Use of timeline in solving time value of money problems

Example 9: Cash flow additivity principle and timeline

Terry wants to have $50,000 in her account at the end of 10 years. Currently, she has $20,000 in her account. How much money does she need to deposit in her account at the end of the third year for that assuming that her account earns 8 percent per annum?

Solution:

We will have two streams of cash flows at the end of five years: FV of the current $20,000 and FV of the amount that she would deposit at the end of three years. The sum of those two payments should be equal to $50,000.

If we subtract the future value of the current $20,000 from $50,000, then that would be the future value of the amount that would be deposited at the end of the third year.

FV of 20,000 after 10 years = 20,000*(1+0.08)¹⁰ = 43,178.50.

The future value required to be made at the end of the third year = 50,000 - 43,178.50 = 6,821.50.

From the timeline, we see that the amount at the end of the third year will accumulate interest for seven years. The amount needed will be equal to the discounted value of $6,821.50 for seven years.

The amount to be deposited at the end of the third year = 6,821.50/1.08⁷ = $3,980.30

Example 10: Using timeline with annuities

Steve wants to invest in a debt security that will provide him equal cash flows of $200 starting from the beginning of the 3rd year. The appropriate discount rate for the investment is 10 percent. He will receive a total of 8 such annual payments at the beginning of each year starting from the third year. What should be the value of such security today?

Solution:

The cash flows occur at the beginning of each year starting from the third year. It looks like a problem of an annuity due but it can be solved using both annuity due and ordinary annuity. If we calculate the present value using annuity due formula in the calculator, then that present value will be equal to the future value at the end of two years (or the start of the third year). We need to discount that value by two years to get the present value.

PMT = 200, I/Y=10, N = 8, CPT->PV = 1,173.68 (using annuity due)

This PV is the future value at the end of the second year. Discounting it to the current date to get the present value.

PV = 1,173.68/1.10² = $969.99

However, if we calculate the present value using ordinary annuity formula in the calculator, then that present value will be equal to the future value at the end of one year (or the start of the second year). We need to discount that value by one year to get the present value.

PMT = 200, I/Y = 10, N = 8, CPT->PV = 1,066.99 (using ordinary annuity)

This PV is the future value at the end of the first year. Discounting it to the current date to get the present value.

PV = 1,066.99/1.10 = $969.99

In both the cases, we get the same solution. That's how we can use the timeline to our advantage in slightly complex problems.

Example 11: Importance of signs: Annuity as cash inflow

Karim has $50,000 in his bank account. The bank is paying him an interest rate of 12 percent compounded monthly. How much money will be in Karim's account at the end of one year if he withdraws $3,000 from his account at the beginning of every month?

Solution:

The annuity of $3,000 is an inflow because Karim gets this cash flow as inflow at the beginning of each month. The present value of $50,000 is an outflow (think of it like that he had to deposit that at the beginning of the year, so it is an outflow from his pocket). The future value at the end of one year will be cash inflow (think of it like he will receive this value at the end of the year).

To solve this problem, the sign of FV and PMT has to be the same, and the sign of PV has to be opposite of that. Let's take negative sign for cash outflow and positive sign for cash inflow (you can take it opposite as well)

We need to change the setting to beginning mode first as it is an example of an annuity due. Press these four buttons to change the settings: [2nd][BGN][2nd][SET].

PV = -50,000
PMT = 3,000
I/Y = 12/12 =1
N = 1*12 = 12
CPT-> FV = 17,913.27

Karim will have $17,913.27 in his account at the end of the first year.

Example 12: Importance of signs: Annuity as cash outflow

Devika has $40,000 in her bank account. She plans to deposit $1,000 at the end of each month beginning from the current month for two years. How much interest rate compounded monthly should she get from the bank so that she has $80,000 in his account at the end of two years?

Solution:

She will deposit money in her account. So, it is a cash outflow similar to the present value. So, the PV and PMT will have the same sign, and FV will have the opposite sign. It is also a problem of an annuity due. We need to calculate I/Y in this case.

PV = -40,000
PMT = -1,000
N = 2*12 = 24
FV = 80,000
CPT->I/Y = 1.44

1.44 is the monthly rate. We need to multiply this by 12 to get the stated annual rate. Stated annual rate = 1.44*12 = 13.73 percent.

Example 13: Funding retirement

Charles is 50 years old. He is going to retire after ten years. He has $100,000 in his retirement account. After retirement, he would stay with his children for two years and then he would live alone. So, he would require $50,000 per annum at the beginning of each year starting from the 3rd year after his retirement. How much amount of money should he put into his retirement account at the end of each year for next ten years so that he can fund his expenses post retirement? Assume that the interest rate earned on his retirement account is 8 percent, and he will live for 20 years post retirement.

Solution:

This problem is a classic real life problem and can be solved in many ways using cash flow additivity principle and timeline.

Let's first break down the problems in cash outflows and cash inflows. The cash inflows will start only the beginning of 3rd-year post retirement. We need to discount all those cash inflows and bring those to a time where we can compare it with the cash outflows.

We need to calculate the present value of annuities post retirement. If we consider the annuity to be an ordinary annuity, then the present value will be calculated at the end of the 11th year from today or at the end of 1st-year post-retirement (because in the case of an ordinary annuity, the present value is one period before the first annuity payment).

PMT = 50,000
I/Y = 8
N = 18 (because he will not need money for first two years post retirement)
CPT-> PV = 468,594.36 (this will be the value at the end of 11th year)

Now we need to discount this value to the end of the 10th year so that we can compare it with the future value of cash outflows.

Value of cash inflow at the end of 10th year = 468,594.36/1.08 = 433,883.66

The future value of the cash outflows has to be equal to 433,883.66 to fund his retirement.

The cash outflows are ordinary annuity at the end of every year for next ten years and $100,000 amount in his account currently.

FV = 433,883.66
PV = -100,000
I/Y = 8
N = 10
CPT->PMT = 15,047.82

Charles needs to deposit $15,047.82 in his account at the end of every year for next ten years to fund his retirement.