Student Notes: CM1-04 Discounting and Accumulating
- anactuary1729
- Jul 16, 2025
- 4 min read
Chapter 4: Discounting and Accumulating
This chapter builds upon our understanding of present and accumulated values for single payments, extending these concepts to streams of cashflows, both discrete and continuous. The core objective is to calculate the value of these cashflows at any given point in time, considering various scenarios of interest and discount rates.
Syllabus Objectives Covered:
Calculating present and accumulated values for streams of cashflows.
Handling equal or varying cashflows, both discrete and continuous.
Addressing deferred cashflows.
Applying constant or varying interest/discount rates.
Section 1: Present Values of Cashflows
The present value (PV) of a cashflow represents the amount of money that needs to be invested today to meet a future financial obligation.
1.1 Discrete Cashflows
Discrete cashflows are payments that occur at specific, distinct points in time.
Single Discrete Payment: The present value of a single cashflow C due at time t years, with a constant effective annual interest rate i, is given by:
PV=Cvt
where v=1+i1 is the discount factor.
Series of Discrete Payments: The present value of a series of payments ct1,ct2,…,ctn due at times t1,t2,…,tn is the sum of their individual present values:
PV=j=1∑nctjvtj
Varying Interest Rates (using v(t)): If the effective interest rate is not constant, we use a time-dependent discount function v(t), which is the present value of 1 due at time t. The formula becomes:
PV=j=1∑nctjv(tj)
Varying Force of Interest (δ(t)): If the force of interest at time t is δ(t), then v(t)=exp(−∫0tδ(s)ds). The present value is:
PV=j=1∑nctjexp(−∫0tjδ(s)ds)
If δ(t)=δ (constant force of interest), this simplifies to PV=∑j=1nctje−δtj.
Example 1: Discrete Payments with Constant Interest Rate To find the sum to be invested today to cover three annual payments of £7,500 at 7.5% effective annual interest:
v=1.075−1
PV (Year 1) = 7,500×1.075−1=£6,976.74
PV (Year 2) = 7,500×1.075−2=£6,489.99
PV (Year 3) = 7,500×1.075−3=£6,037.20
Total PV = £6,976.74+£6,489.99+£6,037.20=£19,503.93≈£19,504
Example 2: Discrete Payments with Constant Force of Interest PV of $250 at t=6 and $600 at t=8 if δ(t)=3% pa:
PV = 250e−6×0.03+600e−8×0.03=$680.79
1.2 Continuously Payable Cashflows (Payment Streams)
These are cashflows where money is paid continuously over a period, rather than at discrete points.
Rate of Payment: If M(t) is the total payment made up to time t, then the rate of payment at time t is ρ(t)=M′(t).
Total Payment over an Interval: The total payment received between time α and β is:
∫αβρ(t)dt
Present Value of a Continuous Cashflow: The present value of a continuous cashflow from time 0 to T (where ρ(t) is the rate of payment) is:
∫0Tv(t)ρ(t)dt
If T is infinite, the upper limit of the integral becomes ∞.
Combined Discrete and Continuous Cashflows: The general formula for the present value of a cashflow with both discrete payments (ct) and a continuous stream (ρ(t)) is:
∑ctv(t)+∫0∞v(t)ρ(t)dt
If the interest rate is constant, this simplifies to:
∑ctvt+∫0∞vtρ(t)dt
Example 3: Continuous Cashflow with Constant Force of Interest PV of a continuous cashflow for 5 years with rate 100×0.8t and constant force of interest 8% pa:
ρ(t)=100×0.8t and v(t)=e−0.08t.
PV = ∫05e−0.08t×100×etln0.8dt=100∫05e(−0.08+ln0.8)tdt≈£257.42
Net Present Value (NPV): When a project involves both income (positive cashflows) and outgoings (negative cashflows), the Net Present Value is the difference between the present value of the income and the present value of the outgoings.
NPV=PVIncome−PVOutgo
Example 4: Net Present Value with Varying Discount Function NPV of continuous income £350 pa for 5 years, outgo £600 at Year 1, £400 at Year 3, with v(t)=1−0.01t:
PV of income = ∫05350(1−0.01t)dt=£1,706.25
PV of outgo = 600v(1)+400v(3)=600(0.99)+400(0.97)=£982
NPV = 1,706.25−982=£724.25
Section 2: Valuing Cashflows at a General Time
Cashflows can be valued at any point in time, not just time 0.
Value at time t1 of a sum C due at time t2:
If t1≥t2: This is the accumulation of C from t2 to t1.
If t1<t2: This is the discounted value at t1 of C due at t2.
In both cases, using a varying force of interest δ(t):
C exp[−∫t1t2δ(t)dt]
If δ(t)=δ (constant force of interest):
Ce−(t2−t1)δ=Ce(t1−t2)δ
Using the discount function v(t):
Cv(t1)v(t2)
This formula implies that to find the value at t1 of C at t2, you can first discount C to time 0 (by multiplying by v(t2)) and then accumulate it to time t1 (by dividing by v(t1)).
General Formula for Valuing a Cashflow at Any Time t1: For a cashflow with discrete payments (ct) and a continuous stream (ρ(t)):
∑ctv(t1)v(t)+∫0∞v(t1)v(t)ρ(t)dt
This formula is powerful as it allows valuation at any arbitrary time point.
Example 5: Valuing Multiple Payments at a Future Date Payments: £100 (1 Jan 2020, t=1), £130 (1 Jan 2021, t=2), £150 (1 Jan 2023, t=4), £160 (1 Jan 2024, t=5). Valuation Date: 1 Jan 2022 (t1=3). Discount function: v(t)=0.92−100t.
Calculate v(t) for relevant times:
v(1)=0.91
v(2)=0.90
v(3)=0.89
v(4)=0.88
v(5)=0.87
Value at t1=3 using Cv(t1)v(t2):
£100 (at t=1): 100×v(3)v(1)=100×0.890.91=£102.25 (Accumulated)
£130 (at t=2): 130×v(3)v(2)=130×0.890.90=£131.46 (Accumulated)
£150 (at t=4): 150×v(3)v(4)=150×0.890.88=£148.31 (Discounted)
£160 (at t=5): 160×v(3)v(5)=160×0.890.87=£156.40 (Discounted)
Total Value at t=3 = £102.25+£131.46+£148.31+£156.40=£538.42
Alternatively, calculate PV at t=0 then accumulate to t=3:
PV at t=0: 100v(1)+130v(2)+150v(4)+160v(5) =100(0.91)+130(0.90)+150(0.88)+160(0.87)=£479.20
Value at t=3: £479.20×v(3)1=£479.20×0.891=£538.43
Key Takeaways:
Time Value of Money: Money today is worth more than the same amount in the future due to its earning potential.
Discounting: Calculating the present value of future cashflows.
Accumulating: Calculating the future value of present or past cashflows.
Discrete vs. Continuous: Payments can occur at specific points (discrete) or flow constantly over time (continuous).
Formulas: Understand and apply the appropriate summation (for discrete) or integral (for continuous) formulas, adjusting for constant or varying interest/discount rates.
Valuation Date Flexibility: Cashflows can be valued at any point in time, not just time zero, by appropriately discounting or accumulating individual payments to the chosen valuation date.
Net Present Value (NPV): A crucial concept for evaluating projects with both inflows and outflows, representing the net value of all cashflows at time zero.
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