14. Engineering Economic Analysis 1

Fundamental Principle of Financial Analysis:

Time is money

A dollar today is worth more than a dollar tomorrow.

Simple/Fixed Interest

Present value $\mathrm{PV} = 1000$ , deposit it at a bank at yearly interest rate (or rate of return or growth rate) $i = 1\%$ .

Future value at time $t$ :

\mathrm{FV}_t = \mathrm{PV}(1 + it)

(this module ignores tax)

This can be shown in a cash flow diagram:

1000    1010   1020             FV_t
  |------|-------|-------|--...--|
t 0      1       2       3       t 
  PV

Cash outflows are negative values; cash inflows are positive. The point of view matters: depositing money (-ve) and receiving it later (+ve) is different from borrowing money (+ve) and returning it later (-ve).

Compounding Interest

Annual Compounding Interest

\mathrm{FV}_t = \mathrm{PV} \cdot (1 + i)^t

$(i + 1)^t$ is called the future value factor.

Varying Compounding Interest Periods

If the compounding interest period is not a year, a modified formula is required. Let $m$ be the number of compounding periods in a year:

\mathrm{FV}_t = \mathrm{PV} \cdot \left(1 + \frac{i}{m} \right)^{mt}

The larger the value of $m$ , the larger the interest.

The effective annual interest rate (EAR) can also be calculated:

\mathrm{EAR} = \left(1 + \frac{i}{m}\right)^m - 1

NB: APR, annual percentage rate, is interest rate per period times the periods per year - simple interest.

NB: in the test, round to the nearest dollar and percentages to the nearest basis point (0.01%).

Continuous Compounding Interest Rule

Limiting case when $m$ tends to infinity:

\mathrm{FV}_t = \mathrm{PV} \cdot e^{it}

Future Value of Multiple Uniform Cash Flows: Annuity

A series of equally spaced, level cash flows that occurs over a finite number of periods.

Disbursements begin at end of period 1 and continue to the end of the period (maturity date).

\mathrm{FV}_\mathrm{annuity} = C\left[\frac{(1 + i)^t - 1}{i} \right]

where $c$ is the annual disbursement, $i$ the interest rate and $t$ the number of years.

Present Value and Discounting

You should discount the future value to find the current value

\begin{aligned} \mathrm{FV}_t &= \mathrm{PV} \cdot (1 + i)^t \\ \therefore \mathrm{PV} &= \frac{\mathrm{FV}_t} {(1 + i)^t} \end{aligned}

The term $(1 + i)^{-t}$ is the present value/discount factor.

$i$ is called the discount rate in this context.

e.g. you need to pay $x$ in $t$ years. Your bank account as an interest rate of $i$ . How much do I need to have now?

Present Value of Uniform Multiple Cash Flows (Annuity)

\begin{aligned} \mathrm{PV}_\text{annuity} = \sum_{t = 1}^n\frac{C_t}{(1 + i)^t} \end{aligned}

Where $C_t$ is the cash flow at time $t$ . Since it is an annuity the value is constant.

For a geometric series whose $n$ th term is $T_n = ar^n$ where $a$ is the initial value, the sum of the first $n$ values is:

S_n = \frac{a(1 - r^n)}{1 - r}

Therefore, $r = \frac{1}{1 + i}$ and $a = \frac{C}{1 + i}$ (TODO $1 + i$ because there is no $t_0$ ?):

\begin{aligned} S_n = \mathrm{PV}_\text{annuity} &= \frac{C}{i + 1} \cdot \frac{1 - \frac{1}{1+i}^t}{1 - \frac{1}{1+i}} \\ &= \frac{C}{i + 1} \cdot \frac{1 - \frac{1}{(1+i)^t}}{\frac{1+i}{1+i} - \frac{1}{1+i}} \\ &= \frac{C}{i + 1} \cdot \frac{1 - \frac{1}{(1+i)^t}}{\frac{1+i}{1+i} - \frac{1}{1+i}} \\ &= \frac{C}{i + 1} \cdot \frac{1 - \frac{1}{(1+i)^t}}{\frac{i}{1+i}} \\ &= C \cdot \frac{1 - \frac{1}{(1+i)^t}}{i} \\ \end{aligned}

Hence, the formula for the present value of an annuity is:

\mathrm{PV}_\mathrm{annuity} = C \left[\frac{1 - \frac{1}{(1 + i)^t}}{i} \right]

Example: Buying House

$450,000 house, $50,000 deposit, borrowing the rest at 6.00% fixed interest rate over 30 years, compounding monthly.

\begin{aligned} C &= \frac{i \cdot \mathrm{PV}_\mathrm{annuity}}{1 - \frac{1}{(1 + i)^t}} \\ C &= \frac{\frac{0.06}{12} \cdot \text{\textdollar}400,\!000}{1 - \frac{1}{(1 + \frac{0.06}{12})^{12 * 30}}} \\ C &= \text{\textdollar}2,\!398 \end{aligned}

Present Value of Uniform Multiple Cash Flows (Annuity) with Constant Growth

The payment increases by a constant rate of $g$ each period with a discount rate of $r$ per period. Only valid where $r > g$ .

