The Time Value of Money (Reading 5)

Learning Outcome Statements (LOS)

a

Interpret interest rates as required rates of return, discount rates, or opportunity costs:

     An interest rate can be interpreted as the rate of return required in equilibrium for a particular investment, the discount rate for calculating the present value of future cash flows, or as the opportunity cost consuming now, rather than saving and investing.

      利率可以理解为均衡收益率，它是计算未来现金流的折现率，或是现在消费（并非是储蓄和投资）的机会成本。

b

Explain an interest rates as the sum of a real risk-free rate, and premiums that compensate investors for bearing distinct types of risk:

     The real risk-free rate is a theoretical rate on a single-period loan when there is no expectation of inflation.

     Securities may have several risks, and each increases the required rate of return. These include default risk, liquidity risk, and maturity risk.

     实际无风险利率是无通胀情况下单期贷款的理论利率。

     证券会有违约风险，流动性风险，和到期日风险，这些风险都会增加证券的要求回报率。

c     

Calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding:

     The effective annual rate when there are m compounding periods is. Each dollar invested will grow to in one year.

     当计算复利的期间为m时，有效年利率是。每投资一美元一年后会变成。

d    

Solve time value of money problems for different frequencies of compounding:

     For non-annual time value of money problems, divide the stated annual interest rate by the number of compounding periods per year, m, and multiply the number of years by the number of compounding periods per year.

     对于非整年的TVM（钱的时间价值）问题，先把名义利率除以每年计复利的期数m，然后把年数乘以每年计复利的期数m得到N。

e  

Calculate and interpret the future value(FV) and present value(PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity(PV only), and a series of unequal cash flows:

     An annuity is a series of equal cash flows that occurs at evenly spaced intervals over time. Ordinary annuity cash flows occur at the end of each time period. Annuity due cash flows occur at the beginning of each time period. Perpetuities are annuities with infinite lives.

     The present (future) value of any series of cash flow is equal to the sum of the present (future) values of the individual cash flows.

     年金是指在一段时间之内，每隔相同时间间隔发生的一系列相同金额的现金流。普通年金指现金的支付或收入发生在每个计复利期间结束的时候。先付年金指现金的支付或者收入发生在计复利期间开始的时候。永续年金是无穷长时间的年金。

     一系列现金流的现值（终值），就是该系列现金流中，每个单独的现金流的现值（终值）的加和。

f

Demonstrate the use of a time line in modeling and solving time value of money problems:

     Constructing a time line showing future cash flows will help in solving many types of TVM problems. Cash flows occur at the end of the period depicted on the time line. The end of one period is the same as the beginning of the next period.  For example, a cash flow at the beginning of Year 3 appears at time t=2 on the time line.

     构建一个时间轴来表示未来现金流有助于解决许多类型的TVM问题。在时间轴上，现金流发生在每个时间段的结束点。前一个时间段的结束点相当于后一个时间段的开始点。例如，在第3年开始时发生的现金流出现在时间轴上t=2的点。

Formulas:

Nominal risk-free rate = real risk-free rate + expected inflation rate

The required rate of return on a security = nominal risk-free rate + default risk premium + liquidity premium + maturity risk premium

FV = PV(1 + I/Y)^N

PV = FV/(1 + I/Y)^N

PV_perpetuity =

Exercise Problems:

A.    $8,421.

B.     $8,056.

C.     $8,026.

 Ans: B;

Use the calculator,

       Change mode to BGN (annuity due)

       N=10×12=120 (monthly payment)

       I/Y=9/12=0.75 (monthly payment)

       PMT=6%×10,000/12=50 (monthly payment)

       FV=10,000

Solve for PV=-8,056

A.    12%.

B.     15%.

C.     18%.

Ans: B;

According to the formula, the required rate of return equals to nominal risked-free rate plus risk premium. The expected inflation rate is already one part of nominal risk-free rate. 

A.     $139,619

B.      $104,712

C.      $147,289

Ans: A;

Use the calculator,

       N=20×4=80 (quarterly payment)

       I/Y=8/4=2 (quarterly payment)

       PV=200,000-50,000=150,000

       FV=0

Solve for PV=-3,774

Nan Chen needs to pay $3,774 per quarter. After 12 payments

       N=12

       I/Y=2

       PMT=-3,774

       PV=150,000

Solve for PV=-139,619

4.      Yang Liu purchased a $1,000, 10-year corporate bond one year ago for $1,050. The bond pays monthly coupon of $5. Over the past year, the bond’s annual YTM has dropped by 1%. What total return did Yang Liu earn over the year?

A.  6.86% 

B.  6.71%

C.  12.6%

Ans: C ; Use the calculator,

       N=10×12=120 (monthly payment)

       PMT=5

       PV=1,050

       FV=1,000

Solve for I/Y=0.446

So the YTM one year ago was 0.446%×12=5.35%. After the drop of interest rate, the price of the bond now should be

       N=9×12=108

       PMT=5

       I/Y=4.35/12=0.363

       FV=1,000

Solve for PV=-1,122

So the total return is

A.    Annual percentage rate of 5% with annual compounding.

B.     Annual percentage rate of 4.9% with monthly compounding.

C.     Annual percentage rate of 4.8% with continuous compounding.

Ans: B :  convert B’s return from APR to EAR:

     =5.01%

Covert C’s return form APR to EAR:

EAR =

6.      The quoted annual interest rate of a bond is 6.17%. The effective annual rate of the loan is 10%. The frequency of compounding periods per year for the bond is closest to?

A.    Quarterly.

B.     Monthly.

C.     Continuous.

Ans: B : 

If interest rate compounding quarterly:

     =6.14%

If interest rate compounding monthly:

     =6.17%

If interest rate compounding continuous:

     EAR =

7.      The following end of month payments of $300, $500, $700, and $1,000 are due. Given a stated annual interest rate of 6%, the minimum amount of money needed in an account today to satisfy these future payments is closest to:

A.   $2,108

B.   $2,368

C.   $2,463

Ans: C : 

The monthly rate is 6%/12=0.5%, so the PV of the future cash flow is

PV