The equation of value | equation of value actuarial

Recommended before this lesson — you can still read and practice below:

Real vs money interest rates (locked)

Problem. Rs. 5,000 is invested at $t=0$ at $i=4\%$ effective. The investor receives Rs. 1,500 at $t=2$ and Rs. $X$ at $t=4$ . For equivalence at $t=0$ , find $X$ .

Solution. Inflow at 0: 5,000. Outflows (from investor's view, or treat 5,000 as outflow if lending — stay consistent):

Value at 0 of receipts:

\frac{1{,}500}{(1.04)^2} + \frac{X}{(1.04)^4} = 5{,}000.

1{,}386.10 + 0.8548X = 5{,}000 \quad\Rightarrow\quad X \approx \frac{3{,}613.90}{0.8548} \approx \text{Rs. } 4{,}228.

Problem. Pay Rs. 10,000 at $t=0$ ; receive Rs. 3,500 at $t=1$ , Rs. 4,000 at $t=2$ , Rs. 4,500 at $t=3$ . Write the equation of value at $t=0$ for unknown rate $i$ .

Solution.

10{,}000 = \frac{3{,}500}{1+i} + \frac{4{,}000}{(1+i)^2} + \frac{4{,}500}{(1+i)^3}.

Solving for $i$ (trial and improvement, or calculator) gives the internal rate of return — the rate that makes the equation hold.

Trial at $i = 5\%$ (left-hand side = PV of receipts):

\frac{3{,}500}{1.05} + \frac{4{,}000}{(1.05)^2} + \frac{4{,}500}{(1.05)^3} \approx 10{,}848.

NPV $\approx 10{,}848 - 10{,}000 = +848$ — rate is too low (receipts are worth more than cost).

Trial at $i = 10\%$ :

\frac{3{,}500}{1.10} + \frac{4{,}000}{(1.10)^2} + \frac{4{,}500}{(1.10)^3} \approx 9{,}869.

NPV $\approx -131$ — rate is too high. The IRR lies between $5\%$ and $10\%$ ; further trials (or a calculator) give $i \approx 9.2\%$ to one decimal.

Problem. Pay Rs. 10,000 at $t=0$ ; receive Rs. 4,000 at $t=1$ , Rs. 4,000 at $t=2$ , and Rs. 3,000 at $t=3$ . NPV is positive at $i = 5\%$ and negative at $i = 6\%$ . Estimate $i$ to one decimal.

Solution. At $5\%$ , NPV $\approx +29$ ; at $6\%$ , NPV $\approx -148$ . The IRR lies between $5\%$ and $6\%$ . Trial at gives NPV ; trial at gives NPV closer to zero. The rate is about to one decimal (calculator: ).

Prerequisites

Teach

Setting up an equation of value

Multiple cashflows

Worked example: Finding an unknown payment

Worked example: Internal rate of return setup

Solving for the interest rate

Worked example: Finding i by trial

What mastery requires

Practice

Practice 1

Practice 2

Practice 3

Practice 4

Practice 5

Practice 6

Practice 7

Practice 8

Practice 9

Practice 10

Practice 11

Practice 12

Synthesis practice

Practice 1

Practice 2

Mastery checks

Equation of value setup

Check 1

Check 2

Check 3

Solving for an unknown payment

Check 1(Complete prior skills first)

Check 2(Complete prior skills first)

Check 3(Complete prior skills first)

Cashflow equivalence

Check 1(Complete prior skills first)

Check 2(Complete prior skills first)

Check 3(Complete prior skills first)

Solving for the interest rate

Check 1(Complete prior skills first)

Check 2(Complete prior skills first)

Check 3(Complete prior skills first)

Unlock the full Theory of Interest Rates module