Time Value of Money

Study guide: Exam 9

The time value of money is the principle that the value of an amount of money depends on when it is paid: because money can be invested at interest, amounts payable at different times cannot be compared until each is accumulated or discounted to a common comparison date. Under Compound Interest at effective rate $i$ , an amount $C$ paid at time $s$ is worth, at time $t$ :

$C(1+i)^{t-s}$

For $t > s$ the amount is accumulated (Accumulated Value); for $t < s$ it is discounted by $v^{s-t}$ , where $v = 1/(1+i)$ is the Discount Factor (Present Value).
The current value of a set of cash flows at a date is the accumulated value of those paid before it plus the discounted value of those paid on or after it.
An Equation of Value equates the values of two sets of payments at one comparison date. Under compound interest every comparison date gives the same answer; under simple interest it does not (see Simple vs Compound Interest).
Here the principle reflects the effect of interest alone; inflation is a separate adjustment (see Real Rate of Interest). The equations that apply it are on Time Value of Money Equations.

Settlement Now or Later A claimant is offered either $10{,}000$ now or $11{,}500$ in 3 years. At an annual effective interest rate of 5%, which is worth more?
Answer Bring the later payment back to today:
 $\begin{align*} PV &= 11500(1.05)^{-3} \\ &= 9934.13 \end{align*}$ 
Since $9934.13 < 10{,}000$ , the payment now is worth more at 5%. The two are equal at $i = (1.15)^{1/3} - 1 = 4.769\%$ ; at any lower rate the later payment is worth more.

Value at a Middle Date A policyholder pays $2{,}000$ now and $3{,}000$ in 4 years. At 6% annual effective interest, find the value of the two payments at time 2, and check it against their value at time 0.
Answer The first payment is accumulated two years and the second discounted two years:
 $\begin{align*} V_2 &= 2000(1.06)^{2} + 3000(1.06)^{-2} \\ &= 2247.20 + 2669.99 \\ &= 4917.19 \end{align*}$ 
At time 0 the value is $2000 + 3000(1.06)^{-4} = 4376.28$ , and $4376.28(1.06)^2 = 4917.19$ : under compound interest the comparison date does not change the answer.