### Discounted Cash Flow Factors

• Calculate present and future values of various payment methods such as single payment, uniform series, gradient series using formulae and vales extracted from table.

• Estimate and compare rate of returns on different investment proposals.

#### Single payment compound amount factor: (F/P, i, N):

• Moves a single payment to N periods later in time.

• A single payment is made after n periods.

• The interest earned at the end of each period is charged on the total amount owed (principal plus interest).

#### Sinking Fund factor: (A/F, i, N):

Takes a single payment and spreads into a uniform series over N earlier periods. The last payment in the series occurs at the same time as F.

#### Uniform Series Compound Amount factor:(F/A, i, N):

• Takes a uniform series and moves it to a single value at the time of the last payment in the series.

• Equal payments, A, occur at the end of each period.

• We will get back (F/A, i, n) at the end of period n if funds are invested at an interest rate i. $F = A + A(1+i) + A(1+i)^2+...+ A(1+i)^{n-1}$

#### Capital Recovery Factor:(A/P, i, N):

• Takes a single payment and spreads it into a uniform series over N later periods. The first payment in the series occurs one period later than P.

• An amount P is deposited now at an annual interest rate i. We will withdraw the principal plus the interest in a series of equal annual amounts A over the next n years.

• The principal will be worth $P(1+i)^n$ at the end of n years. This amount is to be recovered by receiving A every year the sinking-factor formula applies.

#### Uniform Series Present Worth Factor: (P/A, i, N):

Takes a uniform series and moves it to a single payment one period earlier than the first payment of the series.

### Solved Example:

An investor deposits \$1,000 in June in an initially empty account paying monthly 7%, and then makes withdrawals of \$400 and \$500 in July and August. What will be the compound amount of these cash flows in October using the compound amount formula? Solution: Time value of \$1000 after 4 months = $1000 \times \left(1 + \dfrac{7}{100}\right)^4 = \$1310.80$Time value of \$400 after 3 months $= 400 \times \left(1 + \dfrac{7}{100}\right)^3 = \$490.02$Time value of \$500 after 2 months $= 500 \times \left(1 + \dfrac{7}{100}\right)^2 = \$572.45$Net value after all transactions $= \1310.80 - \490.02 - \572.45 = \248.35$ Correct Answer: A ### Solved Example: #### 15-1-02 A company received a buy-out offer where it is offered \$37,000 now or \$45,000 after 3 years. If the interest rate is 6% per annum, which offer should it accept? Solution: We will calculate the present value of \$45,000. $F = P \left( 1 + \dfrac{r}{100} \right) ^t$ \begin{align*} P &= \dfrac{F}{\left( 1 + \dfrac{r}{100} \right) ^t}\\ &= \dfrac{45000}{\left( 1 + \dfrac{6}{100} \right) ^3}\\ &= \dfrac{45000}{(1.06)^3}\\ &= 37,782.87 \end{align*} So, the present value of \$45,000 now is \$37,782.87 which is higher than \$37,000. Hence, the company should accept \$45,000 after 3 years.
Note: Alternatively, you may calculate the future value of \$37,000 after 3 years and compare it with \$45,000.