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# TimeValue

 TimeValuecalculates the time value of a security s at time t for an interest specified by i.
• For a simple amount a and an effective interest rate i, TimeValue gives the future or accumulated value of a at time t.
• TimeValue gives the present or discounted value of a simple amount a for an effective interest rate i.
• Times can be given in abstract units or as dates.
• TimeValue works with arbitrary numeric or symbolic expressions. Symbolic formulas returned by TimeValue can be solved for interest rates, payments, or time periods using built-in functions such as Solve and FindRoot.
• TimeValue computes the time value accumulated or discounted from time to t using interest i. Time serves as a reference point for cash-flow occurrences.
• In TimeValue, the interest i can be specified in the following forms:
 r effective interest rate {r1,r2,...} schedule of rates applied over unit time intervals {{t1,r1},{t2,r2},...} schedule of rates changing at the specified time {p1->r1,p2->r2,...} term structure of effective interest rates function force of interest, given as a function of time EffectiveInterest[...] an EffectiveInterest object
• TimeValue[s, EffectiveInterest[r, 1/n], t] uses a nominal interest rate r, compounded n times per unit period. If times are specified as concrete dates, all interest rates are assumed to be annual rates.
• TimeValue gives the time value of an asset s for an interest rate schedule , where the are interest rates for consecutive unit periods.
• specifies an interest rate in effect before time . This is equivalent to {{-Infinity, r0}, {t1, r1}, {t2, r2}, ...}.
• TimeValue gives the time value of the simple amount a based on the force of interest function f which corresponds to the growth or decay process given by .
• A force of interest specification can be used with any security type.
• The following options can be given:
 Assumptions \$Assumptions assumptions made about parameters GenerateConditions False whether to generate conditions on parameters
 Basic Examples   (14)
Future value of \$1000 at an effective interest rate of 5% after 3 compounding periods:
Present value of \$1000 at 5% over 3 periods:
Future value of \$1000 using a nominal rate of 5% with quarterly compounding:
TimeValue works with symbolic parameters:
Present value at 6% of a 12-period annuity with payments of \$100:
Future value at 6% of a series of cash flows occurring at regular intervals:
Future value in three years' time of \$1000 invested on January 1, 2010 at 7.5%:
Number of periods required to grow \$1000 to \$3000 at a 6% interest rate:
Solve for the interest rate:
Future value after 5 periods using a schedule of rates over unit time intervals:
Present value using a schedule of rates effective at the specified times:
Present value of an amount paid at time 10 using a term structure of interest rates:
Future value using a schedule of rates over irregular time intervals:
Compute the future value after three time periods using a force of interest :
Future value of \$1000 at an effective interest rate of 5% after 3 compounding periods:
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Present value of \$1000 at 5% over 3 periods:
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Future value of \$1000 using a nominal rate of 5% with quarterly compounding:
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TimeValue works with symbolic parameters:
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Present value at 6% of a 12-period annuity with payments of \$100:
 Out[1]=

Future value at 6% of a series of cash flows occurring at regular intervals:
 Out[1]=

Future value in three years' time of \$1000 invested on January 1, 2010 at 7.5%:
 Out[1]=

Number of periods required to grow \$1000 to \$3000 at a 6% interest rate:
 Out[1]=

Solve for the interest rate:
 Out[1]=

Future value after 5 periods using a schedule of rates over unit time intervals:
 Out[1]=

Present value using a schedule of rates effective at the specified times:
 Out[1]=

Present value of an amount paid at time 10 using a term structure of interest rates:
 Out[1]=

Future value using a schedule of rates over irregular time intervals:
 Out[1]=

Compute the future value after three time periods using a force of interest :
 Out[1]=
 Scope   (11)
Symbolic time value computations:
Time value computation using a rate schedule:
Time value based on a force of interest function:
Valuation of cash flows:
A symbolic cash-flow computation:
Valuation of annuities:
A symbolic annuity calculation:
Symbolic solution for the number of periods:
Solve an annuity calculation for the payment amount:
An annuity with a continuous payment flow can be coupled with a force of interest specification:
Hours, minutes, and seconds can be given in date specifications:
 Options   (2)
Some solutions may only be conditionally convergent:
Assumptions can be specified to simplify an expression or to carry out an integration or summation:
 Applications   (14)
Find the amount that must be invested at a rate of 9% per year in order to accumulate \$1000 at the end of 3 years:
Find the accumulated value of \$5000 over 5 years at 8% compounded quarterly:
Find how much time it will take \$1000 to accumulate to \$1500 if invested at 6%, compounded semiannually:
Find the future value of 1 at the end of n years if the force of interest is , where t is time:
Find an expression for the accumulated value of \$1000 at the end of 15 years if the effective interest rate is for the first 5 years, for the second 5 years, and for the third 5 years:
If you invest \$1000 at 8% per year compounded quarterly, find how much can be withdrawn at the end of every quarter to use up the fund exactly at the end of 10 years:
Find the rate, compounded quarterly, at which \$16000 is the present value of a \$1000 payment paid at the end of every quarter for 5 years:
Find the accumulated value of a 10-year annuity of \$100 per year if the effective rate of interest is 5% for the first 6 years and 4% for the last 4 years:
Find the net present value of a \$1000 initial investment producing future incoming cash flows:
Find the internal rate of return of an investment with regular cash flows:
In return for receiving \$600 at the end of 8 years, a person pays \$100 immediately, \$200 at the end of 5 years, and a final payment at the end of 10 years. Find the final payment amount that will make the rate of return on the investment equal to 8% compounded semiannually:
Payments of \$100, \$200, and \$500 are due at the end of years 2, 3, and 8, respectively. Find the point in time where a payment of \$800 would be equivalent at 5% interest:
Another method to solve the problem above:
Find the effective rate of interest at which the present value of \$2000 at the end of 2 years and \$3000 at the end of 4 years will be equal to \$4000:
Since a loan's balance at any time is equal to the present value of its remaining future payments, Annuity can be used to create an amortization table:
Graph the principal payoff over time:
Present value using a schedule of rates over irregular time intervals:
This is equivalent to:
Use Plot and Plot3D to show the dependencies of an annuity on a set of parameters:
Dependence on interest rate:
Dependence on payment growth rate:
Use Plot3D to view the interest rate/growth rate landscape:
When finding interest rate solutions to long-term or high-frequency annuities or bonds, FindRoot may be needed instead of Solve:
In order for TimeValue to determine if there are enough rates in a schedule to reach the valuation period, the valuation period must be numeric:
Use Manipulate to explore the various dependencies a series of cash flows has on a set of variables:
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