MAF101 â FUNDAMENTALS OF FINANCE REVISION
MAF101 – FUNDAMENTALS OF FINANCE REVISION
Topic 2: Financial Maths Define the aim of financial mathematics. • Money has time value - $1 now is worth more than $1 tomorrow • Financial managers compare the marginal benefits and marginal costs of investment projects • Projects usually have a long-term horizon – timing of benefits and costs matters • Understand, compare and calculate interest rate arrangements, including simple and compound interest. • Define and calculate the present value and future value of a lump sum cash flow. • PV: the value today of a cash flow to be received at a specific date in the future. (known as principal of borrowing or lending). 𝑃𝑉 = 𝐹𝑉 × 1 + 𝑟 *+ • FV: the amount an investment is worth after one or more periods of time. (known as principal plus interest at maturity. Simple Interest: 𝐹𝑉 = 𝑃𝑉 × 1 + 𝑖 𝑥 𝑡 Compound Interest: 𝐹𝑉 = 𝑃𝑉 × 1 + 𝑟 + • Time to maturity (t): the duration (usually in years) of the interest rate arrangement • Nominal interest rate (i): quoted interest rate per annum • Interest: monetary return on saving/investment
Interest = Principal × i × t
•
Other variations: If Jack invested $6,000 for 4 months and received $6240 at the maturity how much is the yield? PV = 6000 FV = 6.240 t = 4/12 years 1 ⎛ FV 1 6240 ⎞ ×⎜ − 1⎟ = ×( − 1) 4 t ⎝ PV 6000 ⎠ 12 12 = × (1.04 − 1) 4 = 0.12 = 12%
i=
Define and calculate the present value and future value multiple cash flows, including ordinary annuities and annuity dues. • Nina has just signed a contract for $1 million to be paid $200,000 now, $500,000 in one years time and $300,000 in two years time. Nina has a choice to take the above arrangement or take $910,000 now. Which option should she take assuming the relevant interest rate of 10%p.a. n
PV =
∑ CF (1 + r) j
−j
j =0
PV =
CF0
CF2
+ (1 + r )1 (1 + r ) 2 500,000 300,000 = + + 0 1 (1 + 0.10) (1 + 0.10) (1 + 0.10) 2 = 200,000 + 454,545.45 + 247,933.88 (1 + r ) 0 200,000
+
CF1
= $902,479.34 < $910,000
•
Nina should take the $910,000 now. Joseph has a five-year cash flow saving plan and the amounts to be saved are as below. The saving plan offers interest rate at 10% p.a. compounding annually. How much will Joseph accumulate at the end of plan. Year end
CFj
1
1,000
2
2,000
3
3,000
4
500 n
FV =
∑ CF (1 + r) j
j =0
FV = CF0 (1 + r )
5−0
+ CF1 (1 + r )
5 −1
n− j
+ CF2 (1 + r )5 − 2 + CF3 (1 + r )5 − 3 + CF4 (1 + r )5 − 4 + CF5 (1 + r )5 − 5
= 0 × (1 + 0.10)5 + 1,000 × (1 + r ) 4 + 2,000 × (1 + r )3 + 3,000 × (1 + r ) 2 + 500 × (1 + 0.10)1 + 0 × (1 + 0.10) 0 = 0 + 1,464.10 + 2,662.00 + 3,630.00 + 550.00 + 0
•
= $8,306.10 Annuities § When the periodic cash flows are the same amount, the pattern is defined as annuity. E.g. pension, wages, rental. § Ordinary: cash flows received/paid at the end of each period, such as mortgage payments, wages.
PV = PMT ×
1 − (1 + r ) − n r
E.g. PV of Kate’s income stream A = 30,000 r = 10% n = 20 −20
1 − (1 + 0.1) 0.1 1 − 0.14864363 PV = 30,000 × 0.1 PV = 30,000 × 8.51356372 PV = 30,000 ×
PV = $255,406.91
Kate is indifferent from receiving $255,406.91 NOW or receiving $30,000 per annum at yearend for the next 20 years. Kate’s outstanding balance with the financial institution which promises her the income stream must be no less than $255,406.91. n
FV = PMT ×
(1 + r ) − 1 r
FV of Kate’s Stream
(1 + 0.1) 20 − 1 FV = 30,000 × = 1,718,249.97 0.1
§
Annuity due: cash flows received/paid at the beginning of each period, such as lease payments, rental.
1 − (1 + r ) − n PV = PMT × × (1 + r ) r (1 + r ) n − 1 FV = PMT × × (1 + r ) r
If Kate’s stream is paid every six months and each payment is $15,000, what is PV and FV of the stream PMT = 30,000/2 r = 10% ÷ 2 n = 20 x 2 = 40 −40
1 − (1 + 0.05) 0.05 1 − 0.14204568 PV = 15,000 × 0.05 PV = 15,000 × 17.15908635 PV = 15,000 ×
PV = $257,386.30 40
FV = 257,386.30 × (1 + 0.05) •
= 257,386.30 × 7.03998871 = 1,811,996.65
Deferred Annuity: first payment made in period d (d > 1) Kate decides to defer the collection of her retirement income stream. There will still be 20 payments, but the first payments will be received in year 5. What is the PV of Kate’s retirement income stream as of now? If the first payment of ordinary annuity is paid in period d, the formula present value as of period (d-1) Therefore, result above, $255,406.91 is as of year 4 (5-1) in this case.
•
PV = PMT ×
PV = 255406.91(1 + 0.1)−4 = $174,446.36
provide
Amount of periodic payment: such as mortgage – may not be received/paid annually Steve has bought a house for $800,000 with 20% cash down payment remaining balance is financed by a 25-year mortgage at 5.25% per annum. The mortgage installments are paid monthly and first installment is due in one month. How much is Steve’s monthly mortgage repayment? PV = 800,000 x (1-0.2) = $640,000 r = 0.0525/12 = 0.004375 (0.4375%) n x 25 x 12 = 360
PV = PMT ×
1 − (1 + r ) − n r
1 − (1 + 0.004375) −360 640,000 = PMT × 0.004375 640,000 = PMT × 181.09259248 PMT = 640,000 ÷ 181.09259248 PMT = $3,534.10 •
1 − (1 + r ) − n r
Number of periods (time to maturity)
Suppose Steve obtains a new job right after the purchase of the house. He is able to make $8,000 mortgage repayments per month. When will he pay off the mortgage? PV = 800,000 x (1-0.2) = $640,000 r = 0.0525/12 = 0.004375 (0.4375%) PMT = 8,000
PV = PMT ×
1 − (1 + r ) − n r
1 − (1 + 0.004375) −t ×12 640,000 = 8,000 × 0.004375 80 × 0.004375 = 1 − (1 + 0.004375) − t ×12 1.004375−12t = 1 − 0.35 − 12t × Ln(1.004375) = Ln(0.65) t ≈ 8.22 years
•
Perpetuity: a special ordinary annuity where the cash flow begins at the end of each period and continues perpetually: only has present value no future value
PV =
PMT r
Sam receives constant stream of cash flows worth $1,000 each year with the first payment in one year. What is the value of this income stream if Sam’s required rate of return is 10% p.a.
PV = •
1000 = 10,000 0.1
Growing Perpetuity: cash flows grow at a constant speed and continues perpetually Sam receives a steam of cash flows and the first payment is $1,000 in one year. After that, the annual cash flow will grow at 2% p.a. what is the value of this income stream if the discount rate is 10% p.a. PV =
CF1 r−g
1,000 PV = = 12,500 0.1 − 0.02
Apply the knowledge of present value and future value to evaluate complex cash flow patterns. • Dismantle a mixed cash flow pattern into stages • Each stage can be sorted out by available formulae • Sum up these PVs and obtain the PV of mixed cash flow pattern Michael wants to borrow $5,000 cash from you. The prevailing deposit rate is 3.6% p.a. Michael agrees to repay the money with 12 monthly instalments. The repayment schedule is as below. Should you accept the deal? End of Month Repayment
1
2
3
4
5
6
7
8
9
10
11
12
250
250
250
400
450
550
500
500
500
500
500
500
R = 0.036 ÷ 12 = 0.003 Solution 1 PV0 =
250 1.0031
+
250 1.0032
+
250 1.0033
+
400 1.0034
n = 12 +
450 1.0035
+
550 1.0036
+
500 1.0037
+
500 1.0038
+
500 1.0039
+
500 1.00310
+
500 1.00311
+
500 1.00312
PV0 = 5,040.12 > 5,000, accept Solution 2
1 − 1.003−3 400 450 550 1 − 1.003−6 PV0 = 250 × + + + + 500 × × 1.003− 6 0.003 0.003 1.0034 1.0035 1.0036
PV0 = 745.52 + 395.24 + 443.31 + 540.20 + 2,915.87 = 5040.14 > 5,000, accpet
Apply financial mathematics to make preliminary financial decisions. Peter has $80,000 and his financial advisor recommends a saving scheme. Peter will receive 20% p.a. simple interest rate for initial 2 years. Then the outstanding balance will be transferred to a term deposit account and earn 12% p.a. with monthly compounding for next 8 years. Then the condition of term deposit will be changed to 10% p.a. interest rate compounded quarterly for another 5 years. Besides the saving plan, Peter is expected to receive $1,450 per annum for 10 years as distribution from a family trust for 10 years. The first distribution will be made in six year. These cash flows are assumed to be reinvested at 10% interest compounding annually. How much money would Peter accumulate at the end of 15 years? 1. Future value 2. ‘Divide and Conquer’ One lump sum amount (with multiple stages) + one ordinary annuity
Topic 2: Financial Maths Define the aim of financial mathematics. • Money has time value - $1 now is worth more than $1 tomorrow • Financial managers compare the marginal benefits and marginal costs of investment projects • Projects usually have a long-term horizon – timing of benefits and costs matters • Understand, compare and calculate interest rate arrangements, including simple and compound interest. • Define and calculate the present value and future value of a lump sum cash flow. • PV: the value today of a cash flow to be received at a specific date in the future. (known as principal of borrowing or lending). 𝑃𝑉 = 𝐹𝑉 × 1 + 𝑟 *+ • FV: the amount an investment is worth after one or more periods of time. (known as principal plus interest at maturity. Simple Interest: 𝐹𝑉 = 𝑃𝑉 × 1 + 𝑖 𝑥 𝑡 Compound Interest: 𝐹𝑉 = 𝑃𝑉 × 1 + 𝑟 + • Time to maturity (t): the duration (usually in years) of the interest rate arrangement • Nominal interest rate (i): quoted interest rate per annum • Interest: monetary return on saving/investment
Interest = Principal × i × t
•
Other variations: If Jack invested $6,000 for 4 months and received $6240 at the maturity how much is the yield? PV = 6000 FV = 6.240 t = 4/12 years 1 ⎛ FV 1 6240 ⎞ ×⎜ − 1⎟ = ×( − 1) 4 t ⎝ PV 6000 ⎠ 12 12 = × (1.04 − 1) 4 = 0.12 = 12%
i=
Define and calculate the present value and future value multiple cash flows, including ordinary annuities and annuity dues. • Nina has just signed a contract for $1 million to be paid $200,000 now, $500,000 in one years time and $300,000 in two years time. Nina has a choice to take the above arrangement or take $910,000 now. Which option should she take assuming the relevant interest rate of 10%p.a. n
PV =
∑ CF (1 + r) j
−j
j =0
PV =
CF0
CF2
+ (1 + r )1 (1 + r ) 2 500,000 300,000 = + + 0 1 (1 + 0.10) (1 + 0.10) (1 + 0.10) 2 = 200,000 + 454,545.45 + 247,933.88 (1 + r ) 0 200,000
+
CF1
= $902,479.34 < $910,000
•
Nina should take the $910,000 now. Joseph has a five-year cash flow saving plan and the amounts to be saved are as below. The saving plan offers interest rate at 10% p.a. compounding annually. How much will Joseph accumulate at the end of plan. Year end
CFj
1
1,000
2
2,000
3
3,000
4
500 n
FV =
∑ CF (1 + r) j
j =0
FV = CF0 (1 + r )
5−0
+ CF1 (1 + r )
5 −1
n− j
+ CF2 (1 + r )5 − 2 + CF3 (1 + r )5 − 3 + CF4 (1 + r )5 − 4 + CF5 (1 + r )5 − 5
= 0 × (1 + 0.10)5 + 1,000 × (1 + r ) 4 + 2,000 × (1 + r )3 + 3,000 × (1 + r ) 2 + 500 × (1 + 0.10)1 + 0 × (1 + 0.10) 0 = 0 + 1,464.10 + 2,662.00 + 3,630.00 + 550.00 + 0
•
= $8,306.10 Annuities § When the periodic cash flows are the same amount, the pattern is defined as annuity. E.g. pension, wages, rental. § Ordinary: cash flows received/paid at the end of each period, such as mortgage payments, wages.
PV = PMT ×
1 − (1 + r ) − n r
E.g. PV of Kate’s income stream A = 30,000 r = 10% n = 20 −20
1 − (1 + 0.1) 0.1 1 − 0.14864363 PV = 30,000 × 0.1 PV = 30,000 × 8.51356372 PV = 30,000 ×
PV = $255,406.91
Kate is indifferent from receiving $255,406.91 NOW or receiving $30,000 per annum at yearend for the next 20 years. Kate’s outstanding balance with the financial institution which promises her the income stream must be no less than $255,406.91. n
FV = PMT ×
(1 + r ) − 1 r
FV of Kate’s Stream
(1 + 0.1) 20 − 1 FV = 30,000 × = 1,718,249.97 0.1
§
Annuity due: cash flows received/paid at the beginning of each period, such as lease payments, rental.
