CFA Level I - Quantitative Methods: The Time Value of Money

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The Time value of Money is the idea that a sum of money is more valuable if it is received today than tomorrow because of its capacity to earn interest.
The Time Value of Money problems at CFA level 1 are easy and straight-forward once you've understood the concepts of time value and the use of time line. You won't need a financial calculator, just a normal scientific calculator will do.

This post will cover the following items:

• The use of Time Line
• Interest/Discount rates
• Calculating Effective Annual Rate (EAR)
• Calculating Present Value and Future Value
• Annuity and Perpetuity
• Advanced problems

1. The use of Time Line

Time Line is a diagram, a line that you draw to visualize the problems in finance. It is extremely useful in many problems especially when it comes to calculations related to Time Value. Below is an example of the time line.

Example: Oscar bought a bicycle with $100 and rode it to deliver newspapers every weekend. He earned $25 per week at the end of every week. This is how you present the outflows and inflows of cash with a time line.

 

2. Interest and Discount rates

Interest rate or Discount rate means the same and they are often used interchangeably. There are 5 components that make up the Interest rate:

• (1) Real Risk-Free Rate: is a theoretical rate on a single period loan that has no expectation of inflation.
• (2) Expected Inflation Rate: reduces the purchasing power of the currency
• (3) Default-Risk Premium: is to compensate for the risk that a borrower will not make the promised payment in a timely manner.
• (4) Liquidity-Risk Premium: Liquidity measures how easily a security can be exchanged for cash. The less liquid a security is, the higher premium investors require to hold it.
• (5) Maturity Risk Premium: in the case of bonds, longer-term bonds are more volatile than shorter-term bonds. Therefore, longer-term bonds require higher maturity-risk premium.

 nominal risk-free rate = (1) + (2)

Required interest rate on a security = (1) + (2) + (3) + (4) + (5) 

3. Effective Annual Rate (EAR)

Usually when go to a bank, they would quote the Annual Percentage Rate (APR). For example, a bank will quote the interest rate on a saving account as 10%, compounded quarterly, rather than 2.5% per quarter.
However, the interest rate investors realize is the result after compounding, aka the Effective Annual Rate (EAR)

Example: continue from the example above, EAR of the saving account is calculated as follows:  EAR = (1+0.1/4)⁴ – 1 = 0.1038 = 10.38%

*Continuous Compounding is the limit of shorter and shorter compounding periods  

                                            EAR = e^r – 1

Example: if quoted rate or APR=10%, compounded continuously, 

EAR = e^0.1 – 1 = 0.1052 = 10.52%

4. Present Value and Future Value

PV = Present Value

FV = Future Value

r = Interest Rate or Discount Rate

n = number of compounding periods

Future value of a single sum: FV = PV(1+r)ⁿ 

Present value of a single sum: PV = FV/(1+r)ⁿ

Example: 
Calculate the amount in your bank in 5 years if today you deposit $10000 at an APR of 5%, compounded semiannually. 

number of periods = 5 x 2= 10, interest per period = 2.5%  
FV = 10000 x (1+0.025)^10 = $12800.85

If you want to have $10000 in your bank account 5 years from now, how much do you have to deposit today, assuming APR - 5%, semiannual compounding. PV = 10000/(1+0.025)^10 = $7811.98  


5. Annuity and Perpetuity  
Annuity is a stream of equal cash flows occurring at equal intervals over a given period of time.  An ordinary annuity is an annuity in which cash flows occur at the end of the periods. The future value of an ordinary annuity is calculated as follows:

The present value of an ordinary annuity is calculated as follows:

An annuity due is an annuity in which cash flows occur at the beginning of the periods. To calculate the FV or PV of an annuity due, you simply calculate it if it is an ordinary annuity then multiply the result by (1+r). Think of it this way, if you receive the cash flows at the beginning of the periods, they will be worth more than if you receive them by the end of the periods.

 

Perpetuity is a financial instrument that pays a fixed amount of money at set intervals over an infinite period of time. To calculate the Present value of a perpetuity:

PV = C/r

6. Advanced problems

a. An ordinary annuity beginning later than t=1
Find the present value of an annuity if the first end-of-year payment is to be received 3 years from today at r = 4%

To solve this problem, we first find the present value of the annuity at t=2. Then we discount the whole sum back to t=0  
PV₂ = 100 ×[1−(1+0.04)⁻⁴]/0.04 = 362.99  
PV₀ = 362.99/(1+0.04)²=335.60

b. Present value of a bond's cash flows
Find the present value of a bond which makes coupon payments of $100 (10% of face value) at the end of each year in 5 years. Given the discount rate is 5%

What's different about this problem is that at the end of year 5, you will receive the coupon payment along with the face value of the bond (Face value = 100/0.1=$1000)
To solve this problem, you first calculate the present value of the stream of coupon payment. The second step is to discount the face value to its present value. The sum of the 2 results is the present value of the bond.

PV_coupon = 100x[1-(1+0.05)^-5]/0.05= 432.95
PV_{face value} = 1000x(1+0.05)^-5= 783.53
Present value of the bond = 432.95 + 783.53 = $1216.48 

c. Present value of uneven cash flows
 Find the present value of the stream of cash flow below

To solve this problem, you need to find the present value of each of the single sum every year. The result is the total value of all the present values of these single sums.
Answer = 8347.44