exactly the same as producing$655,771 today. The PV of the cashflows for the fancy restaurant is:

$\displaystyle -\720{,}000 + \120{,}000e^{(.02)(-1)} + \120{,}000e^{(.02)(-2)}+ \cdots + \120{,}000e^{(.02)(-10)}=\621{,}744.99$

Therefore opening the cheap restaurant is a better investment than opening the fancy restaurant.

Unfortunately, in the real world this fictitious bank doesn’t exist. As an alternative we can deposit our money in U.S. government bonds, or if we’re a creditworthy bank we can deposit money with or borrow money from other creditworthy banks. These deposit and borrow rates typically depend on the length of time you’re depositing or borrowing for, and fluctuate daily. We’ll discuss all of these complexities of real-world risk-free interest rates in a future article.

# Interest Rate Math Basics

Suppose you deposit $1000 in a bank account that ears 5% a year. What is your balance at the end of five years? The answer to this question depends on the type of rate that’s being quoted here and can actually range between$1276.28 and $1284.03. When banks quote interest rates on credit card debt, home and auto loans, or savings accounts, they take advantage of this ambiguity to make the rates seem higher or lower to their benefit. Annual Percentage Rate (APR) Most interest rates are quoted as an APR or Annual Percentage Rate. You’ll see APRs quoted when opening a credit card, getting a home or auto loan, as well as investing in bonds. For one of their basic credit cards, Bank of America quotes an APR of 13.49%. If you had$1000 of credit card debt, how much would your debt grow in five years?

When given an APR, you also need to know the compounding frequency, i.e. how many times a year interest is added to your balance. If interest is simply added to your account as a lump sum at the end of every year, your ending balance would be:

$\displaystyle \1000 \times (1+13.49\%)^5=\ 1,882.73$

If interest added semiannually, instead of adding 13.49% at the end of the year, your debt will grow 6.75% every six months. In this case your balance will be

$\displaystyle \1000 \times \left(1+\frac{13.49\%}{2}\right)^{5\times 2} = \1,920.77$

This is slightly higher since interest added earlier accrues more interest itself. Credit card debt usually actually compounds interest monthly, giving us a balance of

$\displaystyle \1000 \times \left(1+\frac{5\%}{12}\right)^{5\times 12} = \1,955.68$

As we continue to increase the compounding frequency indefinitely, the ending balance approaches $1963.05. Using an infinite compounding frequency is called continuous compounding. The balance in an account that earns r% a year compounded continuously after t years is given by the formula $\displaystyle B_t = B_0 e^{rt}$ While continuous compounding is not found in real life, it is helpful in simplifying the math of compounding interest as we’ll see in later on. Annual Percentage Yield (APY) Banks usually advertise the interest rates they offer for their savings accounts as an Annual Percentage Yield (APY), also known as an Effective Annual Rate (EAR). An APY describes the interest accrued over one year regardless of how often it is compounded over the year. One Bank of America savings account offers an APY of .06% with interest added to your account daily. If you deposited$1000 in this account, after five years, your balance would simply be

$\displaystyle \1000 \times (1+.06\%)^5 = \ 1003.00$

However, if we were to rewrite this interest rate as an APR instead of an APY, we would find that the APR is slightly smaller than the APY.

$\displaystyle \left(1+\frac{APR}{365}\right)^{365}=1+.06\%$

$\displaystyle APR = .05998\%$

Alternatively, if we were to rewrite the 13.49% monthly compounded APR as an APY, we would get a much larger number.

$\left(1+\frac{13.49\%}{12}\right)^{12} = 1+APY$

$APY = 14.36\%$

In general, representing an interest rate as an APR makes interest rate look smaller, which is why APRs are used in quoting interest rates on loans or credit card debt. APYs, on the other hand, make interest rates look larger, and are often used in quoting interest earned in savings accounts.

A short proof of the continuous compounding formula

Using binomial expansion we can expand the compounded interest formula in the following way.

$\displaystyle \left(1+\frac{x}{n}\right)^n = 1+\binom{n}{1}\left(\frac{x}{n}\right)+\binom{n}{2}\left(\frac{x}{n}\right)^2 + \binom{n}{3}\left(\frac{x}{n}\right)^3+\cdots$

This is simply an n-degree polynomial on $x$ where the polynomial coefficient of $x^k$ is

$\displaystyle \binom{n}{k}\frac{1}{n^k}=\frac{n!}{(n-k)!k!n^k}$

Using the bounds $(n-k)!(n-k)^k \leq n! \leq (n-k)!n^k$ we can bound this coefficient above and below.

$\displaystyle \frac{(n-k)!(n-k)^k}{(n-k)!k!n^k} \leq \frac{n!}{(n-k)!k!n^k} \leq \frac{(n-k)!n^k}{(n-k)!k!n^k}$

$\displaystyle \left(\frac{n-k}{n}\right)^k\frac{1}{k!} \leq \frac{n!}{(n-k)!k!n^k} \leq \frac{1}{k!}$

Letting $n$ go to infinity, since both the upper and lower bound approach $latex \frac{1}{k!}, so must the coefficient. $\displaystyle \lim_{n \to \infty} \left(\frac{n-k}{n}\right)^k\frac{1}{k!} = \frac{1}{k!}$ $\displaystyle \lim_{n \to \infty} \frac{n!}{(n-k)!k!n^k} = \frac{1}{k!}$ Therefore $\displaystyle \lim_{n \to \infty} \left(1+\frac{x}{n}\right)^n = \sum_{k=1}^\infty \frac{1}{k!}x^k=e^x$ Finally, $\displaystyle \lim_{n \to \infty} B_0\left(1+\frac{r}{n}\right)^{nt}=B_0\left(\lim_{n\to \infty}\left(1+\frac{r}{n}\right)^n\right)^t=B_0e^{rt}$ # Introduction Hello everyone. I’ve decided to start this blog to share some of the more interesting things I have encountered while working in the credit finance world. This blog will be focusing on explaining the mathematics and intuition behind pricing and evaluating risk on a variety credit instruments. So what exactly is Credit? Broadly, credit refers to borrowing or lending activity when there is risk that the borrowing party may not pay back their debt in full. Lending to a home buyer, a public company, or betting with a third party about whether or not public company X will default are all considered to be credit-risky investments. Lending money to the U.S. government, however, would not be considered a credit investment since the U.S. is thought to have no or de minimis default risk.1 Credit instruments include bonds, loans, and a suite of derivatives that are actively traded by banks, mutual funds, and hedge funds. Why is credit important? The debt market is actually much larger than the equity market. In 2010, the total value of all global debt outstanding reached around$158 trillion while the total value of global equities outstanding was \$54 trillion.2

Understanding debt is integral to understanding the equity of credit risky companies.
The equity of a company has no intrinsic value if the company is unable to pay off its debts. Therefore any changes in the creditworthiness of a company must impact the value of its equity as well.

The health of the credit market is linked to the health of the overall economy.
When companies can borrow at low interest rates, they can grow more quickly and boost the economy. When too much credit is available to risky borrowers, as it was in the years preceding the 2008 financial crisis, it can lead to economic collapse.