Time Value Of Money Part 2

ALAN ANDERSON, Ph.D. ECI RISK TRAINING

www.ecirisktraining.com

The time value of money formulas can be used to solve for the appropriate rate of interest or time horizon given the present and future value of a sum.

(c) ECI RISK TRAINING 2009 www.ecirisktraining.com 40

The present and future value formulas can be used to solve for the rate of interest.

41 (c) ECI RISK TRAINING 2009


Suppose that an investor deposits $10,000 in a bank account.

The investor plans to keep these funds in the bank for ten years, with a goal of having $20,000 at the end of that time. What rate of interest would he have to earn to double his money in ten years?



This can be determined algebraically as follows:

FVN = PV(1 + I)N

FVNPV

= (1+ I )N



FVNPV

N = (1+ I )

FVNPV

N −1 = I



In this example,

20,00010,000

10 −1 = 0.07177 = 7.177%



The present and future value formulas can also be used to solve for the time horizon.



Suppose that an investor deposits $5,000 in a bank account that pays 6% interest per year. The investor wants to know how long it will take for these funds to be worth $10,000.



This can be determined algebraically as follows:

FVN = PV(1 + I)N

FVNPV

= (1+ I )N



ln FVNPV

⎛⎝⎜

⎞⎠⎟= N ln(1+ I )

N =ln FVN

PV⎛⎝⎜

⎞⎠⎟

ln(1+ I )



In this example,

N =ln 10,000

5,000⎛⎝⎜

⎞⎠⎟

ln(1+ .06)= 11.896



The Rule of 72 is a quick method for estimating the time horizon or the interest rate needed to double the value of an investment.



Dividing the interest rate into 72 gives the approximate number of years that it would take to double the value of an investment.

For the example of the investor who needs to know how many years it would take to double his money at an interest rate of 6%, dividing 72 by 6 gives a result of 12, which is very close to the actual value of 11.896 years.



Dividing the number of years into 72 gives the approximate interest rate that would be required to double the value of an investment.

For the example of the investor who needs to know what rate of interest is required to double his money in ten years, dividing 72 by 10 gives a result of 7.2%, which is very close to the actual value of 7.177%.



In the case of a stream of cash flows that are not equal, computing the future and present value of the cash flows is a more complex process.



The two basic types of uneven cash flows of interest in finance are:

1) an annuity with an additional payment during the final period

2) a cash flow stream with no pattern, known as an irregular stream of cash flows




The cash flows from most bonds take the form of an annuity with an additional payment during the final period.

Investment projects often generate irregular streams of cash flows to firms.

Suppose that a bond offers investors cash flows of $100 each year for the next three years, with an additional payment of $1,000 at the end of the third year. If the periodic rate of interest is 5%, what is the present value of this stream of cash flows?



In this case,

N = 3 I = 5 PMT = $100 FV3 = $1,000



PVAN = PMT1− 1

(1+ I )N

I

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥



PVA3 = 1001− 1

(1+ .05)3

.05

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

= $272.32



=1,0001.1576

= $863.84

PV =FVN(1+ I )N

=1,000(1.05)3



Combining these results gives the present value of the cash flow stream: PVA3 + PV = 272.32 + 863.84 = $1,136.16



Suppose that an investment project produces cash flows of $200 at the end of the next two years, and $300 at the end of the following three years.



If the periodic rate of interest is 4%, what is the present value of these cash flows?

In this case, the present value of each cash flow is computed using the PV formula; these results are combined to give the present value of the stream of irregular cash flows.



In this case, the present value is:

= 192.31 + 184.91 + 266.67 + 256.44 + 246.58 = $1,146.91

200(1.04)1

+200(1.04)2

+300(1.04)3

+300(1.04)4

+300(1.04)5



Each of the examples considered so far has been based on the assumption that interest is paid annually.

When interest is paid more often than once per year, the present value and future value formulas must be adjusted.



Two adjustments must be made:

1) the periodic interest rate 2) the number of periods



 The periodic interest rate equals:

annual rate / number of periods per year

 The number of periods equals:

(number of years)(number of periods per year)



Suppose that a sum of $1,000 is invested for two years at an annual rate of interest of 4%. Compute the future value of this sum based on the assumption of:

a) annual compounding b) semi-annual compounding



With annual compounding,

N = 2 I = 4 PV = $1,000



Using the future value formula,

FVN = PV(1+I)N

FV2 = 1,000(1+.04)2 FV2 = 1,000(1.081600) FV2 = $1081.60



With semi-annual compounding,

N = 4 I = 2 PV = $1,000



Using the future value formula,

FVN = PV(1+I)N

FV4 = 1,000(1+.02)4 FV4 = 1,000(1.082432) FV4 = $1082.43



The more frequently interest is paid each year, the greater will be the future value of a sum or an annuity.



Compute the present value of $1,000 to be received in four years using an annual interest rate of 6% with:

a) annual compounding b) semi-annual compounding



With annual compounding,

N = 4 I = 6 FV4 = $1,000



Using the present value formula,

PV =

FVN(1+ I )N

=1000

(1+ .06)4= $792.09



With semi-annual compounding,

N = 8 I = 3 FV8 = $1,000



Using the present value formula,

PV =FVN(1+ I )N

=1000

(1+ .03)8= $789.41



The more frequently interest is paid each year, the smaller will be the present value of a sum or an annuity.



As the frequency of compounding increases, the present value of a sum or annuity decreases, while the future value of a sum or annuity increases.



The limiting compounding frequency is known as continuous compounding. In this case, interest is compounded at every instant in time. As a result, the number of compounding periods is infinite.

The present and future value formulas with continuous compounding are:



FVN = eIN

e = 2.7182818......

PV =FVNeIN

= FVNe− IN



The present value of $1,000 to be received in four years with an annual rate of interest of 5% compounded continuously is computed as follows:



PV = 1,000e-(0.05)(4) =

1,000e-(0.20) = $818.73



The future value of $1,000 invested for three years at an annual rate of interest of 4% compounded continuously is computed as follows:



FV3 = 1,000e(0.04)(3) =

1,000e(0.12) = $1,127.50



In order to compare interest rates with different compounding frequencies, they can be converted into the effective annual rate (EAR).



This is done with the following formula:

EAR = 1+ APRM

⎛⎝⎜

⎞⎠⎟M

−1



where:

APR = the annual percentage rate



If a bank charges an APR of 6% per year, compounded quarterly for a loan, what is the effective annual rate?



This can be determined with the formula, as follows:

EAR = 1+ APRM

⎛⎝⎜

⎞⎠⎟M

−1



EAR = 1+ .064

⎛⎝⎜

⎞⎠⎟4

−1 = 0.06136



This indicates that the borrower is actually paying 6.136% per year for this loan.




With continuous compounding, the EAR formula becomes:

EAR = eAPR - 1

If a bank charges an APR of 5% per year, continuously compounded, what is the effective annual rate?

EAR = eAPR – 1 = e.05 – 1 = 0.051271 = 5.1271%


For free problem sets based on this material along with worked-out solutions, write to [email protected]. To learn about training opportunities in finance and risk management, visit www.ecirisktraining.com


Time Value Of Money Part 2

Economy & Finance

Transcript of Time Value Of Money Part 2