RE5003 Chap.4 Time Value of Money and the Six Factors
Time Value of Money and the Six Factors
Why money has time value
The third lecture opens with familiar choices: a dollar today or tomorrow, lending a dollar for a promised repayment, and a coin toss that pays $100 on heads. The answers depend on waiting, on uncertainty and on attitude to risk. The lecture traces time value to inflation, risk and the productivity of capital: money is worth more today because it could be invested at some rate.
A risk-averse investor prefers a predictable payoff to an uncertain one with the same expected value, which is why risk later appears in the discount rate.
Compounding and its intervals
Every compounding problem has four quantities: present value, future value, time and the rate. Given any three, the fourth can be found.
Compound interest adds earned interest to principal so that interest earns interest, and the lecture shows how the gap over simple interest widens with time and with the rate.
When interest is credited more than once a year, the rate per period is the annual rate divided by the number of intervals, and the effective annual yield rises above the quoted rate.
Six factors, three pairs
The lecture's summary arranges the factors on a grid: lump sums or level annuities, moved toward the present or the future. The present and future value factors handle single sums.
The annuity factors value a level series of end-of-period payments, the regular annuity, while an annuity due pays at the start. The mortgage constant converts a loan into the level payment that repays it, and the sinking fund factor converts a target sum into the level saving that reaches it.
Each of those last two is the reciprocal of an annuity factor.
An application to land tenure
The lecture applies the factors to the premium for freehold over leasehold land, comparing the government's conversion table with present value factors at 2.5%, 3% and 3.5%. A lease is an annuity that ends, so its value relative to freehold falls slowly at first and quickly near expiry.
What this chapter covers
- 01
Waiting, risk and risk aversion as sources of time value
- 02
Present value, future value, the rate and the number of periods
- 03
Compounding intervals and the effective annual yield
- 04
Annuity factors for regular annuities and annuities due
- 05
The mortgage constant and the sinking fund factor
- 06
Leasehold against freehold value
Worked example · free
Monthly compounding and the effective yield
- 1Convert to a monthly rate and count the periods.
- 2Compute the future value with monthly compounding.
- 1Compute the effective annual yield.
- 1Compare with annual compounding.
Key terms
- Present Value
- The value today of a sum or series received in the future, found by discounting at the required rate.
- Future Value
- The amount a present sum grows to at a given rate over a given number of periods.
- Effective Annual Yield
- The annual rate actually earned once compounding within the year is taken into account.
- Regular Annuity
- A level series of payments made at the end of each period.
- Annuity Due
- A level series of payments made at the start of each period.
- Mortgage Constant
- The level payment per dollar borrowed that repays a loan over its term, the reciprocal of the present value annuity factor.
- Sinking Fund Factor
- The level saving per dollar of target that accumulates to a required future sum, the reciprocal of the future value annuity factor.
- Freehold Land
- Land held without a time limit on the tenure, valued as a perpetual stream of rent.
Time Value of Money and the Six Factors FAQ
Why is a dollar today worth more than a dollar next year?
A dollar today can be invested to earn a return, and waiting brings inflation and the risk that the payment never arrives. The lecture traces time value to those three sources: inflation, risk and the productivity of capital.
How does more frequent compounding change the result?
Interest is credited sooner and starts earning interest itself, so the effective annual yield rises above the quoted rate. Monthly compounding at 10% gives an effective yield of about 10.47%, for example, rather than 10%.
What is the difference between a regular annuity and an annuity due?
A regular annuity pays at the end of each period and is the default assumption unless a question says otherwise. An annuity due pays at the start, so each payment is discounted one period less and the series is worth more.
When do I use the mortgage constant rather than the sinking fund factor?
Use the mortgage constant when you have an amount today, such as a loan, and need the level payment that repays it. Use the sinking fund factor when you have a target in the future and need the level saving that reaches it.
Why do old leases trade at a steep discount to freehold?
A lease is an income stream that ends. The share of freehold value it carries is one minus the present value of a dollar at expiry, which falls slowly while expiry is distant and quickly as it approaches.
Assessment move
Learn the six factors as three reciprocal pairs on one grid, not as six formulas. For any question, first ask whether the cash is a lump sum or a series and which direction in time it moves; the cell then tells you the factor.
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