BUS202: Principles of Finance

Once you understand the idea of the time value of money, and of its use for valuing a series of cash flows and of annuities in particular, you can't believe how you ever got through life without it. These are the fundamental relationships that structure so many financial decisions, most of which involve a series of cash inflows or outflows. Understanding these relationships can be a tool to help you answer some of the most common financial questions about buying and selling liquidity, because loans and investments are so often structured as annuities and certainly take place over time.

Loans are usually designed as annuities, with regular periodic payments that include interest expense and principal repayment. Using these relationships, you can see the effect of a different amount borrowed ($PVannuity$), interest rate ($r$), or term of the loan ($t$) on the periodic payment ($CF$).

For example, if you get a $250,000 ($PV$), thirty-year ($t$), 6.5 percent ($r$) mortgage, the monthly payment will be $1,577 ($CF$). If the same mortgage had an interest rate of only 5.5 percent ($r$), your monthly payment would decrease to $1,423 ($CF$). If it were a fifteen-year ($t$) mortgage, still at 6.5 percent ($r$), the monthly payment would be $2,175 ($CF$). If you can make a larger down payment and borrow less, say $200,000 ($PV$), then with a thirty-year ($t$), 6.5 percent ($r$) mortgage you monthly payment would be only $1,262 ($CF$) (Figure 4.11 "Mortgage Calculations").

Figure 4.11 Mortgage Calculations

Note that in Figure 4.11 "Mortgage Calculations", the mortgage rate is the monthly rate, that is, the annual rate divided by twelve (months in the year) or $r$ ÷ 12, and that $t$ is stated as the number of months, or the number of years × 12 (months in the year). That is because the mortgage requires monthly payments, so all the variables must be expressed in units of months. In general, the periodic unit used is defined by the frequency of the cash flows and must agree for all variables. In this example, because you have monthly cash flows, you must calculate using the monthly discount rate ($r$) and the number of months ($t$).

Saving to reach a goal – to provide a down payment on a house, or a child's education, or retirement income – is often accomplished by a plan of regular deposits to an account for that purpose. The savings plan is an annuity, so these relationships can be used to calculate how much would have to be saved each period to reach the goal ($CF$), or given how much can be saved each period, how long it will take to reach the goal ($t$), or how a better investment return ($r$) would affect the periodic savings, or the time needed ($t$), or the goal ($FV$).

For example, if you want to have $1,000,000 ($FV$) in the bank when you retire, and your bank pays 3 percent ($r$) interest per year, and you can save $10,000 per year ($CF$) toward retirement, can you afford to retire at age sixty-five? You could if you start saving at age eighteen, because with that annual saving at that rate of return, it will take forty-seven years ($t$) to have $1,000,000 ($FV$). If you could save $20,000 per year ($CF$), it would only take thirty-one years ($t$) to save $1,000,000 ($FV$). If you are already forty years old, you could do it if you save $27,428 per year ($CF$) or if you can earn a return of at least 5.34 percent ($r$) (Figure 4.12 "Retirement Savings Calculations").

Figure 4.12 Retirement Savings Calculations

As you can see, the relationships between time, risk, opportunity cost, and value are some of the most important relationships you will ever encounter in life, and understanding them is critical to making sound financial decisions.