Abstract
You may have heard the expression, “A dollar today is worth more than a dollar tomorrow,” which is because a dollar today has more time to accumulate interest. The time value of money deals with this basic idea more broadly, whereby an amount of money at the present time may be worth more than in the future because of its earning potential. To be self-contained for readers new to finance, the chapter covers: interest rate and return rate; simple interest and compound interest, including a nonintegral number of periods, continuous compounding, and varying interest rates; the net present value and internal return rate; simple ordinary annuities, perpetuities, amortization theory, and annuities with varying payments and interest; applications of annuities; and applications to stock and bond valuation.
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- 1.
Apart from being mindful of leap years, note that banks may use a 360-day year when computing their charge on loans. Any deviation from a 365-day year will be stated explicitly.
- 2.
A dividend does not have to be in the form of cash. It can be a stock dividend—e.g., a company can pay you additional (typically, fractional) shares for each share of company stock you own.
- 3.
This bookkeeping for the cash dividend makes it convenient mathematically when considering reinvesting dividends to buy more units of the investment over consecutive time intervals.
- 4.
Some authors call \(\frac{V (t_{f})} {V (t_{0})}\) the return rate, but we shall not abide by that usage.
- 5.
If a k = 0, then simply apply the same discussion to the lower degree polynomial.
- 6.
Using \(N_{+} \leq N_{\mathrm{sgn}}\) , the reason a nonnegative even integer is subtracted from \(N_{\mathrm{sgn}}\) in the theorem is because N + and \(N_{\mathrm{sgn}}\) have the same parity, i.e., N + is even (odd) if and only if \(N_{\mathrm{sgn}}\) is even (odd). This implies \(N_{\mathrm{sgn}} - N_{+}\) is a nonnegative even number, i.e., \(N_{+} = N_{\mathrm{sgn}} -\mathrm{even}\) . In particular, N + is either \(N_{\mathrm{sgn}}\) , \(N_{\mathrm{sgn}} - 2\) , …, \(N_{\mathrm{sgn}} - 2(n - 1)\) , or \(N_{\mathrm{sgn}} - 2n\) for some nonnegative integer n.
- 7.
By definition, we assume \(r_{\mathrm{IRR}} > 0\).
- 8.
If there is only one period, then \(\mathcal{S}_{1} = \mathcal{P}\) (constant) for all \(r\) since the principal is added only at the end of the first period, but the first interest payment occurs at the end of the second period.
- 9.
That is, \(\mathcal{S}_{n}\) is concave up as a function of \(r\) (it has an increasing slope).
- 10.
While a typical mortgage is a loan used to buy a fixed asset like a house or land, which also secures the loan, a mortgage used to buy movable property such as a mobile home or operational equipment that acts as security for the loan is called a chattel mortgage or secured transaction.
- 11.
Recall that the marketplace is assumed to be in equilibrium, which allows for the required return rate of the stock to be estimated using the CAPM model; see Chapter 4 for an introduction.
- 12.
- 13.
IOU is an abbreviation for “I owe you.”
- 14.
Most corporate bonds are callable. Also, the US Treasury has not issued callable bonds since 1985.
- 15.
For example, such a bond might be issued at a 50% discount from its maturity value.
- 16.
A savings bond offers a fixed rate of interest over a fixed period of time, but cannot be traded after being purchased.
- 17.
It is worth noting that comparing different bonds by their percentage change in price is often misleading since the significance is not the same for an identical percentage price change of bonds with different interest rates. Also, it is important to realize that reinvesting all the coupon payments at the same rate is rather difficult if not impossible in practice.
- 18.
As before, there is no general analytical solution r Y for every n. In most applications, we can only estimate r Y numerically using a software.
- 19.
Mortgages on a house are generally modeled as simple ordinary annuities by lenders.
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© 2016 Arlie O. Petters and Xiaoying Dong
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Petters, A.O., Dong, X. (2016). The Time Value of Money. In: An Introduction to Mathematical Finance with Applications. Springer Undergraduate Texts in Mathematics and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3783-7_2
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