Ch02 Time Value of Money

TIME VALUE OF MONEY

Time Value of Money offers an overview of the information required to calculate the future and present values of individual cash flows, ordinary annuities, due perpetuities and investments with uneven cash flows. TVM is based on the concept that a dollar that you have today is worth more than the promise or expectation that you will receive a dollar in the future. Money that you hold today is worth more because you can invest it and earn interest. After all, you should receive some compensation for foregoing spending. This hand out has been divided into following topics, which will be explained in detail:

  1. PRESENT VALUE
  2. FUTURE VALUE
  3. ANNUITIES
  4. PERPETUITY

PRESENT VALUE

The present value of a future cash flow is the nominal amount of money to change hands at some future date, discounted to account for the time value of money. A given amount of money is always more valuable sooner than later because this enables one to take advantage of investment opportunities.

The present value of delayed payoff may be found by multiplying the payoff by a discount factor which is less than 1. If C1 denotes the expected payoff at period 1, then

Present Value (PV) = discount factor. C1

This discount factor is the value today of $1 received in the future. It is usually expressed as the reciprocal of 1 plus a rate of return.

Discount Factor = 1 / 1+r

The rate of return r is the reward that investors demand for accepting delayed payment. The present value formula may be written as follow:

PV = 1 / 1+r. C1

To calculate present value, we discount expected payoffs by the rate of return offered by equivalent investment alternatives in the capital market. This rate of return is often referred to a the discount rate, hurdle rate or opportunity cost of capital. If the opportunity cost is 5 percent expected payoff is $200,000, the present value is calculated as follows:

PV = 200,000 / 1.05 = $190,476

FUTURE VALUE

Future value measures what money is worth at a specified time in the future assuming a certain interest rate. This is used in time value of money calculations.

To determine future value (FV) without compounding:

  • Save

Where PV is the present value or principal, t is the time in years, and r stands for the per annum interest rate.

  • Save

To determine future value when interest is compounded:

Where PV is the present value, n is the number of compounding periods, and i stands for the interest rate per period.

  • Save

The relationship between i and r is:

Where X is the number of periods in one year. If interest is compounded annually, X = 1. If interest is compounded semiannually, X = 2. If interest is compounded quarterly, X = 4. If interest is compounded monthly, X = 12 and so on. This works for everything except compounded continuously, which must be handled using exponential.

Similarly, the relationship between n and t is:

For example, what is the future value of 1 money unit in one year, given 10% interest? The number of time periods is 1, the discount rate is 0.10, the present value is 1 unit, and the answer is 1.10 units. Note that this does not mean that the holder of 1.00 unit will automatically have 1.10 units in one year, it means that having 1.00 unit now is the equivalent of having 1.10 units in one year.

ANNUITY

An annuity is an equal, annual series of cash flows. Annuities may be equal annual deposits, equal annual withdrawals, equal annual payments, or equal annual receipts. The key is equal, annual cash flows. Annuities work in the following way.

Illustration:

Assume annual deposits of $100 deposited at end of year earning 5% interest for three years.

Year 1: ksh.100 deposited at end of year = ksh.100.00
Year 2: ksh.100 × .05 = ksh.5.00 + ksh.100 + ksh.100 = ksh.205.00
Year 3: ksh.205 × .05 = ksh.10.25 + ksh.205 + ksh.100 = ksh.315.25

There are tables for working with annuities. Future Value of Annuity Factors is the table to be used in calculating annuities due. Just look up the appropriate number of periods, locate the appropriate interest, take the factor found and multiply it by the amount of the annuity.

For instance, on the three-year 5% interest annuity of ksh.100 per year. Going down three years, out to 5%, the factor of 3.152 is found. Multiply that by the annuity of ksh.100 yields a future value of ksh.315.20.

The present value of annuity can be finding out by the following formula: Present value of annuity =

C [1/r1/r(1+r)t]

The expression in brackets is the annuity factor, which is the present value at discount rate r of an annuity of $1 paid at the end of each of t periods.

PEPETUITY

Perpetuity is a cash flow without a fixed time horizon.

For example if someone were promised that they would receive a cash flow of ksh.400 per year until they died, that would be perpetuity. To find the present value of a perpetuity, simply take the annual return in dollars and divide it by the appropriate discount rate.

The present value of perpetuity can be finding out by the following formula:

Present value of perpetuity=C/r

Where C is the annual return in dollars and r is the discount rate.

Illustration:

If someone were promised a cash flow of ksh.400 per year until they died and they could earn 6% on other investments of similar quality, in present value terms the perpetuity would be worth $6,666.67.

Present value of perpetuity= (ksh.400 / .06 = ksh.6,666.67)

Tags: