TimeValue
TimeValue[s,i,t]
calculates the time value of a security s at time t for an interest specified by i.
Details and Options
 For a simple amount a and an effective interest rate i, TimeValue[a,i,t] gives the future or accumulated value of a at time t.
 TimeValue[a,i,t] 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 builtin functions such as Solve and FindRoot.
 In TimeValue[s,…], the security s can be given as a simple amount or as a Cashflow, Annuity, or AnnuityDue object.
 TimeValue[s,i,{t,t_{1}}] computes the time value accumulated or discounted from time t_{1} to t using interest i. Time t_{1} serves as a reference point for cash flow occurrences.
 TimeValue[s,i] is equivalent to TimeValue[s,i,0].
 TimeValue[…,t] is equivalent to TimeValue[…,{t,0}].
 In TimeValue[s,i,t], the interest i can be specified in the following forms:

r effective interest rate {r_{1},r_{2},…} schedule of rates applied over unit time intervals {{t_{1},r_{1}},{t_{2},r_{2}},…} schedule of rates changing at the specified time {p_{1}>r_{1},p_{2}>r_{2},…} 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[s,{r_{1},r_{2},…},…] gives the time value of an asset s for an interest rate schedule {r_{1},r_{2},…}, where the r_{i} are interest rates for consecutive unit periods.
 {r_{0},{t_{1},r_{1}},{t_{2},r_{2}},…} specifies an interest rate in effect before time t_{1}. This is equivalent to {{Infinity,r_{0}},{t_{1},r_{1}},{t_{2},r_{2}},…}.
 TimeValue[security,{r_{1},r_{2},…},t] is equivalent to TimeValue[security,{{0,r_{1}},{1,r_{2}},…},t].
 TimeValue[a,f,{t,t_{1}}] 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
Examples
open allclose allBasic 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 12period 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:
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 :
Scope (12)
Symbolic time value computations:
Time value computation using a rate schedule:
Time value based on a force of interest function:
A symbolic cash flow computation:
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:
Rates can be given as a TimeSeries:
Options (2)
Applications (15)
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 r_{1} for the first 5 years, r_{2} for the second 5 years, and r_{3} 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 10year 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:
Properties & Relations (2)
Possible Issues (3)
When finding interest rate solutions to longterm or highfrequency 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:
Input numeric valuation period:
Specifying rates by a TimeSeries requires the first time to be 0:
Interactive Examples (1)
Use Manipulate to explore the various dependencies a series of cash flows has on a set of variables: