15. Engineering Economic Analysis 2 - Risk and Return

Fundamental Principle of Financial Analysis:

Risk expects appropriate return

Risk: measure of the potential variability of an outcome from its expected value

Return: money earned as a result of money spent

Low risk = low expected rate of return

Low risk + high expected rate of return = scam

The expected return on an investment provides compensation to investors both for the waiting (time value of money) and for worrying (risk of the investment).

90 day bank rate: lend money to bank for 90 days, bank guarantees it will return it with interest (almost the official cash rate).

Interest rates at banks almost nothing at the moment ~0.25% per annum.

Risk and Diversification

Diversifiable/unsystematic/specific/unique risk: risk that you can mitigate by investing in a diverse range of funds. Compare this to non-diversifiable/systematic/market risk where you cannot mitigate the risk.

Markets, low to high risk:

• Bond market (debt market)
  • Government or private bonds
  • Air NZ issued six year, unsecured, fixed rate bonds at an interest rate of 4.25% in 2016. Covid was not kind to them
• Property market
  • Residential or commercial
• Equity market: shares
• Non-equity market: direct investment

Portfolio theory: diversification eliminates specific risk; for a reasonably diverse portfolio (8 or so reasonably different companies), the majority of specific risk is mitigated and only systematic risk matters.

Return on Capital

Two basic sources of capital:

• Equity capital: from the owners of the business
• Debt capital: from lenders of the business

People how provide the capital, debt or equity, needs to decide if the expected return on capital justifies the risk.

Interest or profit available from an alternative investment is called the opportunity cost of using the capital in the proposed undertaking.

Risk, Return and the Cost of Capital

Debt capital usually requires some kind of collateral (e.g. mortgage - repossession) while equity capital does not, making it riskier and hence cost more.

For large companies the cost of capital (CCC) is the weighted average cost of borrowing (debt) and the cost of selling shares (raising equity):

where $\text{C}_i$ is negative if there is an opportunity cost (e.g. sitting in bank account with interest).

If the company is not seeking funding it can set its own minimum acceptable rate of return (MARR).

Example: Weighted Cost of Capital

Buy a house for $500,000 with a 20% deposit ($100,000 of equity).

Assume that at this moment, half of the equity is in a savings account (at 2%) and half in shares (returning 7%).

The remaining 80%, $400,000 is borrowed - debt. The house is collateral in the case that you cannot pay, making it low risk for the bank. Mortgage rates are currently 5.5%.

The weighted cost of capital is $0.1 \cdot 2\% + 0.1 \cdot 7\% + 0.8 \cdot 5.5\% = 5.3\%$.

The present value factor/discount factor (the $(1 + i)^{-t}$ part of $\mathrm{PV} = \mathrm{FV}_t(1 + i)^{-t}$) may be:

• The opportunity cost of capital (OCC)
• The company cost of capital (CCC)
• The minimum acceptable rate of return (MARR)
• The risk free rate, $r_{\text{risk free}}$
  • Venture is basically risk free so should be compared to putting to the interest rate if was put in a bank account

The Treasury has its own set of discount rates to be used in government-funded projects. They are real and pre-tax (e.g. in 2020 the discount rate for infrastructure projects was 6% p.a.)

KiwiSaver: government gives you up to $521.43 per annum, and employers match your contribution (up to a certain point).

Example: $45,000 p.a., 3% ($1350) contributed to KiwiSaver. Matched by employer and eligible for max government contribution: a total of $3,221. In 48 years, assuming an annual average rate of return of 5% this will be worth:

Inflation

Inflation, the rate at which prices as a whole are increasing (measured with Consumers Price Index - CPI), must be taken into account.

NB: the CPI does not take into account things such as house or construction material prices

Real refers to constant dollar - money without inflation i.e. your purchasing power.

Nominal refers to current dollar - money with inflation i.e. the number in your bank account.

Over the last 20 years, the Reserve Bank has attempted to keep annual CPI increases between 1 and 3 percent, with a target of 2% - in the 80s it was in the 10-20% range.

Converting Rates

The Fisher Equation:

Where $i_\text{real}$ is the real inflation rate, $i_\text{nominal}$ is the nominal inflation rate and $\text{inflation}$ is the inflation rate.

For rates under around 10%, the approximation, $i_\text{real} \approx i_\text{nominal} - \text{inflation}$, can be used. Do not use this approximation in the test.

Example: Money in Bank Account

$1,000 lent at 5% compounding interest for 10 years. Assuming 3% inflation, what is the real interest rate?

Using the Fisher equation:

Hence, in ten years time you will be able to buy $1,212 worth of stuff.

Example: KiwiSaver

Using the previous KiwiSaver from above,

Assuming a 2% inflation, $r_\text{real} = (1 + r_\text{nominal})/(1 + \text{infl}) - 1 = 1.05/1.02 - 1 = 2.94\%$.

Hence, the nominal value - equivalent purchasing power when you retire, is $\text{real }\mathrm{FV}_{48} = \text{\textdollar}3,\!221 \cdot 1.0294^{48} = \text{\textdollar}12,\!943$.

Example: Present Value of Infinite Uniform Multiple Cash Flows (Perpetuity)

[from previous lecture]

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?

The example previously used nominal rate of return to get a value of $500,000.

Assuming average inflation of 3%, the real rate of return is 1.94%. With this:

Hence, 13.9 million dollars is required in order to provide $250,000 worth of value every year for perpetuity.

If you want to keep using the nominal rate, you must reflect inflation in cashflow. Using the formula for growing perpetuity:

Example: Nominal vs Real Cash Flow Comparison

Nominal vs. real cash flow does not matter as long as you are consistent in your calculations.

e.g. buying couch at $200 today and two annual payments of $300. Real rate of return of 5%, expected inflation of 3%. What is the actual price?

Using Nominal dollars/rate of return. Rearrange the Fisher equation:

Hence, the nominal rate of return is $\frac{1 + 3\%}{1 + 5\%} - 1 = 8.15\%$

Then we use the standard compound interest formula to find that $\text{Price} = \text{\textdollar}200 + \frac{\text{\textdollar}300}{1 + 8.15\%} + \frac{\text{\textdollar}300}{(1 + 8.15\%)^2} = \text{\textdollar}733.88$.

Using real dollars and rate of return: payments will be $200 initially, $\text{\textdollar}300/1.03 = 291.26$ in the first year, and $\text{\textdollar}300/(1.03)^2 = 282.78$ in the second.

We can then use the standard compound interest formula to find that $\text{Price} = \text{\textdollar}200 + \frac{\text{\textdollar}291.26}{1 + 5\%} + \frac{\text{\textdollar}282.78}{(1 + 5\%)^2} = \text{\textdollar}733.88$.

Example: Discount Rate

Government considering new hydroelectric power plant with capital expenditure of 100 million in year 1 and increasing at 10% per annum for the next two years (currently no inflation therefore real). What is the present value of the cost in real terms?

  PV  -100  -110  -121
  |-----|-----|-----|
  0     1     2     3

Treasury’s real, pre-tax discount rate is 5.0% per annum.

We cannot use the the present value of a growing annuity formula as the growth rate is greater than the discount rate.

Hence, do it manually three times with the formula:

We get $\mathrm{PV} = -\frac{100}{1.05} - \frac{100 \cdot 1.1}{1.05^2} - \frac{100 \cdot 1.1^2}{1.05^3} = -229.54$.

Sensitivity Analysis: these analyses are sensitive to the choice of cashflow and discount rate. We usually want to do best case, worst case and most likely analyses.