# 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 built-in 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,t1}] computes the time value accumulated or discounted from time t1 to t using interest i. Time t1 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 {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[s,{r1,r2,},] gives the time value of an asset s for an interest rate schedule {r1,r2,}, where the ri are interest rates for consecutive unit periods.
• {r0,{t1,r1},{t2,r2},} specifies an interest rate in effect before time t1. This is equivalent to {{-Infinity,r0},{t1,r1},{t2,r2},}.
• TimeValue[security,{r1,r2,},t] is equivalent to TimeValue[security,{{0,r1},{1,r2},},t].
• TimeValue[a,f,{t,t1}] 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 all

## 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 :

## Scope(12)

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:

Rates can be given as a TimeSeries:

## Options(2)

### Assumptions(1)

Assumptions can be specified to simplify an expression or to carry out an integration or summation:

### GenerateConditions(1)

Some solutions may only be conditionally convergent:

## 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 r1 for the first 5 years, r2 for the second 5 years, and r3 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:

## Properties & Relations(2)

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:

## Possible Issues(3)

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:

Input numeric valuation period:

Specifying rates by a TimeSeries requires the first time to be 0: Shift the time series:

## Interactive Examples(1)

Use Manipulate to explore the various dependencies a series of cash flows has on a set of variables: