Question

How long will it take a lump sum to double at 2%? What about at 8%?

2) Find the present value of an annuity that pays $1,000 per month for a period of 10 years. Assume the discount rate is 6%

3) Referring to Question 2, suppose you are asked to find the present value of that same annuity described in Question 2, but the payments do not begin for 25 years.

4) You borrow $65,000 and promise to pay back $98,000 at the end of 15 years. What is the effective annual rate of interest?

5) What is the monthly cost of a simple 30-year home mortgage for $400,000 at 4.25%?

6) What is the monthly cost of a $30,000 balloon loan at 5%. Assume that you will owe $30,000 at the end of 10 years.

7) A bank offers a CD earning 2% compounded daily. What is the effective annual rate (EAR)?

Answer 1

1. You can use the "Rule of 72" to get the time that it will take to double the money at 2% (This will give a approximate value, very close to the answer)

Rule of 72 is, the money will double in 72/k years, where k% is the rate of interest.

Thus at 2%, money will double in 72/2 = 36 years

At 8% , money will double in 72/8 = 9 years

You can also use excel formula for NPER to calculate the time,

Here

Rate = 2%,

PMT = 0 (Used only if its a repeated deposit),

PV = -1 (The negative sign indicates, a cash outflow/ invested. Lets assume a value of 1),  

FV = 2 (Double the investment, the positive sign here is because of cash inflow that will come to you in case of withdrawal)

Type = 0 or 1 (It wouldnt matter here, but 0 indicates payment at the end of period, 1 indicates payment at the beginning of period)

Thus using in excel function NPER, or time to double = NPER (2%,0,-1,2,0) = 35.00 years (very close to 36 years)

If rate is 8%, NPER = NPER(8%,0,-1,2,0)= 9.00

2.We need to find the present value of annuity. We will use excel function of PV to find the value.

Here,

Rate = 6% /12 (Monthly drawings so we find the equivalent of monthly rate by dividing annual rate)

NPER = Time of annuity = 10 years or 10 x 12 = 120 months

PMT = $1000 (Drawings)

FV = 0 (The balance at the end should be zero)

Type = 1, (drawings happen at the start of month)

Using the above values in excel, PV = PV(6%/12, 120, 1000, 0,1) = $90,523.82

The value of annuity is $90,523.82

3. Here, the payments begin after 25 years and continue till next 10 years

In the first step, we will find the PV of drawings of $1000 for ten years (120 months) at 6%. It would be the same value as in question 2.

In the next step, we will find the present value of that annuity with no withdrawals for 25 years

We know,

Rate = 6%/12

NPER = Time for annuity to start = 25 years or 10 x 250 = 250 months

PMT = 0 (No drawings)

FV = $90,523.82 (Opening balance of annuity, 25 years later. Positive because this will be the pool of money that can be drawn, Cash Inflow)

Type = 1 (Deposits happen at the start of month)

Thus putting these values

PV = PV(6%/12, 250,0,90523.82,1) = - $25,938.82 (Negative signs Cash Outflow or amount to be invested)

This is the amount that can be invested now, to get $1000 drawings every month for 10 years, after 25 years

4. You borrow $65,000 and promise to pay back $98,000 at the end of 15 years.

We will use the formula for rate in Excel to find the value

Here,

NPER = 15 years

PMT = 0 (No repeated deposits)

PV = $65,000 (The money borrowed today)

FV = - $98000 (Money that is to be paid back. Negative sign indicates cash outflow)

Type = 1

Thus using above values, we get

Rate = Rate(15,0,65000,-98000,1) = 2.775%