\mathrm{PV}_\text{growing annuity} = \frac{C}{r - g}\left[1 - \left(\frac{1 + g}{1 + r}\right)^t \right]

For future value:

\mathrm{FV}_\text{growing annuity} = C \left[\frac{(1 + r)^t - (1 + g)^t}{r + g} \right]

Example: Land Lease

20 year lease, $100,000 decontamination afterwards.

NPAT (net profit after tax) of $100,000 in first year, average growth rate $g = 5.00\%$ per annum. Cost of capital $r$ is $10.00\%$ .

NB: cost of capital is specific case of opportunity cost, $r$ . The opportunity cost is the delta between the maximum possible gain with the asset versus what you are doing with it now (TODO confirm?)

What is the present value of the cash flows?

        100K   105K  110.25K    265.33K - 100K
  |------|------|------|-----...-----|
t 0      1      2      3             20

\begin{aligned} \mathrm{PV}_\text{growing annuity} &= \frac{C}{r - g}\left[1 - \left(\frac{1 + g}{1 + r}\right)^t \right] \\ &= \frac{\text{\textdollar}100,\!000}{0.1 - 0.05}\left[ 1 - \left( \frac{1 + 0.05}{1 + 0.1} \right)^{20} \right] \\ &= \text{\textdollar}1,\!211,\!208 \end{aligned}

Then subtract the present decontamination cost $\text{\textdollar}100,\!000/1.1^{20}$ to get a value of $1,196,344.

Present Value of Infinite Uniform Multiple Cash Flows (Perpetuity)

The present value of the infinite cash flow is called the capitalized value. This is often used in commercial property as rent is known quantity and they assume it will be paid forever (the difference between 20 years and infinity isn’t that large).

Terms such as “for the foreseeable future” mean for in perpetuity.

\mathrm{PV}_\text{perpetuity} = \frac{C}{r}

NB: annuity formula is derived by modelling it as the difference between two perpetuities.

Example

Starting in one years time, there will be a $250,000 per annum scholarship. Assuming the bank will pay 5.00% interest, what must today’s lump sum deposit be to ensure it can be funded forever?

\mathrm{PV}_\text{perpetuity} = \frac{\text{\textdollar}250,\!000}{0.05} = \text{\textdollar}5,\!000,\!000

A business is making $500,000 of profit a year and is likely to do so for the foreseeable future. Assuming he can invest the sale money at 8% interest, how much should he sell the business for?

\mathrm{PV}_\text{perpetuity} = \frac{\text{\textdollar}500,\!000}{0.08} = \text{\textdollar}6,\!250,\!000

Perpetuity with Constant Growth

$r > g$ so $\lim_{t \to \infty}\left(\frac{1 + g}{1 + r}\right)^t = 0$ and hence, the present value is simply $\frac{C}{r - g}$

Example

Dividends predicted to rise 4.00% per annum. Own 100,000 shares, dividend of 10 cents per share today. Plan to invest all dividends with expected return of 5.00% per year. What is the capitalized value of the dividend stream?

\mathrm{PV}_\text{perpetuity} = \frac{\text{\textdollar}100,\!000 \cdot 0.1}{0.05 - 0.04} = \text{\textdollar}1,\!000,\!000

A $500,000 dollar gift bequeathed to city for construction and maintenance. Maintenance will cost $15,000 a year, with an additional $25,000 every ten years. Interest earned on the balance after construction is 6.00% per annum.

How much will be left for initial construction.

Annual maintenance: $C/r = \text{\textdollar}15,\!000 / 0.06 = \text{\textdollar}250,\!000$

Maintenance every decade: $r = 1.06^{10} - 1 = 0.79084$ , so $C/r = \text{\textdollar}25,\!000/0.79084 = \text{\textdollar}31,\!612$ .

So $\text{\textdollar}500,\!000 - \text{\textdollar}250,\!000 - \text{\textdollar}31,\!612 = \text{\textdollar}218,\!388$ is left for construction.

Annuity Due

If first payment is today, not in one period time (i.e. payments at the start of the period).

The time value of money for first payment does not need to be adjusted, so it can be modelled as the payment value plus the present value of an annuity lasting $t - 1$ years.

Alternatively, you can model it as a standard annuity that you multiply by $1 + i$ (annuity transformation method).

Example: you get 20 payments of $465 dollars every year starting today. You will invest the money in government bonds with an interest rate of 8.00%.

\begin{aligned} \mathrm{PV}_\mathrm{annuity} &= C + C \left[\frac{1 - \frac{1}{(1 + i)^{t - 1}}}{i} \right] \\ &= 465 + 465\left[ \frac{1}{0.08} - \frac{1}{0.08(1 + 0.08)^{19}} \right] \\ &= 4,\!930.674 \end{aligned}