1 − (1 + r ) − n PV = PMT × × (1 + r ) r (1 + r ) n − 1 FV = PMT × × (1 + r ) r
If Kate’s stream is paid every six months and each payment is $15,000, what is PV and FV of the stream PMT = 30,000/2 r = 10% ÷ 2 n = 20 x 2 = 40 −40
1 − (1 + 0.05) 0.05 1 − 0.14204568 PV = 15,000 × 0.05 PV = 15,000 × 17.15908635 PV = 15,000 ×
PV = $257,386.30 40
FV = 257,386.30 × (1 + 0.05) •
= 257,386.30 × 7.03998871 = 1,811,996.65
Deferred Annuity: first payment made in period d (d > 1) Kate decides to defer the collection of her retirement income stream. There will still be 20 payments, but the first payments will be received in year 5. What is the PV of Kate’s retirement income stream as of now? If the first payment of ordinary annuity is paid in period d, the formula present value as of period (d-1) Therefore, result above, $255,406.91 is as of year 4 (5-1) in this case.
•
PV = PMT ×
PV = 255406.91(1 + 0.1)−4 = $174,446.36
provide
Amount of periodic payment: such as mortgage – may not be received/paid annually Steve has bought a house for $800,000 with 20% cash down payment remaining balance is financed by a 25-year mortgage at 5.25% per annum. The mortgage installments are paid monthly and first installment is due in one month. How much is Steve’s monthly mortgage repayment? PV = 800,000 x (1-0.2) = $640,000 r = 0.0525/12 = 0.004375 (0.4375%) n x 25 x 12 = 360
PV = PMT ×
1 − (1 + r ) − n r
1 − (1 + 0.004375) −360 640,000 = PMT × 0.004375 640,000 = PMT × 181.09259248 PMT = 640,000 ÷ 181.09259248 PMT = $3,534.10 •
1 − (1 + r ) − n r
Number of periods (time to maturity)
Suppose Steve obtains a new job right after the purchase of the house. He is able to make $8,000 mortgage repayments per month. When will he pay off the mortgage? PV = 800,000 x (1-0.2) = $640,000 r = 0.0525/12 = 0.004375 (0.4375%) PMT = 8,000
PV = PMT ×
1 − (1 + r ) − n r
1 − (1 + 0.004375) −t ×12 640,000 = 8,000 × 0.004375 80 × 0.004375 = 1 − (1 + 0.004375) − t ×12 1.004375−12t = 1 − 0.35 − 12t × Ln(1.004375) = Ln(0.65) t ≈ 8.22 years
•
Perpetuity: a special ordinary annuity where the cash flow begins at the end of each period and continues perpetually: only has present value no future value
PV =
PMT r
Sam receives constant stream of cash flows worth $1,000 each year with the first payment in one year. What is the value of this income stream if Sam’s required rate of return is 10% p.a.
PV = •
1000 = 10,000 0.1
Growing Perpetuity: cash flows grow at a constant speed and continues perpetually Sam receives a steam of cash flows and the first payment is $1,000 in one year. After that, the annual cash flow will grow at 2% p.a. what is the value of this income stream if the discount rate is 10% p.a. PV =
CF1 r−g
1,000 PV = = 12,500 0.1 − 0.02
Apply the knowledge of present value and future value to evaluate complex cash flow patterns. • Dismantle a mixed cash flow pattern into stages • Each stage can be sorted out by available formulae • Sum up these PVs and obtain the PV of mixed cash flow pattern Michael wants to borrow $5,000 cash from you. The prevailing deposit rate is 3.6% p.a. Michael agrees to repay the money with 12 monthly instalments. The repayment schedule is as below. Should you accept the deal? End of Month Repayment
1
2
3
4
5
6
7
8
9
10
11
12
250
250
250
400
450
550
500
500
500
500
500
500
R = 0.036 ÷ 12 = 0.003 Solution 1 PV0 =
250 1.0031
+
250 1.0032
+
250 1.0033
+
400 1.0034
n = 12 +
450 1.0035
+
550 1.0036
+
500 1.0037
+
500 1.0038
+
500 1.0039
+
500 1.00310
+
500 1.00311
+
500 1.00312
PV0 = 5,040.12 > 5,000, accept Solution 2
1 − 1.003−3 400 450 550 1 − 1.003−6 PV0 = 250 × + + + + 500 × × 1.003− 6 0.003 0.003 1.0034 1.0035 1.0036
PV0 = 745.52 + 395.24 + 443.31 + 540.20 + 2,915.87 = 5040.14 > 5,000, accpet
Apply financial mathematics to make preliminary financial decisions. Peter has $80,000 and his financial advisor recommends a saving scheme. Peter will receive 20% p.a. simple interest rate for initial 2 years. Then the outstanding balance will be transferred to a term deposit account and earn 12% p.a. with monthly compounding for next 8 years. Then the condition of term deposit will be changed to 10% p.a. interest rate compounded quarterly for another 5 years. Besides the saving plan, Peter is expected to receive $1,450 per annum for 10 years as distribution from a family trust for 10 years. The first distribution will be made in six year. These cash flows are assumed to be reinvested at 10% interest compounding annually. How much money would Peter accumulate at the end of 15 years? 1. Future value 2. ‘Divide and Conquer’ One lump sum amount (with multiple stages) + one ordinary annuity
