Interest Rate Theory, Cash Flow Analysis, and Bonds

Interest Rate Theory

Time Value of Money

Interest is the result of the time value of money.

Money received today generally more valuabe than the amout of money received in the future, becasue money available today can earn interest.

For example, if the interest rate is 10% per year, ​\$100 today will become

100 \times (1 + 10\%) = 110

one year later.

Therefore, ​\$100 today is equivalent to ​\$110 one year later.

So, cash flows occurring at different points in time are not directly comparable. To compare them, we need to convert their values to the same point in time using an appropriate interest rate. This is the basic idea behind present value analysis.

货币具有时间价值，在比较不同时间的货币的价值，不能直接比较金额，应该通过 interest rate 转化到同一个时间点进行比较。

Compounding Frequency

Banks may calculate and pay interest more frequently than once a year - for example, quarterly, monthly, or daily. However, the interest rate is usually still quoted on an annual basis.

Suppose the annualized interest rate is ​r and interest is compounded ​m times per year.

  • simple interest - ​A \rightarrow A(1+r)
  • compounding interest - ​A \rightarrow A(1+\frac{r}{4})^4

Annualized Interest Rate vs. Effective Annual Interest Rate

Annualized Interest Rate

The annualized interest rate is the quoted annual rate ​r.

For example, ​r=12\%

with monthly compounding means that the monthly interest rate is

\frac{12\%}{12}=1\%

The annualized rate itself is still ​r=12\%.

Effective annual interest rate

The effective annual interest rate measures the actual percentage increase in wealth over one year after taking compounding into account.

If there are ​m compounding periods per year,

A \rightarrow A(1+\frac rm)^m

after one year.

Therefore,

r_{effective} = (1+\frac rm)^m - 1

Annualized rate 是“报价利率”，effective annual rate 是一年下来实际增长的部分，因为复利的存在，后者通常比前者高。

Continue Compouding

when the number of compounding periods becomes infinitely large:

m\to\infty
\left(1+\frac rm\right)^m\to e^r

Therefore, under continuous compounding, an initial amount ​A grows to

​Ae^r after one year.

The corresponding effective annual interest rate is ​e^r-1.

when

  • Cash flow
  • cash flow stream

Cash Flow Analysis

\text{Cash Flow} \rightarrow \text{Present Value} \rightarrow \text{Compare Investments} \rightarrow \text{IRR}

先表示现金流，再将不同时间的现金流折算到同一个时间点，最后进行投资比较。

Cash Flow and Cash Flow Stream

A cash flow is a net receipt at some time. For example, ​+\$100 means a cash inflow of ​\$100, while ​-\$50 means a cash outflow of ​\$50.

A cash flow stream is a series of flows over several periods. We usually write it as ​(x_0,x_1,\ldots,x_n), where ​x_k represents the cash flow at time ​k. For example, ​(-100,30,40,50).

Present Value

As mentioned above, money at different points in time is not directly comparable. The same applies to cash flows. Therefore, we convert future cash flows into their equivalent values at the present time.

For a future cash flow (X) at time (n),

PV=\frac{X}{(1+r)^n}.

For a cash flow stream,

PV=x_0+\frac{x_1}{1+r}+\cdots+\frac{x_n}{(1+r)^n}.

Present Value Analysis

Choose the investment with the higher present value:

PV(x)\geq PV(y).

All cash flows associated with the investment, both profit and cost, should be included.

Internal Rate of Return

The Internal rate of return (IRR) of a cash flow stream ​(x_0,x_1,\ldots,x_n) is the interest rate ​r such that

0=x_0+\frac{x_1}{1+r}+\cdots+\frac{x_n}{(1+r)^n}.

It is the discount rate that makes the present value of the cash flow stream equal to zero.

Internal Rate of Return Analysis

Economically, IRR can be interpreted as the break-even rate of return of the investment. At the IRR, the present value of future cash inflows exactly offsets the present value of the cash outflows.

For a conventional investment, if the required return is lower than the IRR, the investment has a positive present value; if the required return is higher than the IRR, the investment has a negative present value.

Therefore, IRR can be used to evaluate an investment by comparing the investment's implied rate of return with the required or market rate of return.

Annuity 与 Loan Analysis

Annuity

Bonds

发行人承诺按照事先约定的规则，未来向债券持有人支付钱。到期时会支付面值（face value / par value），很多债券还会定期支付票息（coupon）。

Duration & Interest Rate Risk

Immunization

Dynamic Programming

Valuation of a Firm

Free Cash Flow, FCF