Question 1
Consider John Smith, a new freshman who has just received a study loan and started college. He plans to obtain the maximum loan at the beginning of each year. Although John Smith does not have to make any payments while he is still in school, the 6.5 percent interest per year compounded monthly owed accrued and is added to the balance of the loan.
Study Loan Limits
Freshman $26,250
Sophomore $35,000
Junior $55,000
Senior $55,000
After graduation, John Smith gets a six-month grace period. This means that monthly payments are still not required, but interest is still accruing. After the grace period, the standard repayment plan is to amortize the debt using monthly payments for 10 years.
Required:
Using the standard repayment plan and a 6.8 percent APR interest rate, compute the monthly payments John Smith owes after the grace period.
| Period | 0 | 1 | 2 | 3 | 4 | 4.5 |
|---|---|---|---|---|---|---|
| I/R | 6.5% | 6.5% | 6.5% | 6.5% | 6.5% = 0.0065 | 6.8% |
| FV Factor |
= (1+0.005416)^12
= 1.006972|1.066972|1.066972|1.066972|1.066972|1.066972|FV Factor =
(1 + 0.005667)^6 =
1.03449|
|Freshman|$26,250.00|FV=$26250*1.066972
=$ 28,008.01|FV=$28,008.01*1.066972
= $ 29,883.76|FV=$29,883.76*1.066972
=$ 31885.13|FV=$31885.13*1.066972
= 34,020.54|| |Sophmore|| 35,000.00|$ 35,000 * 1.066972
= 37,344.01| 37,344.01 * 1.066972
= 39,845.01| 39,845.01 * 1.066972
= $ 42,513.51||
|Junior|||$55,000.00|$55,000.00 * 1.066972
= $ 58,683.45|$58,683.45 * 1.066972
= $ 62,613.59||
|Senior||||$55,000.00|$55,000.00 * 1.066972
= 58,683.45|| |Sub Total|||||** 197,831.09**|$197,831.09 * 1.03449
= $ 204,653.35|
6.8% APR = 0.068 per annum 0.068/12 = 0.00567 per month
Total number of repayments = 10 years x 12 months – 120 times
120 – 6 months = 114 months
Present Value = 204653.35
PMT = r(PV) / [1-(1+r) –n]
PMT = 2,442.40
Question 2
Bella Inc. wishes to accumulate funds to provide a retirement annuity for its Vice President of Research, Edward Cullen. Mr. Cullen by contract will retire at the end of exactly 20 years. On retirement, he is entitled to receive an annual end-of-year payment of $35,000 for exactly 30 years . If he dies prior to the end of the 30-year period, the annual payments will pass to his heirs. During the 20-year ‘accumulation period’, Bella Inc. wishes to fund the annuity by making equal annual end-of-year deposits into an account earning 7 percent interest compounded quarterly . Once the 30-year ‘distribution period’ begins, Bella Inc. plans to move the accumulated monies into an account earning a guaranteed 12 percent per year compounded annually . At the end of the distribution period the account balance will equal zero. Note that the first deposit will be made at the end of year 1 and the first distribution payment will be received at the end of year 21.
Required:
How large must Bella Inc.’s equal annual end-of-year deposits into the account be over the 20-year accumulation period to fund fully Mr. Cullen’s retirement annuity?
[8 marks]
N = 30 years
I = 12% / 0.12
PV = $ 35,000.00
PV = C x [(1-(1-I) ^-n/I) = 35,000.00 [(1-(1+0.12)^-30/0.12)] = ** 281,931.44**
How much would Bella Inc. have to deposit annually during the accumulation period if it could earn 8 per cent rather than 7 percent?
[4 marks]
N = 20 years x 4 quarters = 80 months
I = 7% per annum / 4 quarters = 1.75% = 0.0175
P = FV = $ 281,931.44
Ordinary General Annuity Formula
P = PMT [(1 +i)n – 1 / i]
PMT = $ 281,931.44 / [(1+1.75%)20 -1) / 1.75%]
PMT = $ 281,931.44 / (1.017520-1/0.0175)
PMT = 281,931.44/171.794 =** ** 1641.10
How much would Bella Inc. have to deposit annually during the accumulation period if Mr. Cullen’s retirement annuity was perpetuity and all other terms were the same as initially described?
Ordinary Simple Perpetuity Formula
PV = PMT/I
PV= $ 1641.10 / 7%
PV = $ 23,444.2857
Ordinary General Annuity Formula
P = PMT [(1 +i)n – 1 / i]
PMT = P / [(1 +i)n – 1 / i]
PMT = 23,444.29 /[(1.017580-1)/0.0175]
PMT = 23,444.29/171.794
PMT = $ 136.47
together and find a job. Meanwhile, he racks up another 6 months interest for a new outstanding balance of 204,653.35. When LJ negotiates repayment, he has to spread the outstanding balance over 114 monthly payments assuming 6.8% compounded monthly. The payments times the PV factor for 114 months at 6.8%/12 = 0.00567% must equal 204653.35. 114 payments of 2,442.40 have a present value of 204,653 assuming the monthly rate of 0.00567%.