Time Value of Money 货币时间价值

Time value of money(货币时间价值)

TVM is the building block of finance theories
It is important to master TVM basics to have comprehensive
insight of other CFA subjects

Time value of money (货币时间价值)

Money available at the present time is worth more than the
same amount in the future due to its potential earning
capacity.

• Provided money can earn interest, any amount of money is
  worth more the sooner it is received

• It concerns equivalence relationships between cash flows
  occurring on different dates.

Cash flow additivity principle

The amounts of money can only be added on if they are
indexed at the same point in time

Interest Rate

Interpretations of Interest Rate (利率)

Required rate of return (要求收益率)

minimum rate of return an investor must receive in order to accept the
investment.

Discounted rate (折现率)

the rate at which we discount the future amounts to find their value today.

Opportunity cost (机会成本）

the value that investors forgo by choosing a particular course of action.

Example

Selmer Jones has just inherited some money and wants to set
some of it aside for a vacation in Hawaii one year from today.
His bank will pay him 5% interest on any funds he deposits. In
order to determine how much of the money must be set aside
and held for the trip, he should use the 5% as a:

A. required rate of return

B. discount rate

C. opportunity cost

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Answer: B
He needs to figure out how much the trip will cost in one year,
and use the 5% as a discounted rate to convert the future cost
to a present value. Thus, in this context the rate is best viewed
as a discount rate.

components of interest rate

real risk-free interest rate

single-period interest rate for risk-free security without inflation expected

inflation premiun(通货膨胀溢价)

compensating investors for expected inflation risk

$Nominal interest rate (名义利率) = Real risk-free interest rate + Inflation premium + Default risk premium + Liquidity premium + Maturity premium$

risk premium

default risk premium(违约风险溢价)

compensating investors for the possibility that the borrower will fail to
make the promised payments in time and in full amount.

liquidity premium(流动性风险溢价)

compensating investors for the risk of loss relative to an investment’s fair value if the investment needs to be converted to cash
quickly

maturity premium(到期风险溢价)

compensating investors for the increased sensitivity of the market value of debt to a change in market interest rates as maturity is
extended

$risk premium = default risk premium + liquidity premium + maturity premium$

interest rate

simple interest(单利)

The annual interest rate times the principal

compounding interest(复利)

The interest earned on interest is counted in

Example

If the annual interest rate is 10% and the principal is $1000,
what is the interest earned in 2 years under simple interest
and compounding interest?

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Answer:

Under simple interest:Interest earned = $1000 × 10% × 2 = $200
Under compounding interest:
Interest earned = $1000 × (1+10%) ×(1+10%) - $1000 = $210

Stated annual interest rate/Quoted interest rate ($r_s$)

The annual interest rate that does not account for compounding within the year

Compounding frequency (m, 复利频次)

The number of compounding periods per year

Continuous compounding:

the number of compounding periods per year becomes infinite.

Periodic interest rate ($\frac{r_s}{m}$, 期间利率)

Stated annual rate divided by the compounding frequency

Stated annual rate divided by the compounding frequency

The rate by which a unit of currency will grow in a year with
interest on interest included.

$EAR = 1 +Periodic interest rate -1 = (1 + \frac{r_s}{m})^m -1$

For continuous compounding:

$EAR=e^r_s -1$

because

$\lim_{n\rightarrow \infty}{(1+\frac{1}{n})^n} = e$

example

If the stated annual rate is 8%, compute the effective annual
rate with quarterly compounding

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Answer

$EAR = (1 + 8\%/4)^4 – 1 = 1.0824 – 1 = 8.24\%$

annualy compounded 年复利

semi -annually compounded 半年复利——债券

quatterly compounded 季度复利——活期

monthly componded 月复利——贷款

continuously 连续复利——衍生品

Relationships between PV and FV (Cont.)

Present value (PV, 现值)

the value of an initial investment

Future value (FV, 终值)

the value of an initial investment would be worth n periods from today

Present value and future value are equivalent measures
separated in time

$FV = PV (1+r)^n$

or

$PV=\frac{FV}{(1+r)^n}$

where: r = periodic rate, n = number of periods

Relationships between PV and FV (Cont.)

• For a given interest rate, the FV increases with the number
  of periods.
• For a given number of periods, the FV increases with the
  interest rate.
• For a given interest rate, the farther in the future the
  amount to be received, the smaller that amount’s PV.
• Holding time constant, the larger the interest rate, the
  smaller the PV of a future amount.

Example

Suppose a $10,000 investment and a stated annual interest
rate of 8%, compute the future value with monthly
compounding and continuous compounding

Answer

For monthly compounding:

$FV_N = PV * (1+\frac{r_s}{m})^m = 10000 * (1+\frac{0.08}{12})^{12}$

For continuous compounding:

$FV = PV*e^{r_s} = 10000*e^{0.08}=10832.87$

Annuity (年金)

A finite set of constant sequential cash flows

Ordinary annuity (普通年金)

all constant cash flows
occurring at the end of each period\

Annuity due (期初年金)

all constant cash flows occurring
at the beginning of each period

Perpetuity (永续年金)

A set of constant never-ending sequential cash flows
occurring at the end of each period